Lecture 6: Value At Risk Part 2

What we will learn in this lecture

• A recap on the end of last weeks lecture• How we can extend our ideas of mean and variance

of continuously compounded returns to model portfolios of more complicated financial instruments

• We will look at how to model bond portfolios using first order “duration” approximations

• We will also examine option portfolios using first order “delta” approximations

• We will see how these methods are limited and we need a better method: Monte Carlo Simulations.

Value At Risk• Value-At-Risk can be defined as “An estimate, with a given

degree of confidence, of how much one can lose from one’s portfolio over a given time horizon”.

• It is very useful because it tells us exactly what we are interested in: what we could loose on a bad day

• Our previous ideas of mean and variance of return on a portfolio were abstract

• VaR gives us a very concrete definition of risk, such as, we can say with 99% certainty we will not loose more than X on a given day

• “Value at Risk” is literally the value we stand to lose or the value at risk!

The Value Of Risk On A Portfolio

• We are normally interested in describing the value at risk on a portfolio of assets and liabilities

• We know how to describe mean and variance of return on our portfolio interms of the mean, variance and covariance of returns on the assets and liabilities it contains

• We will now use this to describe the stochastic process of the portfolio’s value across time

• From this stochastic process of the portfolio’s value we will estimate the Value At Risk for a given time horizon

Our Method• We can derive the continuously compounded mean and variance of a

portfolio’s continuously compounded return for a portfolio from the expected return and covariance matrix of continually compounded returns for the assets it contains

• Under the assumption that the proportional changes in the portfolio’s value are normally distributed we can translate the mean and variance of these proportional changes to the diffusion of the portfolios value across time

• Using the diffusion process we can put a probabilistic lower bound of the portfolios value across time:

• So for example if we wanted to calculate the value of the portfolio we would only be bellow 2.5% of the time we would use the formula:

)1.(.)( **96.1*0

**96.1*00

TTTT ePePPTVaR

• Where is the mean of returns on the portfolio and is the standard deviation of returns on the portfolio

Portfolio Value Diffusion

PV0

Expected Path For Portfolio

Portfolio Value WillOnly Go Bellow this2.5% of the time

Time

Por

tfol

io V

alue

T

Value At Risk At Time T

Value At Risk For Bond Portfolios

• So far we have been focusing on equity assets

• The principals used to model equities are the building blocks of modelling most financial assets

• More assumptions are need to model more complicated assets

What is a bond?

• A bond can be thought of as a loan in which you put some money in at the start and get certain fixed payments out in the future

• These payments are specified in advanced so where is the risk?

• There is the obvious default risk in that the debtor cannot pay you

• There is a more subtle risk that becomes visible when we talk about the mark-to-market value of a bond.

A thought experiment

• Imagine you have a very reliable friend who borrows £100 from you for one year

• Even though he is your friend you are pretty careful with your money so demand an interest payment at the same rate of interest you would get in your bank account which is currently 10% (annual continuously compounding rate).

• How much does he have to pay you back in 1 years time?

• 100*e0.1 = £110.57

• So your friend signs a piece of paper saying that in 1 year she will pay you £110.57 and you hand over the £100

• A few hours after you sign your contract the interest rate the bank pays rises to 15%!

• How much could you resell the IOU for?• X*e0.15=110.57• =>X = 110.57 / e0.15 =110.57 * e-0.15 = £95.12• Although you will still get your £100.57 back at the

end of the year the current value of the IOU discounted at the prevailing rate of interest has dropped!

• This is called interest rate risk• We don’t actually lose any money if we hold the

IOU to maturity (excluding credit risk)

The market value of a bond

• The market value of the bond is the present value of the discounted cash flows.

• A bond that just pays a lump sum at the end (like our IOU) is called a zero coupon bond (also called discount bonds)

• We will start by just thinking about zero coupon bonds

• Then we will see that any bond, or set of bonds can be thought of as a portfolio of zero coupon bonds

Present Value of Zero Coupon Bond

• Say we have a bond that promises to pay I in T years and the continuously compounded T year interest rate is rT% then we can say that the market value of this bond (P) is:

P=I.e-T.r

• The relationship better P and rT is non linear

• We will make the relationship between P and r linear by using a first order Taylor approximation!

