note cards exam 8 v2

BKM Ch 6 – Risk Aversion & Capital Allocation.................................................................................................................................1

BKM Ch 7 – Optimal risky portfolios....................................................................................................................................................9

BKM Ch 8 – Index models ....................................................................................................................................................................15

BKM Ch 9 – The Capital Asset Pricing Model (CAPM)....................................................................................................................21

BKM Ch 10 – APT and Multi-Factor Models .....................................................................................................................................29

BKM Ch 11 – Market Efficiency ..........................................................................................................................................................37

BKM Ch 12 – Behavioral Finance & Technical Analysis...................................................................................................................39

BKM Ch 13 – Empirical Evidence on Security Returns ....................................................................................................................43

BKM Ch 14 – Bond Prices and Yields .................................................................................................................................................53

BKM Ch 15 – Term Structure and Interest Rates..............................................................................................................................61

BKM Ch 16.1,2 – Interest Rate Risk: Duration and Convexity.........................................................................................................73

Altman – Measuring Corp Bond Mortality and Performance...........................................................................................................85

Section D – Hull text problems Ch 2-7 .................................................................................................................................................89

Hull Ch 2 – Mechanics of Futures Markets.........................................................................................................................................91

Hull Ch 3 – Hedging Strategies Using Futures....................................................................................................................................97

Hull Ch 4 – Interest Rates ...................................................................................................................................................................107

Hull Ch 5 – Determination of Forward & Futures Prices................................................................................................................113

Hull Ch 6 – Day Count Conventions & Quoted Treasury Bond Prices ..........................................................................................121

Hull Ch 7 - Swaps.................................................................................................................................................................................123

Section E – Options..............................................................................................................................................................................131

Hull Ch 8&9 – Options, Markets & Properties of Options Stock Prices ........................................................................................133

Hull Ch 10 – Option Trading Strategies ............................................................................................................................................143

Hull Ch 11 – Option pricing w/ binomial trees..................................................................................................................................147

Hull Ch 12 – Weiner Processes & Ito’s Lemma ................................................................................................................................155

Hull Ch 13 – Black-Scholes-Merton Model .......................................................................................................................................159

Hull Ch 15 – Options on Indices and Currencies..............................................................................................................................167

Hull Ch 16 – Futures Options .............................................................................................................................................................171

Black – How to Use the Holes in Black Scholes.................................................................................................................................175

Fabozzi – Valuation of Bonds with Embedded Options ...................................................................................................................177

BKM Ch 25 – International Diversification ......................................................................................................................................183

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BKM Ch 16.3,4 – Managing Bond Portfolios ....................................................................................................................................189

Noris – ALM for P&C Companies .....................................................................................................................................................195

Feldblum – Asset Liability Matching for P/C Insurers ....................................................................................................................203

Panning – Managing Interest Rate Risk ............................................................................................................................................211

Section H – Financial Risk Management ...........................................................................................................................................221

Hull Ch 17 – The Greek Letters .........................................................................................................................................................223

Hull Ch 20 – Value at Risk..................................................................................................................................................................239

Hull Ch 20 – Credit Risk .....................................................................................................................................................................251

Stultz – Rethinking Risk Management ..............................................................................................................................................265

Butsic – Solvency Measurements for Risk-based Capital Applications ..........................................................................................273

Cummins – Allocation of Capital in the Insurance Industry ...........................................................................................................283

Goldfarb – Risk-adjusted Performance Measures............................................................................................................................289

Cummins – Cat Bonds and other risk-linked securities ...................................................................................................................295

Gorvette – Insurance Securitization: A new asset class....................................................................................................................303

BKM Ch 18 – Equity Valuation Models ............................................................................................................................................307

Risk Aversion & Capital Allocation - BKM

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BKM Ch 6 – Risk Aversion & Capital Allocation

1. Standard risk vs. reward utility function (what are A and U?) What can you do with a given U and A?

2. What is the mean-variance criterion?

3. How do you estimate risk aversion?

4. What is the formula for the optimal portfolio (assuming a person can only invest in the risk-free asset and a risky portfolio)? How is this formula derived?

1. ( ) 2

21 σA A is the risk aversion parameter, higher A -> lower

utility as st. dev. Increases. U is utility. For a given U and A you can create indifference curves. E on y-axis st. dev. On x-axis

rEU −=

2. Portfolio A dominates B if ( ) ( )BA rErE ≥ and BA σσ ≤ and at least one of the two inequalities is strict (rules out equality of the portfolios)

3. Basically you are equating two utility functions, one with a return minus ins cost and no variance, the other with no ins but a variance.

( )2

fp

Aσ You take the derivative of the utility function wrt y (the

allocation to the risky asset)

*

p

rrEy

−=4.

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1. What does y represent in the complete portfolio formula? What is the complete portfolio formula?

2. What is the variance of the complete portfolio? Why? What does the subscript p refer to?

3. What is the CAL? How does it relate to the Sharpe ratio?

4. What is the Sharpe ratio formula?

( ) ( ) ( )pc ryEr fryE −+= 1 1. y is the portion allocated to the risky asset.

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2. St. dev. of the risk-free asset is 0. Remember the formula for the variance of the sum of two random variables or, remember st. dev.s are proportional so

pc y σ= implies 222py σ= . P and C represent the risky and complete

portfolios.

σ cσ

3. CAL = capital allocation line. The Sharpe ratio is the slope of the CAL, it is the reward-to-variability ratio

4.


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1. Graphically, how do you combine the CAL and the indifference curves to find the optimal portfolio?

1. Draw the CAL. Then select a value for A (risk aversion). You want to maximize utility – this is accomplished by finding the highest indifference curve with a point of tangency on the CAL.

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1. What is the CML? Why is it different than the CAL?

2. List 4 common criticisms of indexing and counterarguments

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1. Capital Market Line – CML is using a market index as the risky asset

2.

They’re undiversified Actively managed funds are similarly undiversified – 36% of assets in top ten holdings

They’re top heavey S&P only holds 500 of 6,000 assets, it still represents 77% of market value

They’re chasing performance

This always happens, as a stock’s market value increases all shareholders own more of it

You can do better Once you factor in transaction fees very few investors actually beat the broad indexes.

Optimal risky portfolios - BKM

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BKM Ch 7 – Optimal risky portfolios

1. Two types of risks, what are they? How are they diminished by diversification?

2. What does σ12 represent?

3. Given two risky assets with some level of correlation, how do you calculate the minimum-variance portfolio?

4. What is the formula for the amount to invest in risk 1 to get the minimum variance?

1. Firm-specific and economic. In a portfolio of assets the firm-specific risks tend to cancel each other out (uncorrelated) the economic risks are systemic though and tend to affect all simultaneously.

2. The covariance of risk 1 and 2, ie Cov(r1,r2) = σ1σ2ρ

3. Take the derivative of your variance formula,

, with respect to w1, set the derivative = 0 and solve for w1.

4. be careful, this is not the optimal portfolio when you are talking about also investing in the risk-free asset.

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1. What is the formula for the optimal risky portfolio? How does it compare to the minimum-variance portfolio?

2. What kind of picture does this help draw?

3. After you draw that picture what would you do?

1. You just need to add in the risk premium components like ( )[ ]FrrE −1 on the first four variance terms and remember to add the sum of the risk premiums onto the last covariance term.

2. Notice that you want the highest CAL line that is tangent to the portfolio opportunity set.

3. Go back to drawing utility functions: assign a risk aversion factor and then maximize Utility

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1. Portfolio selection is separated into two distinct tasks, they are…

2. How does this formula show the power of diversification:

3. What is the difference between risk pooling and risk sharing?

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1. 1. Select the optimal combination of risky assets – this involves building an efficient frontier and picking the point tangent to the highest CAL 2. Allocate funds between the risky portfolio and the risk-free asset – this is done by assuming some risk aversion factor and maximizing utility.

2. As N gets large we find the portfolio variance is only dependent on the covariance the of the pieces and that the variance of the portfolio approaches the average covariance of the components.

3. Risk pooling refers more to the growth of an insurers portfolio – the value of the portfolio grows faster than the risk, however the risk is indeed growing. Risk sharing involves keeping the value constant but reducing risk by having smaller pieces of more pies: like a reinsurer with a fixed amount of capital covering smaller shares of more contracts has less risk than a reinsurer with the same capital but larger shares of fewer contracts.

Index models - BKM

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BKM Ch 8 – Index models 1. Under the single-factor model, what are the following formulas for security i:

actual return, variance of return, covariance with another security j? This model also goes by the name _______, or the full-covariance model.

2. What is the main drawback of the single-factor model above? How do we get around this with the Single index model?

3. What are the same 3 formulas as in 1. under the single-index model?

4. What is the portfolio variance under a single-index model?

1.

note, this is the Markowitz model

2. No way to quantify m. Single index model uses an index, S&P 500, as a proxy for m.

3. , here Ri is the excess over risk-free return

;

4. Portfolio variance is the systemic risk plus the weighted sum of the component risks.

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Index models - BKM

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1. How do you estimate the betas in the single-index model?

2. Compare the Markowitz model and the Index model.

3. Why would you use a tracking portfolio? What is the strategy?

1. Use historic returns and the formula , Ri is the excess return over the risk-free rate of the security Rm is the excess return of the index The R’s come out of historical data. Do a regression to get beta and alpha

2. * Markowitz helps us identify more efficient portfolios – however you need to estimate corr. for every pair of risks, that is subject to massive est. error. * Index model assumes covariance are driven by a common factor, although we lose some flexibility – the practical gains are significant, also you are able to separate macro analysis from security analysis

3. To remove market risk. You find some portfolio A with a positive alpha, then you create a tracking portfolio of T-bills and the S&P 500 that has the same systematic risk as A. Then you short the tracking portfolio. If A is well diversified you will be able to earn alpha and have no real market risk.

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Index models - BKM

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Capital Asset Pricing Model (CAPM) - BKM

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BKM Ch 9 – The Capital Asset Pricing Model (CAPM)

1. What are the 6 assumptions of the CAPM?

2. List two important implications of CAPM

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1. 1. No single investor can affect the stock price 2. Investors focus only on what will happen in a single period 3. Number of securities is limited, however they are able to buy/sell unlimited amounts 4. No taxes or transaction costs (bad assumption) 5. All investors select the efficient portfolio using methods from Ch. 6-8 6. All investors use the same assumptions about risk and return of the different securities => all expect the same expectations.

2. 1. Ultimately, all investors will own the same risky market portfolio. Note, this does not imply they will have the same mix of the risk-free and risky assets. 2. Passive strategy is efficient – CAL = CML – Nobody actually needs to do any securities analysis, just own an index fund of risky assets.


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1. What is the risk premium of the Market portfolio?

2. What is the marginal increase in risk from a small change in the weight for security i?

3. 2 leads to formulas for β and E(ri); these are the base definition of the CAPM.

4. What is the SML?

5. What is α? What should be a portfolio manager’s goal with respect to α?

1.

2. Marginal Risk

3. ;

4. Security Market Line – graphs expected return (y-axis) to β (x-axis)

5. α is the amount by which a security’s actual return deviates from its expected return (how much it is above or below the SML). Managers should look for security’s with a positive α (ie, it is under-priced).

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1. 3 main difficulties in applying CAPM

2. Both CAPM and the Single-factor index model produce the same β, what is the formula?

3. What is the difference between noise traders and information traders? Which results in a higher bid-ask spread?

1. 1. CAPM assumes the Market Portfolio contains all risky assets – testing CAPM requires us to know whether an index is mean-variance efficient 2. CAPM is stated in expected returns, but we can only observe actual returns -> we cannot confirm whether the market portfolio has the highest return to variability ratio 3. The linear relationship between expected returns for a stock and market risk premium is impossible to verify since we can’t observe expected returns.

