fixed income zvi wiener 02-588-3049 fixed income 6

Fixed Income

Zvi Wiener

02-588-3049http://www.tfii.org

Fixed Income 6

http://www.tfii.org FI - 6 slide 2

IR derivatives

• Forward and Futures

• Options

• Caps, Floors

• Swaps

• Structured notes

• Hedging

• Mathematical models


Interest rate futures

Legal agreement

settlement, delivery date

quantity and quality of deliverable asset

futures price (it is NOT a price!)

long and short positions

margin requirements


Margin requirements

Initial margin

Maintenance margin - margin call trigger

Variation margin - after a margin call

Mark to market procedure reduces counterparty risk!


Marking to Market

Your balance

time

Initialmargin

Maint.margin

margin call


Futures Contract

T-Bills

T-Notes

T-Bonds

notional is typically $100,000

deliverable bond is unknown, CTD option

timing option (during the delivery month)

wild card option


Options

This type of contract is an obligation of

one side only, but it requires a payment

to purchase the right to choose.


Call Value before Expiration

E. Call

X Underlying


Put Value before Expiration

E. Put

X Underlying

premium

X


IR Options

Call

Put

European

American

Bermudian

Exotic: Asian, Digital, Knock-In, Knock-Out, path dependent and multiple asset options.


Various IR Options

Futures options

Caps

Floors

Exchange options

Swaptions


Cap

Is priced as a sequence of caplets

time

cap


Floor

time

floor


Collar

time

floor

cap


Option pricing

• Time value, intrinsic value

• underlying

• time to maturity

• interest rates

• strike

• coupons

• volatility


Swaps

• Currency swap

• Interest rate swap

• Amortizing swap

• Swaption


Currency swap


Currency Swap

3Y 3Y 3Y 3Y 3Y 3Y

100Y

$5 $5 $5 $5 $5 $5

$130

77

66

55

44

33

221 )1(

100

)1(

3

)1(

3

)1(

3

)1(

3

)1(

3

1

3

YYYYYYY rrrrrrr

77$

66$

55$

44$

33$

22$1$ )1(

130

)1(

5

)1(

5

)1(

5

)1(

5

)1(

5

1

5

rrrrrrr


IR swap


IR Swap

3 3 3 3 3 3

100

L+1 L+1 L+1 L+1 L+1 L+1

100

77

66

55

44

33

221 )1(

100

)1(

3

)1(

3

)1(

3

)1(

3

)1(

3

1

3

YYYYYYY rrrrrrr


IR Swap

L+1 L+1 L+1 L+1 L+1 L+1

100

1 1 1 1 1 1

+ a regular LIBOR loan for one year!

16

65

54

43

32

21 1

100

)1(

1

)1(

1

)1(

1

)1(

1

)1(

1

1

1

YYYYYYY rrrrrrr


Term Structure Models

• Binomial trees

• Short-term based analytical models

• LIBOR based analytical models

• Multi-factor models

• Simulations


Binomial Trees

6%

6.5%

5.5%

7%

6%

5%

7.5%

6.5%

5.5%

4.5%


6%

6.5%

5.5%

7%

6%

5%

7.5%

6.5%

5.5%

4.5%

83.97

88.2

89.8

93

94

95

100

100

100

100

Interest rates

Bond prices


Typical yield curves

time to maturity

yield increasing

decreasing

humped


Analytic Term Structure Models

dBbdtdrMerton )73(

dBdtrdrVasicek )()77(

rdBbrdtdrDothan )78(

dBrdtrdrCIR )()85(


Analytic Term Structure Models

Hull, White

Black-Karasinsky

Black-Derman-Toy

Heath-Jarrow-Morton

Affine TS modles

Gaussian models


Arithmetic BM dX = dt + dW

time

X


Geometric BM dX = Xdt + XdW

time

X


Mean Reverting Process

dX = (-X)dt + XdW

time

X


Ho and Lee Model

Rates are normally distributed.

All rates have the same variability.