What is a first order approximation

• Often in modelling we have a complicated function with a lot of high order terms that we want to simplify it

• The greatest simplification is to just take the slope of a function at a point and draw a straight line to approximate the complicated function

• This straight line approximation is called a first order Taylor expansion

First Order Approximation

Q

Complicated Function

Linear ExpansionAbout Point Q

P

r

r

P

We estimate P for a r using the straight line approximation

First Order Approximation Of Bond Price

• This makes the relationship between P and r very simple, perhaps too simple!

• But remember it is only an estimation, for large changes in r it wont give a very accurate change in P.

TrTeITdr

dP ....

TTTr

T rPTreITrdr

dPP T ....... .

The Stochastic Behaviour Of The Interest Rate Across Time

• It is a common assumption that there is no trend in interest rates and as such we can say that the change in interest rates (rT) has a mean of 0

),0( is 2rTT Nr

rT Normally distributed with mean 0 and variance

rT

• If we treat rT as a stochastic variable then we can say that:

TrPTP ~..~

Value At Risk For A Bond

• Since P is the change in the market value of the bond it directly measures the Value At Risk for the bond

• It is important to notice the negative relationship between the interest rate and the bond price

• If the interest rate goes up the bond price goes down• If we want to calculate the lower boundary for the the

bonds value then we must calculate the upper boundary for the value of r

Value At Risk For The Bond

Because of the negative relationship between P and r we need to estimate the upper boundary of r to estimate the VaR of P

r

P

VaR For The Bond• Since the mean is zero the 2.5% upper boundary for the

interest rate will simply be 1.96*rT

• So the VaR on P, the value of the bond will be:

rTPTPVaR *96.1**)~

(

• So we can say that we will only loose more than VaR(P) 2.5% of the time.

• As will all VaR calculations we can change the number of standard deviations from the mean to change the level of confidence

• In general we say that VaR implies a negative value so we drop the negative:

rTPTPVaR *96.1**)~

(

Portfolios of Zero Coupon Bonds• If we want to describe the risk of a portfolio of bonds we have to

describe the stochastic behaviour of the interest rates which effect it

• Bonds of different maturities can have different interest rates• These interests rates can move independently but are likely to be

highly correlated• We want to express the change in the portfolio of bonds value in

terms of the change in the various interest rates that effect the bond prices

• We will use our first order approximation to express the change in the portfolio’s value in terms of the changes in the various interest rates

VaR of a Bond Portfolio• Let PV be the change in the portfolio’s value, T be the

time in year to maturity for the bond, PT be the amount invested in the zero coupon bond of maturity T, rT be the change in the change in the continuously compounded annualised interest rate for bonds of maturity T.

N

iiii rPTVP

1

~..~

• In the later case we divide everything by PV (Portfolio Value) to get the proportional change in PV in terms of the investment weights (w) in the various bonds

N

iiii

N

ii

iiP rwTrPV

PT

PV

VPR

11

~..~..~

Or

• So if we want to express the variance of return in the bond portfolio in terms of the variances of the various interest rates it contains we simply time weight the portfolio weights!

• Notice that this will mean that the sum of weights will no longer equal 1

T1.w1

T2.w2

T3.w3

Var(R) Cov(R1, R2) Cov(R1, R3)

Cov(R1, R2) Var(R2) Cov(R2, R3)

Cov(R1, R3) Cov(R2, R3) Var(R3)

C =W =

• Note we drop the minus since it will cancel in the variance calculation

• Var(Rp) = WT.C.W

• E(RP) = 0 from our assumption that R is always zero

• Note the scaled weights will not add to one

• Once we have calculated Var(Rp) we can then simply calculate the lower 2.5% confidence boundary on the change in the portfolio’s value as

)(*96.1)( PP RVarRLower

• So the 2.5% VaR:

)(*96.1*)(*)( PP RVarPRLowerPPVaR

• Now

VPPVPV

VPPVRp

~.