2. 3. Noise traders are not likely to have any special information – they are just

rebalancing a portfolio… Information traders trade based on specialized information. Information traders.

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1.

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1.

APT and Multi-factor models - BKM

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BKM Ch 10 – APT and Multi-Factor Models

1. What is the difference between a multifactor factor model and a multifactor SML?

2. This crucial result of the APT model tells us what about well-diversified portfolios?

3. How would you use the formula in 2 to get the result of CAPM?

1. The model is a statement about what causes actual returns to deviate from their expected values; the SML is a statement, based on some underlying theory, of what the expected returns are.

2. In equilibrium all well-diversified portfolios must have the same risk premiums relative to their betas.

3. Assume portfolio A is the market portfolio, then β = 1 and

4.

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1. CAPM and APT product the same single-factor model, but the assumptions are different. How are the APT assumptions simpler?

2. What is the two-factor expected return model for APT?

1. APT just assumes that a single factor model can be used for security prices and arbitrage opportunities will be exploited.

2. ; RP is the risk premium over the risk free rate for that Factor portfolio… need to do some problems to know what formulas I need. This is actually simpler than it looks.

3.

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4.


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5.

Market Efficiency - BKM

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BKM Ch 11 – Market Efficiency

1. Explain a random walk – in particular the randomness in the stock price and the randomness in the price change

2. Describe the three version of the Efficient Market Hypothesis

3. What is the difference between Technical analysts and Fundamental analysts?

4. … lots and lots of talk about different investment strategies and performance anomolies, no likely test questions.

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1. The price changes are random, the prices themselves are not

2. Weak – current prices reflect all information that can be obtained from records of trading history,…ie costles and easily available information is already incorporated Semi-strong – All publicly available data is reflected in the price Stron – All information out there is already reflected in the price

3. Technical analysts go off the idea that a stock is worth what others will pay for it, Fundamentalists build off the concept that a stock has intrinsic value

4.

Behavioral Finance & Technical Analysis - BKM

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BKM Ch 12 – Behavioral Finance & Technical Analysis

1. How would you calculate the Trin Statistic?

2. Describe the confidence index

3. Describe the Put/Call ratio

4.

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1. Average volume of declining stocks/Average volume of advancing Average volume = volume of group / number in group TS > 1 indicates bear market due to net selling pressure

2. Uses the bond market to guage investor sentiment: high grade corporate bond yields to mid-grade corp bond yields. Optimism will push down the default premium (ie risk premium investors charge to mid grade) this will push the ratio closer to 1

3. Ratio of outstanding put options to outstanding call options Normally about .65. An increase indicates more put options which indicates more investors are hedging positions against a fall in the market.

4.

Behavioral Finance & Technical Analysis - BKM

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1.

Empirical Evidence on Security Returns - BKM

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BKM Ch 13 – Empirical Evidence on Security Returns

1. What is the book’s 2-stage (4-step) process to test the CAPM?

2. Who/How would you test the validity of CAPM using the assets of Human Capital and Business Cycles?

3.

1. Stage 1 – 1. Gather 5 years of historical data for 100 stocks. Use S&P 500 index as ‘the market’. 2. Estimate the Security Characteristic Line = build a single variable regression of each stock against the Market – you are estimating the betas. Result: 100 betas, 100 est. of avg. excess return, 100 est. of var of residual, 1 est. of avg. excess return of the Market. Stage 2 – 1. Estimate the Security Market Line = do a second regression, 1 data point for ea. stock. . You are estimating the ’s. 2.

Interpret results = 0 should be 0, 1 should be , and 2 should be 0.

2. Jaganathan & Wang. CAPM assumes all assets are tradable – that is not true. Human Capital and the Business Cycle affect security prices. You can use aggregate labor capital for Human Capital and high grade corp bond yields as a measure of business cycle. This creates a conditional CAPM model which fits significantly better to actual returns

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1. What are the 2 main indexes in the Fama/French model?

2. What are the results of the model?

3. How do you explain the results? Two camps: Risk-based and Behavioral explanations.

4.

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1. SMB (small,medium,big) and HML (high,medium,low). SMB refers to the size of the company measured by market capitalization. HML refers to the ratio of book value to market value. SMB and HML are indexes, not actual values.

2. As size increases the sensitivity to that variable decreases. As book/market ratio increases the sensitivity to that variable increases.

3. 1. Size and value factors are proxies for risks not captured by CAPM. They seem to predict GDP changes, ie business cycle. Also, the factors seem to capture the fact that betas change under different market conditions. 2. Investors may irrationally prefer glamour stocks. They extrapolate good performance too far into the future and drive up prices, resulting in poor returns. Low book/market ratio stocks are called growth stocks. The investors are generally disappointed when earnings on these stocks are announced.


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1. What is the Equity Premium Puzzle?

2. What are some of the explanations?

3. What is this equation and what are the parameters? . What is a useful rearrangement of this formula?

1. A few studies have shown that in recent decades investors have been excessively rewarded in terms of risk premium, ie they have earned more than they should expect to based on CAPM.

2. Theories are that maybe 1. investors have been lucky 2. Excess returns are real but affected in the US by a survivorship bias 3. Perhaps some extensions of CAPM can solve the puzzle 4. Various behavioral biases lead to irrational behavior which creates the equity premium puzzle

3. P=price, Div1 = dividend, k = E(return), g = E(dividend growth)

, E(return) as a function of last div and last price + E div growth

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1. What is the formula for the actual return?

2. If you rearrange and take the difference from the expected return you find that the difference between expected and actual is what?

3. What does this have to do with Fama and French?

4. Fama and French argue that the Dividend Discount Model (DDM) produces more reliable predictions of future than historical returns for 3 reasons:

5. What is the survivorship bias argument?

1.

2. : the actual return minus the expected dividend growth.

3. They noticed that prior to 1950 the expected and actual were pretty close, but after that the actual returns outpaced the expected.

4. 1. realized returns (for firms 1950-1999ish) couldn’t be representative of real expectations unless firms were engaging in negative NPV projects. 2. measurements from DDM are measured more accurately 3. DDM estimates are more consistent with stable investor risk aversion

5. All tests are based on the US markets. They have not been shut down or failed. This is exceptional in the world where other markets have been suspended or failed.

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1. What are the behavioral explanations for the high returns?

2.

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1. People tend to consider risky decisions in isolation rather than in the context of a portfolio. Combine this with risk aversion and people tend to require hire than necessary risk premiums. ie their effective risk aversion is higher than what objective measures indicate.

2.

Bond Prices and Yields - BKM

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BKM Ch 14 – Bond Prices and Yields

1. Describe a couple of key differences between US Treasury Bonds and Corporate Bonds

2. Corporate Bonds come in three flavors (with respect to timing their termination)

3.

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1. US-T bonds have no credit risk – therefore no risk premium whereas Corp bonds may have very high risk premiums. Corporate bonds are less liquid so the quoted prices are rarely what you would get if you actually bought one.

2. Callable – the company may pay off the debt; this happens when interest rates drop and co’s refinance their debt Convertible – bondholder (investors) may exchange bonds for preferred stock at a predetermined ratio Putable – bondholder may redeem the bond prior to maturity at some set price

3.


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1. What is the annuity factor for a series of n payments?

2. What are stripped bonds? Give an example.

3. Define default risk (in context of bonds).

4. Prob: A bond has par value of 100 and a 6 mo term. 10% annual bond equiv. yield and a 5% chance of default. What is the price?

5. What is the risk premium?

1. 2. When the coupon payments are stripped from the principal payment and sold

as separate securities. US Treasury STRIPS are sold this way. US T-Bills are short-term bonds which have no coupons, only a principal payment.

3. The risk that the issuer will not make coupon or principal payments. This risk must be factored into the pricing of expected cash flows.

4. if there is no default risk, with risk:

5. this implies an 11.07% risk premium.

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1. Give an example of the difference between bond equivalent and annual effective yields.

2. Describe the three types of yield: Yield to Maturity, Current Yield, Yield to Call

3. Describe the four types of bond indentures: sinking funds, subordination, dividend restrictions, collateral

4.

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1. Bond coupons are typically paid every six months and yields are quoted as annual rates. Usually people just multiply the six month yield by 2 to get what is called a bond equivalent yield, compounding is ignored. If you want the annual effective yield then you must do the compounding.

2. Yield to Maturity: sets the PV of cash flows = current price. assumes you will hold the bond to maturity Current Yield: divide the coupon amount by the price (price determined by discounting all cash flows at Yield to Maturity rate) Yield to Call: Same as yield to maturity except assume issuer prepays face value as soon as he can (call date)

3. sinking funds: the issuer buys back the bonds over time. subordination: the first investors in get their money first. dividend restrictions: dividends to shareholders cannot be paid unless coupons are paid first. collateral: the firm pledges assets to be sold off to cover principal repayment


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4. asdf

5. asdf

Term Structure and Interest Rates - BKM

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BKM Ch 15 – Term Structure and Interest Rates

1. What are the two main risks a bank takes by issuing a mortgage and how are they hedged?

2. Describe the basic structure of a pass-through mortgage pool. How is this an example of diversification, risk pooling, and/or risk sharing?

3. Describe CMOs and MBSs, as well as the MBS variants IO and PO. Which have values that increase with interest rate increases? Why is that attractive to investors?

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1. Default risk – the homeowner may stop paying. the bank then takes the house and sells it at market value. still risk it may not sell enough to cover principal Prepayment risk – homeowner pays off loan faster than expected.

2. A bunch of similar mortgages are pooled together. An investor ‘buys’ some portion of the principal, say 12%. They have bought 12% of the future cash flows – 12% of all interest and principal payments. The only real risk is prepayment. ie You get all your money back too soon and you don’t get that cash flow stream. The risk is diversified by having so many mortgages in the pool, the risk is shared by having other investors take a piece of it.

3. CMO = tranch structure. Tranch A receives it’s share of interest, it also receives all the principal until his investments is paid off. MBS = interest and principal payments are separated for different investors. IO means interest only – as interest rates climb so does the value of the IO. This would diversify a portfolio bonds whose values decline as interest rates climb.


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1. Describe the pure yield curve. What makes it so useful? How does it compare with the on-the-run yield curve?

2. Define short rate and spot rate. Show the mathematical relationship using r’s for short rates and y’s for spot rates to get $1000 on 12/31/10 in 1/1/09 money.

1. It is a plot of zero coupon treasury bond yields (spot rates) against their maturity periods. It is useful because it can be used to create the present value for any stream of cash flows. The other yield curve includes the cash flows from coupon payments and so the overall yield is a complex average.

2. Short rates are 1-yr interest rates. Say we’re at 1/1/09. The r2 is the rate to use to discount a $1000 cash flow from 12/31/10 to 1/1/10. To get the cash flow to

today you’d use: . Spot rate is the rate you’d use today to bring that single cash flow all the back

to today:

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1. What is a forward rate?

2. Which short rates do we know the value of?

3. Sometimes E(r2)<f2 and other times E(r2)>f2. Why is this? Give a quick explanational explanational.

1. The forward rate is calculated by assuming you have $1000 to invest today for n years, but instead of an n-year bond you buy an n-1-year bond and invest the cashout for 1 more year to get you to year n. The rate you would need to reinvest at to get the same return as an n-year bond is the forward rate.

2. Only the ones for this year. Any future short rate can only be an expected

value.

3. Remember . Then for the 2-yr investor , because 1-yr bond sellers will offer a premium for t

uncertainty on r2, this leads to he

For the 1-yr investor =>

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1. Under the Expectations Hypothesis, why would the yield curve be upward or downward sloping?

2. How are yield curves influenced under the Liquidity Preference theory?

3. Describe Segmentation Theory.

4. Interpreting the term structure is difficult because of the various forces at play. However, a downward sloping yield curve does most likely indicate…

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1. The 1-yr spot rate = the 1-yr short rate. If bond buyers expect short rates to rise the yield curve will be upward sloping and vice versa. Because for y2 > y1 you have to have r2 > r1.

