exam_9_formula_sheet.pdf

8/17/2019 EXAM_9_Formula_Sheet.pdf

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EXAM 9 Formula SheetBKM 6

1. Utility Function2. CAL3. Sharpe Ratio4. Equation for the optimal ratio,

1. 2. |

3.

4. Take the derivative of U w.r.t with the |


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1. Equation to determine the total variance (risk) of a portfolio of equally weighted assets2. Expected Value of two assets3. Variance of two assets4. Proportion perfect hedging with perfectly negative correlation assets5. Weight for minimum variance portfolio —Two risky assets6. Weights for optimal investment proportions —Two risky assets7. Weights for optimal investment proportions —Two risky assets and a risk free asset8. Formulas for the expected return any risk portfolio9. Formulas for the Variance of any risk portfolio10. Risk Premium, Standard Deviation, and Sharpe Ratio of Risk Pooling11. Risk Premium, Standard Deviation, and Sharpe Ratio of Risk Sharing

1. 2. 3. 4.

5.

6. 7.

8. 9. 10. Risk Pooling: Portfolio A has y in and (1-y) in (risk free), then adds z in (new asset) creating a new portfolio Z

a. Risk Premium: i.

b. Standard Dev: i.

c.

Sharpe: and d. We know no risk premium for a risk free asset, setting with z=y, , and we get

the following:i. Higher Returns

ii. Higher Riskiii. Higher Sharpe Ratios

11. Risk Sharing: Portfolio A has y in and (1-y) in (risk free), then sell z of and add z in (new asset) creating a newportfolio Z

a. Risk Premium: 0i.

b. Standard Dev: i.

c. Sharpe: and

d. We know no risk premium for a risk free asset, setting with z=y/2, , and we getthe following:

i. Same Returnsii. Lower Risk

iii. Higher Sharpe Ratios: same as the risk pooling


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EXAM 9 Formula Sheet

BKM 81. Single Factor Model2. Single Index Model3. Variance of Returns for a Single Factor/Index Model4. Covariance of Returns for a Single Factor/Index Model

5.

Correlation of Returns for a Single Factor/Index Model6. Expected Return-Beta relationship of a Single Index model7. Optimization Procedure for the Single Index Model

1. 2. 3. 4.

5.

6. 7. Optimization

a. Compute the initial position of each security in the active portfolio

b. Scale those initial positions

c. Compute the alpha of the active portfolio: d. Compute the residual variance of the active portfolio:

e. Compute the initial position in the active portfolio:

f. Compute the beta of the active portfolio:

g. Adjust the initial position in the active portfolio: h. Optimal risky portfolio has weight: i. Calculate the risk premium:

j. Compute the variance:


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1. Derivation of CAPM2. Zero Beta CAPM3. Extension of CAPM that accounts for labor income4. ICAPM Expected Return-Beta Equation

1. We have n stocks and market portfolio M, find :For any stock :

where The reward to risk ratio for investments in can be express as:

The reward to risk ratio for investments in can be express as:

(AKA the market price of risk)Given the basic equilibrium principle, all investment should off the same reward to risk. We get:

We set and get the CAPM:

2.

3.

Where

4. Where is the beta on the market index portfolio and is the beta on the k th hedge portfolio


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1. Two Factor model2. Total Risk Premium for portfolio with different factor sensitivities3. Well diversified Portfolio4. Well diversified portfolio with expected return and that deviations of its return from

expectation can serve as the systematic factor

1. 2. Where is the

risk premium3. A portfolio that is diversified over a large enough number of securities, with each weight, , small

enough that for practical purposed the nonsystematic variance, negligible. where

4. we actually expect then

BKM 11/121. Abnormal return from an event study2. Trin Statistic3. Confidence Index4. Put/Call Ratio

1. --Solve for

2.

3.

4. Ratio of outstanding put options to outstanding call options


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EXAM 9 Formula SheetBKM15/Hull 4

1. Equating the 2. Forward Rate3. Forward Rate for Continuous Distribution4. Instantaneous Forward Rate

5. Liquidity Premium6. Cash flow and Payoff with a FRA(X is lending to Y)7. Value of a FRA8. BootStrap Method9. Par Yield on the a bond

1.

2.

3.