The model has an analytic solution.

dzdtttdr )()(


Bond Prices under Ho and Lee

Where

)(),(),( tTreTtATtP

22 )(2

1

),0(log)(

),0(

),0(log),(log

tTt

t

tPtT

tP

TPTtA


Option Prices under Ho and Lee

A discount bond matures at s, a call option matures at T

)(),()(),( PhNTtXPhNstPCall

)(),()(),( hNstPhNTtXPPut P

2),(

),(log

1

)(

P

P

P

XTtP

stPh

tTTs


Monte Carlo

-1 -0.5 0.5 1

-1

-0.5

0.5

1


Monte Carlo Simulation

10 20 30 40

-15

-10

-5

5

10

15


Callable Bond

PayoffStraight Debt

Debt

Callable Bond

Value of the firm’s call option


Convertible Bond

PayoffStock

Stock

Straight Bond

Convertible Bond


Protective Put

Payoff

X Underlying

X

Put

Protective Put

Stock


Covered Call

Payoff

X

X

Written Call

Covered Call

Stock


Straddle

Payoff

X

X

Straddle Call

Put

Fixed Income

Zvi Wiener

02-588-3049http://pluto.mscc.huji.ac.il/~mswiener/zvi.html

DAC


Life Insurance

• yearly contribution 10,000 NIS

• yearly risk premium 2,000 NIS

• first year agent’s commission 3,000 NIS

• promised accumulation rate 8,000 NIS/yr

• After the first payment there is a problem of insufficient funds. 8,000 NIS are promised (with all profits) and only 5,000 NIS arrived.


10,000 NIS

Risk2,000 NIS

Client’s8,000 NIS

Agent3,000 NIS

• insufficient funds if the client leaves

• insufficient profits


Risk measurement

• The reason to enter this transaction is because of the expected future profits.

• Assume that the program is for 15 years and the probability of leaving such a program is .

• Fees are – 0.6% of the portfolio value each year

– 15% real profit participation


Obligations

• The most important question is what are the

obligations?

• The Ministry of Finance should decide

• Transparent to a client

• Accounted as a loan


One year example

Assume that the program is for one year only

and there is no possibility to stop payments

before the end.

Initial payment P0, fees lost L0, fixed fee a%

of the final value P1, participation fee b% of

real profits (we ignore real).

Investment policy TA-25 (MAOF).


Liabilities (no actual loan)

)1,,()1(

)1( 010

0

0

10 XXCall

X

aPb

X

XaP

Assets (no actual loan)

0

100 X

XLP


Total=Assets-Liabilities

)1,,()1(

010

0

0

100 XXCall

X

aPb

X

XLaP

Fair value

)1,,()1(

00

0

000 XXCall

X

aPb

X

XLaP t

t


Liabilities (actual loan)

RteLXXCallX

aPb

X

XaP 001

0

0

0

10 )1,,(

)1()1(

Assets (actual loan)

0

10 X

XP


Total=Assets-Liabilities (loan)

Rtt

t eLXXCallX

aPb

X

XaP 00

0

0

00 )1,,(

)1(


2 years liabilities (no actual loan)

)2,,()1(

)1( 020

20

0

220 XXCall

X

aPb

X

XaP

2 years assets (no actual loan)

0

200 X

XLP

In reality the situation is even better for theinsurer, since profit participation fees oncetaken are never returned (path dependence).


2 years fair value, no loan

)2,,()1(

)1(1

020

20

0

20

20

XXCallX

aPb

X

XLaP


2 years liabilities (with a loan)

ReLXXCallX

aPb

X

XaP 2

0020

20

0

220 )2,,(

)1()1(

2 years assets (with a loan)

0

20 X

XP


0.5 1 1.5 2 2.5 3 3.5

-0.1

0.1

0.2

0.3

Stock index

Profit

No loan With a loan10 years, L0=7%


Partial loan - portion q

nRn

n

nn

qLenXXCallX

aPb

X

XLqaP

),,()1(

)1()1(1

00

0

000

Theoretically q can be negative.


Mixed portfolio

When the investment portfolio is a mix one should analyze it in a similar manner. Important: an option on a portfolio is less valuable than a portfolio of options.

Another risk factor - leaving rate should be accounted for by taking actuarial tables as leaving rate.


Conclusions

It is a reasonable risk management policy not to take a loan against DAC.

Up to some optimal point it creates a useful hedge to other assets (call options and shares) of the firm.

Intuitively DAC is good when the stock market performs badly and profit participation is valueless. DAC performs bad when the market performs well.