2. If there are more long-term investors then there will be a higher risk premium on short-term investments and therefore the yield curve will slope down; if there are more short-term investors then the risk premiums will be higher on long-term investments and the yield curve will slope up.

3. Investors don’t readily switch their preference between short-,mid-, and long-term investments. The yields in the different ranges are a function of supply and demand in the range.

4. expected declines in interest rates, due to either declines in the inflation rate or the real rate.


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1. Suppose the 1-yr yield is 8% and the 2-yr yield is 9%. What is the 1-yr forward rate? ie y1 = 8%, y2 = 9%, f2 = ?

2. Suppose you want to borrow $1000 a year from now and pay it back 2 years from now, and you want to ‘lock’ the lending rate at the current forward rate. How do you work the system to do that?

3.

1. Assuming , f2 = 10.01%

2. You would buy $1000 par value 1yr bonds and sell $1100 par value 2yr bonds. Do the math. You should find that to buy/sell prices match at $925.93 today. So there is 0 cash flow net today, 1000 net in in 1 yr, and 1100 net out in 2 years.

3.

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4.

Interest Rate Risk (Duration and Convexity) - BKM

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BKM Ch 16.1,2 – Interest Rate Risk: Duration and Convexity

1. 6 things you must memorize about interest rate sensitivity

2. Formula for the Macaulay Duration. It is denoted D.

3. Describe the formula.

1. 1. Bond prices and yields are inversely related 2. Increase in yield has a lower impact (on price) than decrease in yield 3. Long term bonds more sensitive to Δyield than short term bonds 4. Sensitivity increases, at a decreasing rate, as maturity increases 5. High coupon bonds are less sensitive to Δyield than low coupon bonds 6. Sensitivity of a bond’s price to Δyield is inversely related to current yield.

2. 3. It is the weighted average of the time to each payment. Times are 1,2,3,…n,

and the weights are

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1. How does D relate the change in price to the change in yield?

1. , m is the compounding frequency. So if you are dealing with continuous

compounding you end up with

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1. What is the modified duration, D*?

2. Can you prove the relationships in the formulas on the previous card? Start

with

3. How do you use your calculator to approximate the modified duration?

4. What is a bond value under continuous compounding?

5.

1. It is D/(1+y/m). If we take that then we get

2. (do the proof)

3. Define P as the current price of the bond, then calculate the price if the yield goes up or down by a small amount Δy = ±.1%, define P‐ as the price when yields go down and P+ as the price when yields go up. Then

4.

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1. What is the Macaulay Duration, D, under continuous compounding?

2. What is the modified duration under continuous compounding?

3. Why?

4. What is the Convexity of a bond with annual compounding?

1. 2. Same as the D, ie D* = D.

3. The only difference is the frequency of compounding. When compounding is

semi-annual D* = . So if you compound k times D* =

4.

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1. What is the convexity with continuous compounding?

2. What is the approximation to convexity if you use the calculator?

3.

1.

2. ,

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Interest R

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ate Risk (Duration and Convexity) - BKM 1. What is this

chart?

1. It shows 4 bonds A-D. Bond prices sensitivities change relative to the yield to maturity and coupon rate. Low coupon -> highly sensitive.

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Measuring Corp Bond Mort’y and Perf. - Altman

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Altman – Measuring Corp Bond Mortality and Performance

1. Describe how Altman recommends calculating credit risk.

2. What is the formula for the CMRt , (credit mortality rate?)

3. If MMR 1,2,3 are .03, .04, .05 then what is CMR3?

4. What is the average recovery rate on a defaulted bond? How does it change wrt term and initial bond rating?

5. The paper presents a study comparing excess returns by rating and period. The finding is that investors are overcompensated for the risk of default. What are a few reasons this may be?

6.

1. He suggests we use the same form of model as life contingencies. Look at a cohort of bonds (ie. all B-rated bonds) and calculate the probability of survival.

2. , the product 1-MMR bit is the product of all the one year survival rates to get you to t years.

3. 1-(1-.97)(1-96)(1-.95)

4. 40%, term and rating don’t really matter.

5. 1. Perhaps the market just overcompensates for default risk 2. Perhaps investors are being compensated for other sources of risk such as liquidity risk and reinvestment risk 3. The recovery rate may vary more in lower rated bonds 4. Inv may be taking on more systematic risk ie default corr to market 5. Due to access restrictions, low rated bonds may have lower demand.

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Measuring Corp Bond Mort’y and Perf. - Altman

87

1. What are the three main risks investors care about in fixed income securities?

2. What are the four ways a bond can exit the population?

3. Altman checked two factors for whether they affected the recovery rate, what were they?

4. Altman found that even after accounting for higher default rates, lower rated bonds were delivering excess returns over Treasuries. Give four possible reasons to explain this.

88

1. Interest rate risk Default risk Liquidity risk

2. Default Call Sinking fund Maturity

3. Time since issue and Rating at issue

4. 1 – investors compensated for assuming liquidity risk 2 – compensation for assuming systematic default risk 3 – comp. for uncertainty in default rates 4 – investors taking advantage of supply/demand imbalances

Section D Text problems

89

Section D – Hull text problems Ch 2-7

Ch 2: 3, 8, 9, 10, 11, 16

Ch 3: 1, 2, 3, 5, 6, 7, 16, 18

Ch 4: 1, 3, 5, 6, 11, 13, 16, 17, 22, 23

Ch 5: 1, 2, 5, 6, 9, 10, 14, 15

Ch 7: 1, 2, 3, 5, 7, 16

Mechanics of Futures Markets - Hull

91

Hull Ch 2 – Mechanics of Futures Markets

1. What is the difference between a forward price and a delivary price?

2. What is a futures contract?

3. What is the difference between a market order and a limit order?

4. How does a margin account work? In particular, what are the functions of the initial margin, mark to market, maintenance margin, margin call, and variation margin?

92

1. The forward price is for a new contract, the delivery price is for an existing contract.

2. It is the normal case of a forward. It is a forward contract that is so standardized that it can be traded on an exchange (like oil, rice…)

3. In a market order you take the current price. In a limit order you specify some threshold at which you will buy or sell, when the limit comes the transaction happens

4.


93

1. On March 1 you enter into a long position for 5 April gold futures. Size is 100ounces. Price that day is $350. Closing prices are 345, 340, and 345, at closing on 3/1, 3/2, and 3/3. Initial margin is $1500/contract, maint. margin is $1000 per contract. What is the balance each day in the Margin acct. and when is there a margin call?

2. as

1. Initial margin = $7500.

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95

1. What are the most important aspects of the design of a new futures contract?

2.

96

1. Underlying asset, size of the contract, delivery arrangements, and delivery months.

2.

Hedging Strategies Using Futures - Hull

97

Hull Ch 3 – Hedging Strategies Using Futures

1. Arguments for and against hedging. (3 areas)

Against For

Shareholders should be able to hedge their own risks

Hedging in bulk should be cheaper. Also, the company still needs to hedge its own risks.

Competitors in a common industry most are not hedging. If you did then both your profits and losses would be higher.

Hedging involves losing money when things go well. If mgt doesn’t understand what you are doing then you may look foolish.

98


99

1. What is basis risk?

2. Which formula describes the case where our basis risk is minimized?

3. How does this formula lead to the result above?

4. How can you measure hedge effectiveness?

5.

1. It is the risk that the asset you are trying to hedge and the asset underlying the futures contract are the same or have the same delivery date.

2. , h is the basis risk, S is for the asset and F is for the future

3. Assume it is basically impossible for the difference to be 0. So take the variance of the difference

Take the derivative wrt h, set = 0, then solve for h.

4. The proportion of the variance eliminated by hedging or the R2 of the regression of ΔS against ΔF.

5.

100


101

1. How do you convert the hedge ratio in to a number of contracts?

2. What is the formula for Tailing the Hedge? How does it relate to the previous one? Why would we want to use it?

3. How is the contract size of a Stock Index future determined?

1. , QA is the number of units of the asset you have and QF is the number of units in each future contract. Then round N to the nearest whole number.

2. , VA = QA*spot price of the asset, VF = QF*spot price of the futures contracts. Futures contracts have daily settlement.

3. The contract size is the value of the index times a multiplier. Multipliers are $50 or $250. If you have 1 $50 future of the S&P 500 and it turns out the index is 1,000 then you would owe $50,000.

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103

1. What is the formula you would use to calculate the number of index futures contracts you would need to hedge and equities portfolio?

2. If you buy the suggested number of shares, then what happens to the beta of your stock portfolio?

3. What formula for N would you use to avoid 2? ie to get to some target beta

4. What is the essential difference between a futures contract and a forward?

1. , P = portfolio value. F is the value of a single futures contract (50 or 250 times the index). beta is the portfolio beta against the index.

2. It will be reduced to 0.

3. , beta* is the target.

4. You separate the commitment to buy or sell the asset from the side bet on the price change of the asset.

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105

1. How did Metallgesellschaft (MG) lose 1.3b ?

2. You own an equities portfolio worth $10m, beta to S&P 500 is 1.1. You want to complete hedge the market risk buy buying index futures. S&P index is currently 800, futures contracts are each valued @ $250 * Index. How many futures do you buy?

3.

106

1. In 1990’s they sold ~160m barrels of oil/gasoline at fixed prices for delivery 5-10 yrs in the future. They pulled in $3-5/barrel more than current prices. MG wanted to hedge by buying oil futures. But futures contracts had shorter dates than their commitments to people, so they had to roll over their futures contracts. The hedge worked, as prices fell the value of their consumer contracts rose. But the futures money was all marked to market, so they had to recognize massive losses. Mgt made them cancel the hedging program which forced them to also cancel the consumer contracts.

2. N = beta * P/F = 1.1 * 10,000,000 / (250 * 800) = 55

Interest Rates - Hull

107

Hull Ch 4 – Interest Rates

1. What is meant by par yield or swap rate?

2. What is the process for determining Zero Rates?

3. Why do derivatives transactions use LIBOR rates to develop the zero curve?

4. Knowing the stuff from 2, how do you then calculate the forward rate?

1. Both terms refer to the same thing. It is the coupon rate on a semi-annual bond whose price = it’s par value.

2. 1. Assume you know the price of a 6-mo bond w/ principal 100

, solver for R1. Note it is stated on an annual basis. 2. Now you can calculate the 1-yr coupon rate by solving

, for R2

3. Typically the counterparty in a derivatives transaction is a highly rated financial institution – so some element of credit risk is appropriate.

4. Solve for RF

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109

1. What is a Forward Rate Agreement?

2. What is the formula for the Value of an FRA and what are the terms?

3.

1. It is an agreement to earn a set amount of interest for a set time period on a set amount of principal.

2. RK and RF are the agreed on rate and the forward rate. Note RK > RF for this exercise to be worth anything. T2 and T1 are the end and beginning points in time The e^ is the PV term where R2 is the continuous discount rate to get from T2 to present.

3.

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111

112

4.

Determination of Forward & Futures Prices - Hull

113

Hull Ch 5 – Determination of Forward & Futures Prices

1. What is the formula relating the forward price of an asset to it’s spot price?

2. Hull proves the formula in 1 using the Arbitrage Argument, give a summary.

3.

1. , often the subscripts are left out. r is the continuously compounded risk free rate and T is time to expiration of the forward.

2. If F0 > S0erT or vice versa then you would be able to enter into a hedging strategy. Go long on the forward and borrow money to buy the stock. But, other investors would compete for the forwards and price would balance out

114


115

1. The previous argument showed that two alternative strategies should be equal. What is a purchasing strategy that should produce a 0 cash flow now and at T, but would produce a return at T if there were an arbitrage opportunity?