4. = where is the zero rate for maturity 5. 6. Company X is agreeing to lend money to Company Y for the period of time between

Define:

Extra Interest that it earns by entering the agreement is Cash flows to Company X at time is

Cash flows to Company Y at time is Payoff for Company X at time is

Payoff for Company Y at time is

7. If If If FRA can be valued if we:

Calculate the payoff on the assumption that forward rates are realized(i.e. Discount this payoff at the risk free rate

The Risk Free rate is the Zero Rate for an investment at time 8. We need to solve for each , the Zero Coupon Rate, iteratively:

9. It is the coupon rate that causes the bond’s price to equal its par value


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EXAM 9 Formula SheetBKM16/Hull 7

1. Macualay Duration with Compounding2. Modified Duration with Compounding3. Modified Duration with Continuous4. Duration of a Perpetuity5. Effective Duration6. Convexity with Compounding7. Convexity with Continuous8. Bond Price change with Duration and Convexity9. Interest rate Swap —Transforming a Liability10. Interest rate Swap —Transforming a Asset11. Comparative Advantage12. Value as an Exchange of Bonds

1. 2.

3.

4.

5. where

is the price of the bond if interest rates rise is the price of the bond if interest rates fall

6.

7.

8.

9.

10.

11. 12.

FinancialInstitution

Y X

LIBOR LIBORLIBOR -N

I-FI+FI-M

FinancialInstitution

Y X

LIBOR LIBORLIBOR +N

I+FI-FI+M


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EXAM 9 Formula SheetFeldblum Asset

1. Duration of Reserves

1.

Noris1. Duration Market Value of Surplus/Duration Gap of Surplus2. Stock duration3. Duration of Total Return on Surplus4. Duration Gap of Leverage

1.

2. 3. 4.


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EXAM 9 Formula SheetPanning

1. Untaxed Net Income2. Formula for Premium3. Current Economic Value4. Franchise Value(perpetuity)5. Franchise Value(Fixed Periods)6. Target Rate of Return7. Franchise Duration8. Duration of Total Economic Value

1. 2.

3.

4. ----

5. 6.

7.

8.


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EXAM 9 Formula SheetHull 23

1. Probability of Default2. Unconditional Default Probability3. Survival Probability Function4. Conditional Default Probability5. Approximate Calc for Default Probabilities from Bond Prices6. Black Sholes Merton Formula7. Credit Value Adjustment8. Gaussian Copula9. Factor Based Correlation Structure(Probability of default at time T for a given F)10. Gaussian Copula Model for Credit VaR

1. 2. 3.

4. --Default intensity for period

5. Where Then use:

6. Where

The risk neutral probability that the company will default is .From Ito’s Lemma:

We can solve for simultaneously using the equation for and Ito’s Lemma 7. where

and=value today of an instrument that pays off the exposure on the derivative under consideration at time

is the value of the derivative, this means than the expression 8. Define:

=the time to default for company 1=the time to default for company 2

Transformation:

9. ---correlations are assumed to be all equal10. then


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EXAM 9 Formula SheetHull 24

1. CDS-Bond Basis2. CDS Spread3. Value of CDS to the seller( is negotiated) 4. Value of CDS to buyer( is negotiated)

1.

2. Assume defaults can always happen halfway through the year | | | Three steps:Calculate the PV of expected Payments (Buyer’s Payments):

1) We assume defaults happen halfway through, so we calculate the PV of accrual payments (Buyer’sPayments):

2)

Calculate the PV of expected payoff (Seller’s Payments) :3) Set the PV of expected and accrual payments equal to the expected payoff and solve for : (1)+(2)=(3)Value of the swap should be zero

3. 4.


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EXAM 9 Formula SheetButsic

1. EPD Discrete Assets Certain2. EPD Discrete Losses Certain3. EPD Continuous Certain Assets Uncertain Liabilities4. EPD Assuming Normal distribution5. EPD Assuming LogNormal distribution6. Square Root rule

1.

2.

3. 4. 5.

Where:

6.

Where:

7.


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EXAM 9 Formula SheetCummins Capital

1. EVA2. EVAOC3. Exceedence Probability4. Allocating Capital with Exceedence Probabilities5. Target RAROC for Multiple Periods

6. EPD Ratio as an Insolvency

1. 2. 3. 4.

5.

Goldfarb1. RAROC2. Myers Read Method3. Set price such that RAROC is above a specified target4. RAROC over multiple periods

1. 2. 3. =

4.