Fixed Income

Zvi Wiener

02-588-3049http://pluto.mscc.huji.ac.il/~mswiener/zvi.html

Risk Management


Qualitative Requirements

• An independent risk management unit• Board of directors involvement• Internal model as an integral part• Internal controller and risk model• Backtesting• Stress test


Quantitative Requirements

• 99% confidence interval• 10 business days horizon• At least one year of historic data• Data base revised at least every quarter• All types of risk exposure• Derivatives


Types of Assets and Risks

• Real projects - cashflow versus financing

• Fixed Income

• Optionality

• Credit exposure

• Legal, operational, authorities


Risk Factors

There are many bonds, stocks and currencies.

The idea is to choose a small set of relevant economic

factors and to map everything on these factors.

• Exchange rates

• Interest rates (for each maturity and indexation)

• Spreads

• Stock indices


How to measure VaR

• Historical Simulations

• Variance-Covariance

• Monte Carlo

• Analytical Methods


Historical Simulations

• Fix current portfolio.

• Pretend that market changes are

similar to those observed in the past.

• Calculate P&L (profit-loss).

• Find the lowest quantile.


Returns

year

1% of worst cases


-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

Profit/Loss

VaR

1% VaR1%


Variance Covariance

• Means and covariances of market factors

• Mean and standard deviation of the portfolio

• Delta or Delta-Gamma approximation

• VaR1%= P – 2.33 P

• Based on the normality assumption!


Variance-Covariance VVVaR 33.2%1

2.33

-2.33

1%


Weights

Since old observations can be less relevant, there is a technique that assigns decreasing weights to older observations. Typically the decrease is exponential.

See RiskMetrics Technical Document for details.


Monte Carlo

• Distribution of market factors

• Simulation of a large number of events

• P&L for each scenario

• Order the results

• VaR = lowest quantile


Example

Your portfolio consists of two positions.

The first one is a zero coupon bond maturing in 1 year with current market value of $10M.

The second one is a zero coupon bond maturing in 10 years with market value of $1M.

Which position contributes more to the risk of the portfolio?


Real Projects

Most daily returns are invisible.

Proper financing should be based on risk

exposure of each specific project.

Note that accounting standards not always reflect

financial risk properly.


Example

• You are going to invest in Japan.

• Take a loan in Yen.

• Financial statements will reflect your

investment according to the exchange rate

at the day of investment and your liability

will be linked to yen.

• Actually there is no currency risk.


Airline company

• fuel - oil prices and $

• purchasing airplanes - $ and Euro

• salaries - NIS, some $

• tickets $

• marketing - different currencies

• payments to airports for services


Airline company

• loans

• equity

• callable bonds


Airline company

Base currency - by major stockholder.

Time horizon - by time of possible price change.

Earnings at risk, not value at risk, since there is too much optionality in setting prices.

One can create a one year cashflow forecast and measure its sensitivity to different market events.


Reporting

Division of VaR by business units, areas of

activity, counterparty, currency.

Performance measurement - RAROC (Risk

Adjusted Return On Capital).


How VaR is used

• Internal Risk Management

• Reporting

• Regulators


Backtesting

Verification of Risk Management models.

Comparison if the model’s forecast VaR with

the actual outcome - P&L.

Exception occurs when actual loss exceeds

VaR.After exception - explanation and action.


Backtesting

Green zone - up to 4 exceptions

Yellow zone - 5-9 exceptions

Red zone - 10 exceptions or more

OK

increasing k

intervention


Stress

Designed to estimate potential losses in abnormal markets.

Extreme events

Fat tails

Central questions:

How much we can lose in a certain scenario?

What event could cause a big loss?


Unifying Approach

• One number

• Based on Statistics

• Portfolio Theory

• Verification

• Widely Accepted

• Easy Comparison


Board of Directors(Basle, September 1998)

• periodic discussions with management concerning the effectiveness of the internal control system• a timely review of evaluations of internal controls made by management, internal and external auditors• periodic efforts to ensure that management has promptly followed up on recommendations and concerns expressed by auditors and supervisory authorities on internal control weaknesses• a periodic review of the appropriateness of the bank’s strategy and risk limits.


pluto.mscc.huji.ac.il/~mswiener/

• Useful Internet sites

• Regulators

• Insurance Companies

• Risk Management in SEC reports

Risk Management resources


Risk Measuring Software• CATS, CARMA• Algorithmics, Risk Watch• Infinity• J.P. Morgan, FourFifteen• FEA, Outlook• Reuters, Sailfish• Kamacura• Bankers Trust, RAROC• INSSINC, Orchestra

fixed income zvi wiener 02-588-3049 fixed income 6

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