2. The formula for the forward price of an asset is a bit more complicated if the asset produces cash flows while you own it. What are the formulas to account for the PV of pmts or the pmts as a percent of asset value?

3. What is this formula:

1.

2. , , I is the present value of pmts and q is the div yield or coupon rate. Note: the first formula only works if the payment for the dividends is due in a lump sum at T.

3. K is the delivery price on the future, f is the price of the forward at any point in time. S is the spot rate at the point in time of f. K is the only fixed quantity.

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117

1. What causes the difference in price between forwards and futures?

2. What do you need to watch out for when valuing futures on foreign currency?

3. There are 3 formulas to remember for commodity forwards

4. What is the cost of carry?

5.

1. The accounting for futures. Futures that you have on your books must be marked to market on a daily basis. Marking to market results in cash flows. In the short term it can be assumed they are equal.

2. rf is the risk free rate in the foreign land, not in dollars.

3. , , , U represents storage costs for the goods, and y is the convenience yield, a measure of people’s reluctance to sell their actual goods for arbitrage.

4. If the inflation (interest) rate quoted in the problem is said to include the cost of

holding or dividends then it is to be used alone as the inflation rate

5.

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119

1. Why are futures prices not necessarily = to expected future spot rates?

2. The term structure is usually upward sloping. ie long-term rates are higher than short-term rates. How is this consistent with the Liquidity Preference theory?

3.

1. Usually your risk adjusted disc’t rate will exceed your risk free rate. Since the future spot rate is an unknown you would discount it at the risk adjusted rate to get it to PV. (then inflate it at r to compare to F)

, usually F will be a bit lower than E(ST), but if people are really speculative then you could get the reverse.

2. Investors tend to preserve liquidity by investing funds for short periods of time. This leads to a situation in which forward rates > expected future zero rates.

120

Day Count and T-bond Prices - Hull

121

Hull Ch 6 – Day Count Conventions & Quoted Treasury Bond Prices

1. Describe the 3 standard day counting conventions

2. Describe the price quotes for T-bills

3. A 6mo T-bill sells for 9,600 with a face value of 10,000. What is the quoted rate?

4.

122

1. * Actual/Actual – used for US Treasury bonds * 30/360 – used for Corporate and Municipal bonds * Actual/360 – used for T-bills and other money market instruments

2. Discount rate reflects interest relative to face value, not price; rate is annualized using a 360-day calendar and simple interest.

3. The formula used would be (Face-Price)/Face * (360/Period) = (10,000 – 9,600)/10,000 * (360/180) = 8%

Swaps - Hull

123

Hull Ch 7 - Swaps

1. Describe a basic interest rate swap. What are the usual pieces?

2. What is in it for the two parties?

3. What is the role of an intermediary in one of these transactions?

124

1. Notional amount X. Party A agrees to pay fixed interest on X to B, and Party B agrees to pay floating interest on X to A. The floating rate will be some function of LIBOR.

2. Transforming a Liability: Suppose Party B had incoming cash flows that were well correlated with interest rates. What if Party B borrowed money at a fixed rate? Then his cash flows would not tie well to his obligation. So he does a swap with A, now he can have a steady income to pay back his loan and make payments that fluctuate with the market. Transforming an Asset: Basically, Party A has now transformed a fixed rate asset into a floating rate asset.

3. The intermediary assumes the credit risk. He also charges a few basis points to each party for facilitating the deal.

Swaps - Hull

125

1. What is the Swap Rate?

2. What is the comparative advantage argument for the existence of swap markets? Two flaws the book states in this argument.

126

1. Swaps dealers publish bid and offer rates. Bid is the fixed rate they will pay to receive LIBOR, Offer is the fixed rate they want to get in exchange for paying LIBOR. The Offer < Bid. Avg(Bid,Offer) = Swap Rate.

2. The parties each borrow in the most efficient way offered to them in their respective markets and then enter into a swap since they actually wanted the deal offered in the other market. Flaw 1 – swaps markets are so big now that those arbitrages should no longer exist Flaw 2 – the apparent arbitrage is actually a result of the credit risk burden the fixed rate lender faces. the floating rate lender adjusts every 6 mos.

Swaps - Hull

127

1. Describe the two ways of valuing a swap agreement.

2. When does credit risk exist? See, sometimes you benefit from the swap and sometimes you don’t.

128

1. Exchange of Bonds: Suppose you buy a floating rate bond Bfl and sell a fixed rate bond Bfix. Then the value of the transaction is Vswap = Bfl - Bfix note that on the date the coupon rate is reset, the floating rate bond will always trade at par Series of Forward Rate Agreements: We can value each cash flow of the swap by assuming the forward rates are realized and calculate the present value of the net cash flows. Remember: you will need to calc the continuously compounded LIBOR forward rate then use that to determine the semi-annual compounded forward rate to determine swap cash flows.

2. Credit risk exists when the value of the swap is positive, ie if the counterparty doesn’t pay then you are short. If the value of the swap were negative you would just keep your money if they refused to pay – but of course they wouldn’t in that case, if they were that short on cash they would borrow just enough to get you to comply and then pay back their loan.

Swaps - Hull

129

1. What does tenor refer to?

2. What is a forward swap?

3. What is a compounding swap?

4. What is a LIBOR-in-arrears swap?

5. What is a diff swap or quanto?

6.

130

1. payment frequency

2. the payments don’t start until later

3. payments on one side or the other are compounded until the end of the deal and then a lump sum is paid.

4. the LIBOR rate observed on the payment date is used to calculate the pmt due on that date – normally you would use LIBOR at the start of the period for a payment at the end of a period.

5. a rate observed in one currency is applied to a principal amount in a different currency. example: 3mo US LIBOR is swapped for 3mo UK LIBOR both based on 10m GBP.

Section E - Options - Hull

131

Section E – Options

Ch9: 1, 4, 7, 8, 11, 14, 15

Ch10: 1, 6, 7, 10

Ch11: 1, 5, 6, 11, 15

Ch12: 3, 5, 10

Ch13: 1, 4, 5, 6, 7, 8, 15, 21, 23, 24, 25

Ch15: 2, 3, 9, 16, 17

Ch16: 3, 4, 7, 20

Options: Markets & Properties of Stock Prices - Hull

133

Hull Ch 8&9 – Options, Markets & Properties of Options Stock Prices

1. What is the difference between an option and a future?

2. What is the difference between an American option and a European option?

3. Is selling a call and buying a put the same thing?

4. What happens when there is a stock split?

5. What are three types of options that, when exercised, result in new stock issues?

6.

134

1. The transaction specified in a future must be carried out; the transaction in an option is only carried out if profitable for the owner.

2. American you may exercise the option any time up to the expiration date. A European option may only be exercised on the expiration date.

3. They both gain when prices drop. However, when you sell an option you have no control over the exercise of it; when you buy you have control.

4. The option values will adjust accordingly so that you are in the same position.

5. Warrants, employee stock options, and options imbedded in convertible bonds.


135

1. 6 key factors that affect the value of an option

2. What are the Wash Sale rules?

136

1. * Current stock price: call↑ put↓ * Exercise price or Strike price: call↓ put↑ * Time to expiration: The longer the better, except with European options since dividends paid may reduce value * Volatility: call↑ put↑ - this is because high volatility can only work in your favor – in case of a call if price drops you don’t exercise, if it increases you do. * Risk free rate: tricky – as r↑ so does price so call↑, but higher r implies higher discount rate so downward pressure on call. Mostly call↑ though. For puts both effects push down price put↓ * Dividends: higher dividends means higher stock price so call↑ put↓

2. Tax laws which state you cannot claim a loss on an asset if turn around and buy it again < 30 days


137

1. What are the Constructive Sale rules?

2. Provide 2 examples of Constructive Sales

138

1. Tax laws – if you make a trade which eliminates substantially all exposure to price movement (up or down) for another asset then you have triggered taxes

2. a short sale on an identical asset a future or forward to deliver an identical asset at a fixed price //any other transaction that eliminates substantially all of the potential for gain or loss qualifies


139

1. What are S0, ST, K, r, P, p, C, c

2. In terms of 1, what are the upper bounds for P,p,C, and c ?

3. What are the lower bounds?

4. Draw the put-call parity chart: @t=0 PortA: 1call and cash = Ke-rT, PortB: 1put and 1share. What are the final cash states if ST<K, ST>K? The results should lead to put-call parity, ie the cost of PortA = PortB.

1. S0 = initial price; ST = price at time T; K = strike price, r = risk free rate; P = value of Amarican Put; p = value of European Put; C, c = American/European call

2. C <= S0, c<= S0, P<=K, p<=Ke-rT

3. , C≥S0 – K,

4.

140


141

1. What is the main put-call parity formula? What are the assumptions hidden in here?

2. When should you exercise American calls early?

3. When should you exercise an American put early?

4. What happens to all the previous formulas if you include dividend payments: D = present value of dividends? What if the dividend is paid continuously?

5. What is the American Put-Call Parity w/ Dividends (present value of Dividend is D)

1. 2. Almost never. The only time is when dividends are paid early and you expect

the stock price to fall. You never exercise early because if you need cash you would just sell the option.

3. You may want to exercise early to get the cash and invest it, so an American put will be worth more than a European put and P ≥ K ‐ S0.

4. c ≥ S0 – D – Ke-rT, p ≥ D + Ke-rT-S0, put-call parity: c + D + Ke-rT = p + S0

cont. div: S0-D = S0e-qT =>

5. S0-Ke-rT > C – P > S0 – D – K

142

Option Trading Strategies - Hull

143

Hull Ch 10 – Option Trading Strategies

1. Profit positions – how should you plot data? Plot the profit diagram for a bull spread: you buy a call for $5 with strike $20 and sell one for $3 with strike $30.

2.

1. These are charts: Stock price / Profit (x, y). Always plot where stock price is 0 and a couple points above and below the strike price. If there is more than one strike price then do both.

144

Option Trading Strategies - Hull

145

1. Define: Bull spread, Bear spread, Butterfly spread, Calendar spread, Diagonal spread

2. Define: Straddle, Strip, Strap, Strangle

146

1. Bull spread: buy a call with low K, sell a call with high K Bear spread: buy a call with high K, sell one with low K Butterfly: buy a high K call and a low K call, sell two intermediate K calls Calendar: strike prices are the same but expiration dates differ. When plotting show the payoffs at the earliest expiration date, assuming that the option that has not yet expired is sold at that time Diagonal: both strike prices and expiration dates differ

2. straddle: buy a put and a call with the same strike and expiration strip: buy one call and two puts all with same strike and expiration strap: buy two calls and one put strangle: buy one put and one call, same T, different K

Option Pricing w/ Binomial Trees - Hull

147

Hull Ch 11 – Option pricing w/ binomial trees

1. The binomial model makes a fundamental assumption, what is it

2. How do you build a risk-free payoff model for pricing an option?

3. How do you figure out the multiple for the number of shares to include in your initial portfolio?

4. What is Goldfarb’s way of thinking about the ‘replicating’ portfolio?

5. You must understand the strategies above. You should memorize two formulas to get the answer more quickly for f = the call price at time0. What are the two formulas?

1. The stock price at time T can only have one of two values: S0u or S0d

2. Construct a portfolio that is a mix of the stock and call options (usually you sell one option and buy a multiple of the stock), whether the stock price rises to S0u or S0d the value will be the same. You discount to time0 and set = to the price for your portfolio. Solve for the call price and you are done.

3. 4. At time0 borrow money that will grow at a known rate to ΔS0d at time T and

buy the Δ number of shares. That will grow to be equal to the call value at T.

5. , , notice how you can almost think of p as a probability, think of it as an artificial probability.

148


149

1. What is an alternative way to calculate p? (much more intuitive) What are the assumptions to remember to rebuild this?