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EXAM 9 Formula SheetFeldblum IRR

1. Model Setup

1. 0 1 2 3

1PremiumPayment

2 Loss Payment3 Expense Payment

4=1-Sum(2:3) U/W Cash Flow

5=i*Ending Assets Investment Income6 Tax Payment

7=5-6 Total Other

8=7+4 Total Cash Flow

9= Total 2 - Pd toDate Reserves10=Some Basis Required Surplus

11=sum(9:10) Required Assets

12=8+14 Prior Beginning Assets13=11-12 Equity Flow14=12+13 Ending Assets


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EXAM 9 Formula SheetFerrari

1. Return on Equity Formula

2. Roth

1. Actual Surplus Change2. Required Surplus Change3. Actual Rate of Return4. Required Rate of Return

1. Actual Surplus change=Retained Return on Capital +Surplus paid in2. Required Surplus change=Expense & claims inflation + Increase in demand for insurance

+ increase in reserves3. Actual Rate of Return=Stockholder dividends + Retained Return on Capital4. Required Rate of Return=Required Surplus change + Stockholder Dividends - Surplus

Paid in


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EXAM 9 Formula SheetRobbin1. Policyholder Supplied Funds2. Calendar Year Investment Offset Procedure3. Present Value offset Procedure4. Calendar Year Return on Equity Method5. Present Value of Income over the Present Value of Equity Method

6. Present Value Return on Cash Flow Model7. Risk Adjusted Discounted Cash Flow Method8. The Internal Rate of Return on Equity Flows Algorithm

1.

2.

3. where and 4.

,

,

5. 6.

whereEQ=Equityr =Target Rate of Returni =investment rate used for discounting cash flow

7. 8.


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EXAM 9 Formula SheetMango

1. Modeled Event2. Expected Loss Occurrence Loss distribution3. Variance of Occurrence Loss distribution4. Covariance of Occurrence Loss distribution5. Total Variance of two Accounts6. Marginal Surplus Method7. Marginal Variance Method8. Generalized Covariance Share Formual

1. Employ a Binomial approximation with probability of occurrence

2. 3. 4. 5. 6. ;

;

is the needed surplus; y is the return on Marginal surplus; is a distribution percentage point corresponding

to the acceptable probability that the actual result will require even more surplus than allocated

Marginal Surplus Requirement is then:

Based on the required return, , on that marginal surplus(which is based on Management goals, market forcesand risk appetite) the MS risk load would be:

7. For an existing portfolio and new account , the MV risk load would be: Where is a multiplier similar to from the MS method although dimensioned to apply to variance rather

than standard deviation.Footnote: A variance based risk load multiplier can be developed by converting a standard deviation-basedmultiplier using the following formula:

In example , to simply the difficulty in selecting a variance based multiplier.

8. Use a weight to determine an account X’s share of the mutual covariance between itself and anotheraccount Y for event i:

Then Y’s share of that mutual covariance would simply imply:

Total Covariance share allocation for account X over all events would be:

Where


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EXAM 9 Formula Sheet

Bault1. Derive the Marginal risk approach from the ruin theory equation2. Leverage Approach

1.

2.

-Prob. Of RuinStandardizing produces:

(R+V)/SEquation for Marginal Surplus required for a new risk x:2.4:

2.5:

| | |

Consider the following Return on Equity equation:

where


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EXAM 9 Formula SheetKreps

1. Premium Equation2. Loss Safety Constraint3. Investment Variance Constraint4. Asset equation for Loss Safety Constraint-Swap Technique5. Risk Load for Loss Safety Constraint-Swap Technique

6. Asset equation for Variance Constraint-Swap Technique7. Risk Load for Variance Constraint-Swap Technique8. Asset equation for Loss Safety Constraint-Put Technique9. Risk Load for Loss Safety Constraint- Put Technique10. Asset equation for Variance Constraint- Put Technique11. Risk Load for Variance Constraint- Put Technique12. High Exess Layere/Finite rate event, non-zero rate on line

1.

2. : The final value of the initial investment (F) at year end must be large enough to

cover the losses at a specified safety level (s)3. , because the reinsurer does not want higher volatility from the contract compared to the

target return

4. 5. 6. 7.

8. 9. Where

10. Where

11.

12.

Option Case: Rate on line in the limit as goes to zero

Set

Swap Version:Let

exam_9_formula_sheet.pdf

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