2. Draw the Binomial tree picture from T=0 to T=2

1. Solve this formula for p: ; Investors are risk neutral -> stock earns risk free rate Then use the p in the call version of the formula:

2.

150


151

1. How do you modify the tree model to deal with American options?

2. How does the delta change for a binomial tree? (what is the delta?)

3. Now, suppose the stock has volatility σ, how do they define u and d?

4. How does this formula change if the stock pays dividends at a continuous rate q ? (also, use delta t instead of T and the new definitions of u and d.)

1. Check at each node whether the value from the formula is > or < the value of the options of exercised right then and use the larger one.

2. Same delta we talked about before: spread of option values/spread of stock value. The difference to notice is that delta can change over time, so be careful to recalculate it. If you recalculate it what does that mean? It means you will have a different mix of stock and calls in your portfolio – you have rebalanced your portfolio.

3. , , note once we define these the Su = Su…

4. =>

, think of it as equating the future values.

152


153

1. What is the formula you solve for p if dealing with foreign currency options?

2. What about solving for futures?

1. , rf is the risk free rate in the foreign currency. (This could all be presented in the same structure as the others.

2. , F is the current value of the futures contract.

3.

154

Weiner Processes & Ito’s Lemma - Hull

155

Hull Ch 12 – Weiner Processes & Ito’s Lemma

1. What is the basic formula for Geometric Darwinian motion?

2. What is dz?

3. What is the general Weiner process equation? Derive the variance of the process.

4. How does an Ito process differ? What is the formula?

1. 2. dz = normal random mean 0 variance dt

3.

Var(Δx) =

4. The Ito process defines the a and b terms as functions of the underlying variable and time

156

Weiner Processes & Ito’s Lemma - Hull

157

1. If you have a function , how do you express dG as Taylor expansion?

2. Now, substitute in the ds from the previous card. (Since dt is so small you can throw out any piece with a dt to a power greater than 1 and the entire portion of the Taylor expansion beyond the 2nd order)

3. Now assume it’s ok let and , then you get Ito’s Lemma:

1.

2.

3.

158

Black-Scholes-Merton Model - Hull

159

Hull Ch 13 – Black-Scholes-Merton Model

1. What are mean and st. dev. of total return? Annual return? What is the distribution of LN(ST)?

2. How do mu and sigma fit in?

3. How is Volatility defined?

4.

1. mean: St dev:

mean: St dev: , notice all you did was multiply the total return by 1/T. Also mu and sigma are for the Stock price

Normal: mean=ln(S0)+ , st dev =

2. , the assumptions of the other chapters result in the stock price having a log-normal distribution.

3. It is helpful to always define it on an annual basis: Annual volatility = Daily volatility * 252.5

160


161

1. What is the intuitive derivation of the Black-Scholes formula? Note: this is only valid for a European call.

1. Remember, we are trying to value a call option. A formula in the book is

, f is the call price @ t=0. Now imagine the first delta and the term in parenthesis. It turns out that the parenthesis bit is = to a multiplier times the strike price. We are going to describe delta and that multiplier with normal distributions.

162


163

1. The formulas on the previous card are only valid for a European call. How do you price American calls?

2. How do you get the value of a put?

3. What are warrants?

4. How do you value warrants?

5. When is the stock price affected by a warrant? At exercise or at time of sale? Why? What does that mean about the problems you’re given?

6. What expected rate of growth does a risk neutral investor use?

164

1. So long as there is no dividend then it works for American calls as well.

2. You should solve for the call with same strike and time and then use put-call parity formula to get the put price.

3. A warrant is like a call but when it gets exercised the company issues new stock and collects the strike price from the warrant holders. Assuming there were N shares already out there if M warrants are exercised then the warrant value is N/(N+M) times the value of a similar call.

4. N/(N+M) * call price

5. At time of sale. The price of the stock will be adjusted to reflect the possible future dilution. Need to use the stock price before issue of the warrant.

6. risk free rate, regardless of the mu given.


165

1. How do you value a European call option w/ Dividends?

2. American call option with dividends?

3. What is Black’s approximation for the American call with dividends?

166

1. Use the regular formula but subtract the PV of the dividend payments from the stock price. Use risk free rate and ex-dividend dates in PV formula. Only include dividends that are paid before expiration of call.

2. If you exercise early you get to collect the dividends so it may make sense. It works out that it is really only the ex-dividend date prior to the final dividend before expiration that would lead to early exercise.

3. Value a European call with same strike and time as the American call. Then value another one with same strike but expiration time is on the ex-dividend date of the last dividend. Whichever is greater is the value of the option. NOTE: if there are dividends be sure to take the PV of them out of S0.

Options on Indices and Currencies - Hull

167

Hull Ch 15 – Options on Indices and Currencies

1. How do you modify Black-Scholes for stocks w/ continuous dividends?

2. How does put-call parity change w/ continuous dividends?

3. What is the common approach in both 1 and 2?

4. How about currencies and indices?

5. To value options on indices you need to know 1 of two things, what are they? Which is usually available?

1. ;

2. 3. You use Se-qT in place of S

4. Currency options can be viewed as assets that pay a dividend – just use the formulas above. Slide the risk-free rate for the foreign curr in place of the q, or use the Forward rate F in the calc for d1.

5. Forward rates or dividend yield. Forward rates.

168

Options on Indices and Currencies - Hull

169

1. What is the formula for the Forward price of an index? How is that derived from put-call parity?

2. What will you need to do for American options?

3. How would you insure your asset portfolio using index options? Suppose you want to floor your portfolio so it won’t drop below 90% of current value, what do you do? (list assumptions)

4.

1.

Substitute into and do the algebra to solve for F.

2. Instead of solving for F you will solve the put-call formula for q

(don’t bother memorizing)

3. You will buy puts. Assuming your portfolio beta is 1.0 and the dividend payout is similar – you just need to buy put options on the index with a strike price = 90% of current value. You calculate how many contracts based on ratio of portfolio value to value of a put contract.

170

Futures Options - Hull

171

Hull Ch 16 – Futures Options

1. What are the two dates involved in a futures option? What are you buying?

2. How do futures and forwards behave differently?

3. When valuing options on futures it is ok to use the Black-Scholes formula with F in place of S. What assumption must work? How is the assumption not met? What is the practical condition that keeps the broken assumption from being a problem?

172

1. The option maturity and the delivery date. The option buys you a bit of breathing time: today is 1/1, you are looking to buy a future with delivery 7/1 with price of $50. Or you can enter into a call option with maturity in two weeks which will lock in a future price of $52, same delivery date.

2. A future is marked to market daily – so you get the cash difference between the current futures price and the delivery price on your contract. => if the futures price goes down then you lose money.

3. F must follow Geometric Brownian Motion with constant variance. This isn’t really met since the contract value will converge to S on the delivery date. Since the maturity date is usually much less than the delivery date it is fairly safe. Note – this may be referred to as the Black Model since he did this one in a separate paper from the one with Scholes.

Futures Options - Hull

173

1. How is a futures contract like a stock that pays continuous dividends?

2. If you swap out the S for F and the q for r in the Black-Scholes formula what do you get for futures options? (include a formula for d1)

3. How does put-call parity look if we make the same substitution?

4. When is a futures option and a spot option the same?

5. Why is an American futures call worth more than a European futures call option?

1. When you do a futures contract there is no up front cash exchange. So the money you save gets to earn interest at risk free rate – like a constant dividend.

2.

3. 4. If you are dealing with European options and the maturity = delivery

5. Because it can be exercised early.

174

How to Use the Holes in Black-Scholes - Black

175

Black – How to Use the Holes in Black Scholes

1. List all ten of the unrealistic assumptions made by the model and Black’s suggestion of how to take advantage when they are not met.

176

1.

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Valuation of Bonds w/ Embedded Options - Fabozzi

177

Fabozzi – Valuation of Bonds with Embedded Options

1. What is a callable bond? What is the basic idea in valuing it?

2. How do you build the binomial tree for bonds?

1. The issuer has the option to prepay the principal (has the effect of lowering the value of the bond). You value the non-callable bond and SUBTRACT the value of the call option.

2. You do it for interest rates and then work back from the par value:

178


179

1. How do you calculate rH and rL?

2. How do you calibrate rL?

3. How do you deal with coupons?

4. Now, how do you deal with callable bonds? What about putable bonds?

5. Be sure to do Goldfarb’s sample problems for Step-up Callable Notes and Range Notes. see problems past page 355 in guide.

1. I guess one of the two will be given then

2. I’m glad you asked. You take a non-callable bond with a known price and par value and derive the embedded rL. Usually you would use recently issued (on the run) US Treasury bonds.

3. After determining the value at the end of each period then add in the coupons

4. At each point in time on the tree you cap the bond value at the strike price. Assume the call will occur after the coupon payment unless stated otherwise on the exam. Same thing for putable bonds, you just need to check at each point whether the bond is worth more than the put.

180


181

1. What is the OAS? How would you calculate it?

2. What are the formulas for duration and convexity?

1. The Option Adjusted Spread is the number of basis points you would need to add to the actual price of a callable bond in order for it to match the theoretical price the Fabozzi procedure describes. You calculate it by iteratively adding to rL (to get rL*) until your theoretical price matches the actual one. OAS = rL* - rL

2.

182

Sect. F: International Diversification - BKM

183

BKM Ch 25 – International Diversification

Most portfolios exhibit a homecountry bias

1. What are two additional risks you face w/ int’l investments?

2. How do you find the forward rate of a stock using native and foreign risk-free rates?

3. What is the best model to use when assessing the benefits of global diversification?

4. What happens in a bear market?

1. Exchange rate risk and country-specific risk (probably correlated right?)

2. Continuous:

Annual:

3. Use a global CAPM estimate est of expected returns, not realized returns, to build your efficient frontier.

4. Markets become much more highly correlated – all markets are impacted by some common factor. You can only diversify country-specific risk.

184

Sect. F: International Diversification - BKM

185

1. How can a passive investor benefit from international diversification?

2. How does the active investor benefit?

186

1. He can’t really.

2. Her goal is to find mispriced assets. By expanding the universe of assets her chances of finding the mispriced ones only gets better.

Sect. G: Asset – Liability Management

187

BKM Ch 16 – 7, 10, 12, 15

Managing Bond Portfolios - BKM

189

BKM Ch 16.3,4 – Managing Bond Portfolios

1. List and describe the two main types of passive bond investment strategies: main difficulties? solution?

2. What is Net Worth Immunization?

3. What is Target Date Immunization?

190

1. Indexing mirror the overall market by tracking an index. difficulties due to bond calls and maturities – lots to keep up with, must break an index into broad maturity / sector categories and match those. Immunization you want to cancel out the effects of market changes, various strategies

2. Net Worth Immunization: Net worth = ability to meet future obligations; both liabilities and assets are affected by rate changes, goal here is to make it so that they move together, to accomplish this you match the dollar durations of assets and liabilities.

3. Target Date (Holding Period) Immunization: as interest rates rise the resale value of a bond falls but the value of reinvested coupons rises. Since we care about total future value we want these effects to cancel out. They do if your target date equals the Macaulay duration of the bond.


191

1. If you are engaging in an immunization strategy, why must you rebalance?

2. So why not just buy bonds in such a way that asset cash flows match expected payment cash flows?

3. Why would do a Substitution Swap?

4. Why would you do an Intermarket Spread Swap?

5. Rate anticipation swap?

192

1. Immunization works only for small changes in rates. As soon as rates change durations change, which means you need to rebalance portfolio.

2. Although interest rate risk is eliminated your portfolio options are very limited – maybe even impossible if liability cash flows are very far in the future.

3. replace one bond with another of basically equiv cash flows except a lower price

4. Eg: The spread between corp and gov’t bonds is expected to narrow so an investor may want to swap gov’t for corporate bonds.

5. You would do this to protect against an expected change in interest rates. If rates are going to drop you may try to get into bonds with longer durations now. And the other way: if you expect rates to rise you will want to be in short duration bonds so that upon expiration you get into higher yielding bonds.


193

1. What is a pure yield pickup swap? Why do it?

2. What is contingent immunization?

3.

194

1. Replace low yield bonds with high ones – no real mispricing you would typically be accepting higher risk. You could do this by replacing short term bonds with longer term bonds when the yield curve is upward sloping.

2. Invest regularly in a passive strategy so that it grows to a target value at a specified time(s). Then actively manage the portfolio. So long as the value stays above the target value then you continue active inv. as soon as it dips below then active mgt ceases.

ALM for P&C Co’s - Noris

195

Noris – ALM for P&C Companies

1. What does Portfolio Equity represent?

2. What are the three valuation methods? describe them

3. What rate does Noris use? Why?

4. What is Franchise Equity?

196

1. Net value of currently booked assets and liabilities already written.

2. Book Value (Stat Surplus) – bonds discounted according to yields at time of purchase, stocks are valued at market value and liabilities are not discounted. Current Value Surplus – Values all assets at current value and the liabilities on an undiscounted basis. Market Value Surplus – Assets are current market value, discount liabilities to PV based on: hist yield rates, curr yield on assets, some made up rate.

3. Yield on municipal bonds. It is a good proxy for current after-tax yield.

4. Value of business not yet booked. (ignored in the paper – too hard to price)


197

1. What is the formula for Duration of Market Value Surplus?

2. What does a Duration of 0 indicate?

3. What is the Duration Gap of Surplus? What does the sign indicate?

4. What are the calculations for DMVA? (Note: stocks use the modified duration of stocks formula)

5.

1. 2. That there is no sensitivity to interest rate changes

3. It is the amount by which DMVS exceeds 0. If positive then MVS falls as rates rise; negative then MVS rises as rates rise.

4. Macaulay Duration for Bonds; Modified duration for Stocks = 1/(dividend yield); dividend yield = Div / Stock Value

5.

198


199

1. What are three types of Duration Gap Targets? For each one specify strengths and weaknesses.

2.

1. Duration Gap of Surplus – set this to zero (set the formula above to zero and solve for ?). Immunizes form the effects of rate changes. Results in large fluctuations in earnings and is unduly restrictive. Duration Gap of Total Return – People may want to achieve a target return for a specific holding period. If H is the desired period then solve

Duration Gap of Leverage – Set duration of Assets = duration of Surplus

200


201

1. Inflation is a source of unexpected loss development. Why can’t duration matching immunize against it?

2. What are the 3 approaches Noris gives for dealing with this?

3. asd

202

1. Duration matching uses fixed income assets – but these have cash flows stated in nominal terms. So your asset cash flows will stay in 2009 dollars but your claims cash flows will grow with inflation.

2. 1 – Invest more heavily in stocks and real estate, both inflation sensitive. Unfortunately this introduces too much non-interest rate volatility 2 – Hold assets which roll over more frequently. Unfortunately this requires holding more short-term assets and keeps you from doing duration matching 3 – Overstate your liabilities in the form of contingency reserves. ie – hold higher assets to account for the higher expected cash flows.

Asset/Liability Matching for P/C Ins’s - Feldblum

203

Feldblum – Asset Liability Matching for P/C Insurers

1. What are 5 ways that P/C Ins’s and Life Ins’s differ that impact Asset/Liab strategies?

2. What are Feldblum’s statements on Cash Flow matching and Duration matching?

3.

204

1. a. Life liab cash flows tend to be stated in nominal terms b. P&C liab tend to have shorter duration than life so duration matching results in inv in short-term assets c. Equity cash flows and P&C liab’s share two sensitivities: inflation and considerable risk other than interest rate risk d. P&C ins’s do not face disintermediation risk (policyholders move funds to other inv products as rates rise) as do Life ins’s. This forces Life Ins to sell assets ahead of time e. Long-term bonds are risky for P&C Ins’s because the payout time of liab is less certain. P&C Co’s must focus on real (market) return of investments.

2. Cash flow: cumbersome, inefficient, and costly Duration: seems like this is ok, follows same as BKM


205

1. Feldblum’s Loss Reserve Duration calculation; describe the key assumptions: pmt time within each year treatment of pmts in yr incurred pmt of balance at end of yr 10 discount rate

2. How does inflation impact the duration calc?

3. What is the formula for Duration of Equity presented by Feldblum?

4. How does the Dur. of Equity formula differ from Norris’s?

1. pmt time: mid year incurred yr pmts: not included – we are specifically interested in the payout of reserves since that is what we are holding assets for yr10 balance: paid out evenly in yrs 11-15 disc’t rate: current (accident year of reserve) yield on asset portfolio

2. If you adjust the disc rate for inflation you will end up with something close to no discounting which results in a duration of 0.

3. 4. The g=0 in Norris’s equations. g is the dividend rate, k is cost of capital or

discount rate.

206


207

1. How do you derive the Dur of Equity formula?

2. The dividend is sensitive to the interest rate. What 3 reasons does Feldblum give to explain common stock prices fall in short run but rise in long run when interest rates rise?

1. Start with value of firm equity: take derivative wrt k:

expressed as a % of value:

= , just ignore the neg

2. * Value of Assets: if int rates rise due to inflation then the value of real assets will also rise * Supply & Demand: higher int is higher cost, if demand doesn’t increase then dividend and stock value will drop * Market Demand for Stock: higher int rates drive inv to bonds, decreasing demand for stocks -> prices drop

208


209

1. Feldblum shows the corr of stock prices to inflation. what is the relationship in same year? what is the relationship to inf lagged 1 year?

2. What are 4 other costs to consider when managing interest sensitivity?

210

1. negative; positive, ie stock values rise if there is positive inflation in the year prior.

2. a: Yields – shorter asset durations involve giving up the higher yields assoc with long-term bonds. of course you have less risk too, so maybe not much of a loss b: Transaction costs – much higher with stocks and short term bonds since you have trading costs and analysis costs c: Disintermediation – technically, this only affects when a loss is recognized. the bond value loss already occurred when rates rose d: Cash flow risk – a P&C ins is unlikely to become insolvent just because rates rise. long-term bonds can be held at ammort cost and curr premiums used to pay old claims, so your asset portfolio can set yield as a higher priority than duration matching.

Managing int rate risk - Panning

211

Panning – Managing Interest Rate Risk

1. What is the main point in Panning’s paper?

2. Why is the market value of a firm not the same as the accounting (book) value?

3. If your book / market value ratio is 0.667 what does that indicate about your Franchise value?

4. Panning notes 3 things not to ignore about Franchise value

212

1. That interest rate risk can be managed by adjusting pricing strategies and that may have advantages over traditional methods of just managing that risk with the inv portfolio

2. Book value does not account for future business or Franchise Value.

3. It means market / book is 1.5 which indicates $50 out of $150, or roughly 1/3 of your market value is Franchise Value.

4. 1. the point above – it is a significant portion of firm value 2. franchise value is exposed to interest rate risk because it is the present value of cash flows on future (renewals and new) business 3. it is often unmeasured, unreported and unmanaged by the firm


213

1. What are the 9 basic assumptions in Panning’s simple model?

2. What are P,E,L,y,S,k,cr,F, and C?

3. Panning assumes we set prices so that

, this means P equals:

4.

1. bus written 1/1; expenses paid immediately; claims paid in full at eoy; exp and exp losses are same ea year; maintains same surp ea yr (pay out profits and raise capital to cover net losses); no risk of bankruptcy; no taxes; term structure of int rates is flat; calcs done 1/1

2. Premium, Expenses, E(Loss & LAE), risk free rate, Surplus, target return on surplus, client retention, Franchise value, Current econ value of assets and liab on balance sheet

3.

214


215

1. C =

2. Derive (or just show) Panning’s base formula for Franchise value.

1. 2. Notice that F is the present value of future premiums less expenses and

claims. Remember claims are paid at eoy so disc them one extra period. If you

apply retention and discount factor

216


217

1. Panning’s goal is to have P fluctuate as a function of y. So he builds a linear model ________________ to reflect the fact that the target return is a function of y.

2. Rewrite the formula for F to include the function for k

3. Don’t bother with the derivation of the Duration. Just remember it is the derivative wrt y. What is the result:

1.

2.

3.

218


219

1. Assume your curr econ value = 54.76 w/ duration of 1.0 and franchise value = 28.57 w/ duration of 17.62 What is the duration of the total economic value?

2. What is the problem with trying to manage Franchise duration with investment policy?

3. What is the main difficulty w/ Panning’s approach?

4. What is the key virtue?

1. 2. In this case, even if you bring inv duration to 0 that only reduces the tot econ

value duration to 5.18. Maybe you could do some negative duration assets. Unfortunately investors only see the balance sheet and it will look mismatched and risky.

3. It is difficult to maintain a rigid relation between target return on surplus and target duration of econ value for anything other than small changes in rates. But this is true of all duration matching strategies.

4. implementing a pricing strategy to help manage duration of franchise and total econ value is invisible to external audiences as is the franchise value you are trying to protect.

220

Section H – Financial Risk Management

221

Section H – Financial Risk Management

Hull Ch 17 - 2, 3, 5, 10, 11, 14, 16, 17, 22

Hull Ch 20 - 1, 2, 3, 9, 10

Hull Ch 22 - 1, 3, 5, 7, 14

The Greek Letters – Hull Ch 17

223

Hull Ch 17 – The Greek Letters

1. Which is the most important letter to hedge?

2. Lets say you sell a European call and now you face risk. What is the risk if you do nothing? What is the risk if you practice a cover strategy? What is the risk if you practice a stop-loss strategy?

3. What does delta represent?

224

1. Delta

2. Do nothing – if stock price rises the counterparty will exercise and you will have to buy a very expensive stock and absorb the cost S-K Cover – you buy the stock at inception so you have it to give for K if price goes up. But if price drops far below K then you are left with a stock w/ little value Stop-loss – Own shares whenever S>K and don’t own them of S<K. This results in high trading costs though. Also you’ll end up buying high and selling low.

3. rate of change in the value of an option as the price of the underlying security changes.


225

1. Suppose you own a call for 100 shares of stock w/ a delta of .6. What is the delta portfolio to hold?

2. What problem do you run into maintaining a delta hedge?

3. If you are using Black-Scholes, what is the formula for delta on a call? on a put? (Note: European options without dividends)

4. If you have a portfolio of options on the same security what is the delta for the portfolio? Why is it helpful to look at it this way?

5.

1. Sell short .6*N shares of the stock. So that when price goes up $1 option value goes up .6*$1*100=$60, but the short sell loses $60. So you net 0.

2. The delta changes so you end up having to rebalance the portfolio of short sells. Expensive.

3. Call ; Put

4. Weighted average: , you can execute a single hedging transaction in the underlying asset – you save transaction costs.

5.

226


227

1. What is Gamma? Why is it important?

2. Call Gamma formula and Put Gamma formula

3.

1. Rate of change of Delta. ie 2nd partial derivative of option value wrt asset value Gives you a feel for sensitivity of Delta, tells you how often to rebalance

2. , same for the Put

3.

228


229

1. What is Vega?

2. Vega formula:

3. What is Theta?

4. Theta formulas for Calls and Puts

1. The change in value of the option with respect to the volatility of the underlying asset.

2. 3. Theta measures the sensitivity of the option value to changes in the time to

expiration.

4.

230


231

1. How is Theta hedged against? Why?

2. What is rho? Call and Put formulas?

3.

1. Theta does not really represent a risk: time to expiration is a known thing. So Theta is used 1: to determine how quickly your option position will decline 2: Theta is a proxy for Gamma

2. Change in value of option wrt interest rate.

,

3.

232


233

Call option w div yield q Put option

Call option w div yield q Put option

same

Don’t bother same

same

234


235

1. What is the delta of a Forward contract? why?

2. What is the delta of a Futures contract? give a quick explanation.

3. Suppose you need to hedge N shares of a stock. How many Futures contracts do you get into?

4. Why do you need fewer futures contracts than shares of the stock?

5. What if the future is on a dividend paying stock, what is the delta? why?

6. What if it is a currency future?

1. 1, , so the derivative wrt S is just 1.

2. erT remember the futures price is F=SerT. So derivative of F wrt S is erT.

3. You buy e-rT futures.

4. Because the delta is greater than 1. For each dollar of stock value change, the future value will change by more than a dollar.

5. e(r-q)T because F=Se(r-q)T

6. replace the q with rf, just like in the other chapters.

236


237

1. Suppose you own $100m of S&P 500 index and you want to insure with a put. But no puts are for sale. How do you create a synthetic put with short sells?

2. What happens to delta as the value of the portfolio falls? How does that affect your strategy? How is that expensive?

3. How could you use futures instead?

4. What happened in 1987 that presents a problem for this strategy?

238

1. You would short sell delta * $100m worth of stock and invest proceeds at risk free rate.

2. It gets more negative. You end up doing bigger hedging transactions. You are buying high and selling low.

3. You just use futures, really the same process. Hopefully the transaction costs are a bit lower.

4. Like most investment ins strategies it relies on the market existing for the things you will need to sell. In 1987 so many people were using the same put strategy for portfolio insurance that when the index started coming down everyone was trying to sell it even harder that put even more pressure downward…

5.

Value at Risk – Hull Ch 20

239

Hull Ch 20 – Value at Risk

1. What is the easiest way to calculate VaR?

2. How do you convert between annual and daily volatilities?

3. Assume you have $10m of a stock. Find the 10day 99% VaR. daily volatility is 2%, each day is independent, expected daily price changes are small – assume 0, change is normally distributed.

1. Historical Simulation: Grab a few hundred days of historical daily rate changes in various market variables. Then use those changes applied to the current market variables to calculate the impact on your portfolio. Rank the outcomes and grab the X% value. That is your X% VaR.

2. 3. 10day price change is normally distributed with mean of 0 and

st dev = . From the Normal tables we get

. So we’re 99% sure the portfolio will not decrease more than in 10 days.

240


241

1. What is the formula for variance in the linear model for calculating VaR when multiple assets are in the portfolio?

2. So what would be the 99% VaR?

3. What two formulas do you use for the Duration Approach of calculating VaR – this is for assets subject to interest rate fluctuations? What is the simplifying assumption?

1. , note corr parameter is different for each i,j pair.

2.

3. , , P is the value of the portfolio

and is the size of the parallel shift in rates. The assumption is that all yield curves are shifting up or down to gether.

242


243

1. How does the Cashflow Mapping Approach work?

2. How do you match the st dev in the mimic portfolios?

3. Explain the components of this formula for a portfolio of options:

4. What is the variance of delta P ?

1. You look at the cash flows for your entire portfolio and map them to a small number of zero coupon bonds with stated maturities. Bundle cash flows so that each cash flow can be mimicked with a portfolio of zero coupon bonds. So a 7mo cash flow would be mimicked with a portfolio of a 6mo and 12mo bond. You must match the value and st dev.

2. Solve for alpha in

3. The change in price = sum of amount of stock * stock delta * percent change

in stock

4.

244


245

1. Change in portfolio value formula using little delta and gamma for a bunch of options, each subject to just one market variable.

2. What is in these sections?

3. What is wrong with using the Delta-normal approximation for calculating the VaR of the value of a call option we bought (were long in)?

1.

2. It is

3. You will overstate VaR. The delta of an option changes quite a bit wrt underlying stock value. The dist is skewed to the right.

246


247

1. What is the more accurate method for calculating the VaR of an option?

2. What if we sold (short) the call option? Compare the two VaR calc methods of the option value.

3. What could we do instead of assuming distributions for the portfolio and option values and picking percentiles off those distributions?

4. Pros and Cons of Historical simulation

5. Pros and Cons of Model approach (picking a distribution)

1. Estimate the new Black-Scholes option price at the VaR point using normal-approximation method on the portfolio. The difference between this price and the option price at the regular point is the VaR.

2. Now the normal approximation understates the VaR.

3. We could do Monte-Carlo modeling

4. lets us used actual observed data, but computationally slow and volatility updating schemes cannot be used

5. calculates quickly and we can use sophisticated volatility estimation, typically assumes is normally distributed which isn’t true in reality.

248


249

1. Why do companies have to do stress testing?

2. What is the value of back testing?

3. What is principal component analysis?

250

1. VaR modeling does a good job showing what can happen in normal market conditions. But extreme events seem to happen more often than models predict. So mgt needs to build special stress cases of specific market movements.

2. Pretend you were building your model and making decisions using it in the past. How well would it have performed?

3. Identify key factors which explain movement in market variables. Then use those factors to estimate portfolio risk. (this is a lot like the predictive modeling that Craighead and I did, it is also similar to what the RECA does)

Credit Risk – Hull Ch 22

251

Hull Ch 20 – Credit Risk

1. Given this table of cumulative default rates by rating and year since issue, what is the conditional probability of default for a Baa rated bond in yr 3?

2. The result from 1 gets formalized into an instantaneous hazard rate

. Using this, what is the cumulative probability at t formula?

3. In a year with high rates of default, what will happen to recovery rates?

4. What is the simple formula for relating spread, default hazard, and recovery rate?

1. Think that of mortalities. 100 people start the race. If you make it to start of yr 3 you are in a group of 100-1.03=98.97. 1.62 out of that group will fall out in yr 3. So conditional probability is 1.62/98.97 = 1.64% chance of falling out given you made it that far.

2. 3. They will go down. Recovery rates are negatively correlated with default rates.

4.

252


253

1. Do at least one example of the more exact calculation from the chapter notes. What is the basic idea?

2. In an Asset Swap who assumes default risk? Who holds the interest rate risk? What happens if the bond defaults? How is the risk-holder compensated?

3. How could you use 2 to calculated default probabilities?

4. Why are estimates of default probabilities based on historical data much lower than estimates based on bond prices?

254

1. You calculate Q by first finding the LGD for each possible period of default. LGD is loss given default.

2. The swap buyer assumes default risk. The swap seller assumes interest rate risk. If the bond defaults that buyer must keep making the coupon payments to the seller, the seller makes LIBOR+Spread payments to the buyer, but the bond stops paying coupons to the buyer. So the buyer must get compensated for the credit risk as part of the Spread payment.

3. Get the PV of the spread pmts. Use this in place of G-B in the process in 1. Then solve for Q like normal.

4. Risk Neutral vs Real World probabilities. Historical probabilities are real world (discounting done w/ risk adjusted rates) while those calculated from bond prices are risk neutral – derived by discounting at the risk free rate.


255

1. Why do the different rates and probabilities imply?

2. Why should traders be expecting a return? (4 reasons)

3. So which rates should be used and when? ie pricing verses making probabilistic statements.

4. When are risk-neutral rates most commonly used?

256

1. That investors are building in an expected return.

2. a: liquidity premium – corporate bonds are somewhat illiquid and so traders are paid for that risk b: conservatism – traders may be using higher default probabilities than historical rates suggest c: systematic risk/credit contagion: default rates fluctuate, perhaps systematically – so they must be compensated for that d: skewness: bond returns are skewed: limited upside, large downside. more difficult to diversify the risk

3. It depends what you are trying to do. For pricing you will use either risk-neutral prob’s and risk-free disc rates or real-world prob’s and risk-adj disc rates and the price will be the same. But for making statements about probabilities you need to use real world probabilities.

4. When calculating prices for derivatives.


257

1. What is wrong with using credit ratings in your risk analysis?

2. In the Merton Model there are two import variables VT and D. How are they related? What is ET?

3. Why is ET = 0 when VT < D?

4. Using the Black-Scholes formula, what is E0?

5. So what is the probability of default?

1. They are not updated often enough.

2. VT is the company value at T. D is the outstanding debt. ET is the equity = max(VT-D,0).

3. If VT < D then the owners of the firm (equity holders) would prefer to just let the debt holders have the assets. So equity is never really negative.

4. , notice that D (debt) is the strike price and V (asset value) is the stock price. So use the normal d1 and d2 formulas too

,

5.

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259

1. The model on the previous card needs V0 and σV. The strategy will be to get another formula and solve two equations with two unknowns. What is the other formula we use?

2. What kind of contracts can be both an asset and a liability? (ie, you can lose if the counterparty defaults, or counterparty loses if you default)

3. What are 3 methods for managing credit risk?

4.

1. Ito’s Lemma:

2. A 1yr Forward contract to sell $100 for AUD 150. If the counterparty defaults then we won’t get the deal. Same is true if we default.

3. *Netting – transactions w/ a single counterparty are netted to reduce overall credit risk *Collateral requirements *Downgrade triggers

4.

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261

1. What is the Gaussian copula formula for default probability?

2. What is the formula for the percentage of defaults in a portfolio? (similar)

3.

1.

2.

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263

1. Use the formula on the previous card to solve this: portfolio value: $100m. 1-yr prob of def: 2% (for ea. asset) recovery rate: 60% copula coeff: .10 What is the 1-yr 99.9% credit VaR? (ie the quantity which we will be above w/ 99.9% certainty)

2. The CreditMetrics model is more like what we will be doing in RAPC. What additional behavior does it model?

1.

2. CreditMetrics uses transition matrices so that the bonds change in rating.

264

Rethinking Risk Management - Stultz

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Stultz – Rethinking Risk Management

1. Since firms cannot gain much by actively managing an investment portfolio they sometimes seek to in order to manage risk. But finance theory suggests shareholders can hedge financial risk themselves, they don’t need the company to do it for them. So, why should the company manage risk or try to diversify better?

2. What are 3 examples of costs (associated with cash flow volatility) that can be reduced through risk management?

3. What is the primary objective of risk management?

4. The statement in 3 indicates that who should be hedging more? (large or small firms?) What do we see in the market?

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1. Sometimes by managing volatility of cash flows you can reduce costs. So that is an area to work on.

2. * Reduce bankruptcy costs * Reduce payments to stakeholders * Reduce taxes

3. eliminate costly lower tail outcomes, minimizing the likelihood of financial distress, thereby preserving financial flexibility to carry out investment objectives.

4. Small firms should be hedging more, but we don’t see that. We see larger firms hedging (maybe because of computers and personnel)


267

1. What is speculative hedging? What is a danger?

2. As debt to equity ratios increase a company is exposed to greater risk of financial distress. How should the following companies use risk management as a replacement for equity: 1 – firm with very low debt to equity 2 – firm with low credit rating and near financial distress 3 – firm already in financial distress

3. VaR is not useful as a risk measure over longer periods of time (1 year or more) for two reasons:

268

1. It is when a company due to its business has a good understanding and expectation of market movements. They should probably make the most of that knowledge. The problem is that knowledge may not actually give them an advantage in trading.

2. 1 – doesn’t need to worry about hedging. may want to consider increasing debt ratio for tax benefits, management incentives, and concentrated ownership 2 – engage in risk management 3 – already in distress, don’t engage in risk mgt because benefit will just go to debt holders. instead you should probably be speculative in investments

3. 1 – you can’t calibrate or test your measures for reasonability because you can collect enough data over time 2 – VaR typically assumes some normal distribution which just doesn’t fit reality.


269

1. What does Stultz recommend instead of VaR? What are the benefits?

2. Stultz makes two suggestions for management incentives to encourage some risk taking:

270

1. Cash flow simulations (DFA). Measure risk over longer time horizons, capture correlations among variables, use non-normal distributions

2. 1 – gains from risk taking should be measured on a risk-adjusted basis 2 – managers should not be rewarded simply for taking risk, only for earning excess risk adjusted returns, ie only for increasing shareholder wealth

VaR: Uses and Abuses – Culp, Miller, & Neves

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1. Three main strengths of VaR as a risk measure that made it popular

2. List some of the uses of VaR

3. Two main risk management objectives:

4. They go over four cases where VaR may have helped, but because the companies were intentionally assuming risk VaR probably just would have put a number on the risk and not really scared them away. What are 3 other risk measures that could have been used?

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1. *Consistent – expressing risk in $ terms made it consistent across firms *Probability based – mgt can specify their tolerances *Common time horizon

2. Risk reporting, Risk control, Risk management, Capital allocation, and Exposure monitoring (monitor external fund managers)

3. *value risk managers: watch over total firm value *cash flow risk managers: watch over cash flow variability

4. *Cash flow risk – DFA – set debt levels or establish contingent funding *Risk-based capital – VaR just shows the downside. RBC may help to factor in potential gains. Very difficult though because cost of capital factors must be thrown into the mix. Shortfall risk – focus on the shortfall relative to a doomsday level. This is more objective and based on a real target.

Solvency Meas. for RBC - Butsic

273

Butsic – Solvency Measurements for Risk-based Capital Applications

1. The important part of this paper is the calculation of:

2. Three criteria for an effective risk-based capital model

3. In words, what is the EPD?

4. What is Butsic’s basic formula and what are the components?

1. Expected Policyholder Deficit

2. *same for all classes of insureds *objectively measured *discriminate between quantifiable measures of risk

3. The expected value of the difference between the insurer’s obligation to pay the claimant and the actual amount paid. In other words, it is the expected shortfall in payments that result from inadequate insurer capital.

4. , DL is the EPD. A is the asset amount, x is the loss (claim), p(x) is the discrete prob of x or the prob density of x.

274


275

1. Calculate the EPD and the EPD Ratio in the following situation:

Scenario Asset Loss Probability 1 12,000 6,900 0.10 2 15,000 10,000 0.80 3 10,000 13,100 0.10

2. What is wrong with just managing the probability of ruin?

3. How is EPD used in practice?

276

1. EPD = 310. EPD Ratio = 310/10,000 = .031 (see pg 483 or excel notes) Note that “Capital” is the expected asset – expected loss

2. It doesn’t consider the severity of the ruin – which directly impacts the policyholders. Some strategies may cause probability of ruin to decrease but the magnitude of ruin may increase.

3. You set some target EPD then use a solver to find the initial asset requirement – ala TAS:P/C


277

1. In the complicated formulas what are: C,c,A,L,cA,k, and kA

2. What is the relationship of EPD to ruin probabilities?

1. Capital, Capital/Expected(Liabilities) Expected(Assets), Expected(Liabilities), Capital/Expected(Assets), coefficient of variation for liabilities, coefficient of variation for assets. coeff of var = st dev / mean

2. Ruin Prob . Note that for each c/k there is a specific ruin prob, but each EPD may have many possible c, k combinations.

278


279

1. If you can assume that assets and liabilities follow normal distributions what are the formulas for dL and dA ?

2.

1. 2.

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281

1. What are the formulas if you assume Liab or Assets are lognormal?

2. Since the capital requirement by EPD is roughly proportional to the standard deviation of the risk element, Butsic provides a square root rule to calculate C, what is the formula:

1.

2. , Ci is the capital for the risk element i. If i and j are on opposite sides of balance sheet then the corr coeff is negative.

282

Allocation of Capital in the Ins Indus. - Cummins

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Cummins – Allocation of Capital in the Insurance Industry

1. Cummins give three reasons for allocating capital:

2. Cummins approaches risk management on behalf of the shareholder. What is the goal?

3. What is involved in calculating RAROC?

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1. 1-pricing, underwriting, and other decisions may be enhanced by thinking of capital as allocated – even when it isn’t 2-it can help tie together financial decisions and risk-based capital rules 3-RAROC and EVA (Econ value added) make use of capital allocation for performance measurement.

2. To maximize shareholder value of the firm. Economic (PV) profits must be weighed against capital requirements.

3. Somehow allocate capital to the business units (or some other segmentation of your company), preferably incorporating some risk measure. Then calculate the after tax pv of net income for each segment. Divide that by the allocated capital to get your RAROC. RAROC for each segment should be compared to some target – above is good, below is bad.


285

1. How does EVA work?

2. What is meant by Economic Cost of Capital?

3. Cummins has a bunch of conclusions: which is better EPD or VaR?

4. What are the options models? Which of them is better?

5. How should decision-making impact data?

6. How will capital allocation impact the company?

286

1. You don’t calculate a ratio. Income is reduced by product of allocated capital and target rate. Target rate is typically called cost of capital.

2. These are real costs. Agency costs, Double taxation, and Regulatory costs

3. EPD. Both should be calculated though – and at various thresholds

4. Myers-Read and Merton-Perold. The Myers-Read is better

5. Decision-making should dictate data needs – not vice versa

6. It will lead to better pricing, underwriting and strategy decisions – thus increasing shareholder value


287

1. What are the two methods of marginal capital allocation?

2. What is the difference between the two?

3. Why is M&R more appealing?

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1. Merton-Perold and Myers-Read

2. M&P method is like what Doug McKenzie is doing. Calc the total capital requirement. Then run the model without a line of business. The decrease in capital requirement is that line’s marginal capital. M&R make a small adjustment to increase size of a LOB (measured in E(loss)) Then you solve for the capital needed by each line so that they each have the same marginal impact on the firm’s overall insolvency. Key is that they look at the value of the Put option the shareholders have on the policyholders

3. The LOB capital requirements sum to equal the firm capital requirement.

Risk Adjusted Performance Measures - Goldfarb

289

Goldfarb – Risk-adjusted Performance Measures

1. What are four standard income measures used in RAROC calcs? (income is the numerator in a RAROC calc)

2. What are some of the drawback to using Economic profit as income?

3. What happens to the value of the put option in the M&R method as capital levels increase for the same level of exposure?

4. Why does the M&R method require greater computation? What is another big math problem with the M&R method?

290

1. * GAAP Net Income * Statutory net income * IASB (Fair value basis) net income * Economic profit

2. * unfamiliar to management * does not take into account future expected revenues (neither do the other measures of income though, so why is this here?) * if measures cannot be disclosed to investors, regulators and rating agencies then management may have a hard time explaining their decisions

3. Put option value decreases since you are less likely to exercise it.

4. You need to calculate the value of the default option. When you change the loss level of a line of business it changes the prob. distribution of the overall risk – does not exhibit homogeneity.


291

1. What is the co-measures approach? Benefits?

2. What is the cost of risk capital factor about?

3. List the 4 broad key risk sources in the article:

292

1. It is the shortfall method that we were doing at TMRe except it includes the market risk and everything. You set your threshold (99% TVaR or whatever) then run the 50k simulation. You look at the 1% worst scenarios and work out each risks contribution in those scenarios. (it really is exactly what we did at TMRe) + it is additive so allocation is easy

2. It is the return required for the capital that the line soaks up to support its business.

3. Market Risk, Credit Risk, Insurance Risk (uw, catastrophe…), Other (operational…)


293

1. List the 4 main methods used to allocate capital to risk sources

294

1. * Proportional by some risk measure – the risk measure is applied to each line in isolation. Then the relative ratios are used to allocate agg capital. * Incremental allocation – determine the impact that each risk has on the overall capital and allocate proportionally * Marginal allocation – this is the Myers&Read method, determines the impact to overall capital of a small change in each risk source. Then uses ratios * Co-measures – described above – (TMRe method)

2.

CAT Bonds and Other Risk-linked Securities - Cummins

New Paper 295

Cummins – Cat Bonds and other risk-linked securities

1. What were the failings of the CBOT Futures and the Nationwide Contingent Surplus notes?

2. What are the 4 key ways listed that a CAT Bond differs from traditional reins?

296

1. CBOT – insurers investing in these would end up harming relations w/ reinsurers, and faced high basis risk since the securities were based on industry losses and not some type of mimick portfolio Surplus Notes – Nationwide could trigger them for things other than property cat, investors didn’t like that. Also, investors didn’t like facing the credit risk associated with Nationwide.

2. 1 – insurer faces no credit risk since the contract is fully collateralized 2 – CAT Bonds typically issued for multiple periods, 3 years is common: locks in pricing and helps to amortize issuance costs 3 – typically cover high layers 1-5% probability 4 – since the investors shouldn’t already have cat exposure the insured expects to face a lower risk margin, this doesn’t really play out. The investors benefit though because they face pure cat risk and none of the risk of insurer operations.


New Paper 297

1. What is the purpose of the total return swap in a Cat Bond?

2. What are the types of Loss Triggers used?

3.

298

1. The SPR invests in Treasuries and other secure bonds which pay fixed interest. But SPR needs to pay investors LIBOR+Premium. So the swap is such that the counterparty pays LIBOR to SPR and SPR pays fixed interest to counterparty. This removes the market risk from SPR.

2. Indemnity – trigger levels tied to actual losses of insurance co, more difficult for investors since they need to learn about reins. Index – trigger based on industry index loss triggers, rely on PCS to put out number in case of event, or other modeled loss triggers, rely on modeling company to produce numbers in case of event. Parametric – trigger based on specific characteristics of the event itself: wind speed, central pressure, eq magnitude


New Paper 299

1. What is a sidecar? How is it actually two products?

2. What is a Cat-E-put?

3. Catastrophe risk swaps?

4. What are the benefits and structure of an ILW?

300

1. Much like a CAT Bond but it is a quota share deal. There is a debt component and an equity component. The debt works like the CAT Bond. The equity retains a stake in the SPR after the Bond expires and can benefit from earnings on residual assets.

2. Private transaction between insurers and reinsurers to provide financing after an event. The insurer got funds from the reinsurer in case of catastrophe – it was a put on its own equity in the form of preferred stock. The main point is that there is no cash up front really, just terms agreed to and the cash flow happens after an event.

3. Just like the swaps we did at TMRe

4. Transparency. The structure is like a reins contract for an insurer – usually there is an indemnity trigger as well as the industry trigger because reins requires indemnification.


New Paper 301

1. Why aren’t Cat Bonds cheaper for issuers than buying regular reinsurance? (two main reasons)

2. What could be changed to increase use of Cat Bonds?

302

1. 1 – expenses of issuance are very high 2 – investors are providing a service to issuers and are charging a premium for that which reflects issuer demand more than investor demands

2. * Improved reporting of insured losses could help create better indices * Regulatory capital requirements could recognize the credit risk in reins recoveries, since bonds are fully collateralized they would get preference * Personal lines rates deregulated and insurers locking in multi-period reins rates could be given more credit * something about ERISA rules for CAT Bonds

Insurance Securitization - Gorvette

New Paper 303

Gorvette – Insurance Securitization: A new asset class

1. Definition of Insurance Securitization

2. From the issuers perspective – what is the primary goal of insurance securitization?

3. There are 3 potential economic benefits of risk management:

4. What is so important about 3 with respect to Modigliani and Miller?

304

1. * transformation of underwriting cash flows in to tradable securities * transfer of underwriting risks to the capital markets

2. manage underwriting risk

3. * minimize taxes by reducing earnings volatility * minimize the costs of financial distress * avoid underinvestment problem associated with financial distress

4. M&M said that risk management doesn’t really change the value of the firm, it just changes who is holding what piece of risk. They did this by assuming away the 3 benefits in Gorvette’s paper. Risk management is really a tool for pre-lubricating the financial gears for when times are tough.

Section I - Valuation

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Equity Valuation Models - BKM

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BKM Ch 18 – Equity Valuation Models

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Equity Valuation Models - BKM

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note cards exam 8 v2

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