~.

• The loss is normally expressed as a positive figure:

)(*96.1*)( PRVarPPVaR

Value At Risk On Option Portfolio

• An simple option is basically the right to buy or sell an asset at a fixed price at a future point in time

• There are 2 types of options a call option and put option

• A call option is the right to buy at a fixed price at a future date

• A put option is the right to sell at a fixed price at a future date

Thought Experiment

• Imagine you have a deal with your friend to have the option (not obligation) to buy a bar of gold in 1 years time for £100

• You will only want to exercise the option if the market value of the bar of gold is more than £100 (ie you can make profit)

• You pay £5 to have the option to buy this bar of gold

• What will the your profit look like 1 year from now?

Our Call Option Payoff Diagram

Price of GoldIn 1 Years Time

£100

£105-£5

Profit

If price of gold is less than £100 we throw the option away

If Price of gold is greater than £105 then we make more than the initial cost of £5

£115

£10 profit in 1 year!

How to we value this contract?!?

• Pricing of options is very complex

• A lot of what you read in textbooks (Black-Scholes etc) is not used in practice

• We will just say that the option price (OP) is a complex function of Time T, Stock Price S and Stock Price Volatility (V):

),,( VSTfOP

Delta Approximation

• We will take a first order approximation of the option pricing function with respect to the price of the asset it is written on:

DeltaS

VSTf

S

OP

),,(

• Delta is essentially how the option price moves with the price of the underlying asset at a given point in time

• Delta is always between –1 and 1

SDeltaOP .

A crucial insight

• The delta approximation lets us make an equivalence between holding an option on an asset and holding some fraction of the asset itself

• The delta tells us what fraction of the asset is equivalent to holding the option at a point in time

• We can model a portfolio of options in terms of a portfolio of the assets those options are written against

• Our task is to work out the amount of ‘exposure’ to the underlying assets our options imply

Estimating an Equity Option Investment Into An Equity Investment

• 1) Calculate the total value of equities the options are written upon

• 2) Multiply this value by the Delta of the option• 3) We can then calculate the VaR on the stock

holding and multiply it by the option delta to get the option holding VaR!

An Example

• We have 100 call options on Manu Shares• The current price of Manu Shares is £2.50• The delta of the option is 0.5• The portfolio of 100 call options is equivalent to a

portfolio of :2.50 * 100 * 0.5 = £125 worth of Manu Shares, or 50 shares

• We need to convert to a value because sometimes the option might be to buy more than 1 share

A More Formal Definition: A Portfolio of Options & Shares

• The value of the portfolio can be expressed as the sum of the value of Options and Shares it contains:

NS

ii

NO

ii SOPPV

11

• We can say that the change in the portfolio value is derived from the change in the value of the options and shares the portfolio contains:

NS

ii

NO

ii SOPPV

11

• Our next step is to just re-express this as:

j

jNS

jj

NO

i i

ii S

SS

S

OPS

PV

PVPV

...11

• Where Si is the price of the share the option is written on

• We know from our earlier discussion that:

SDeltaOP .

• So we can re-express this as:

j

jNS

jj

NO

i i

iii S

SS

S

SSDelta

PV

PVPV

....11

• Finally expressing this in the familiar ‘weight’ format:

j

jNS

j

jNO

i i

iii S

S

PV

S

S

S

PV

SDelta

PV

PV

...11

j

jNS

jj

NO

i i

iii S

Sw

S

SDeltaw

PV

PV

...11

• Now we might have an option on the same asset that we have a share holding in, so lets join these two sums:

N

i i

iiii S

SwsDeltawo

PV

PV

1

)..(

• Where woi is the investment weight of any options invested in the ith stock and wsi is the investment weight of any direct investment in the ith stock

An Example

• We have a portfolio of 2 assets a 15 call options on Abbey National each with a delta of 0.3, and 20 shares in Manchester United. The current price of Abbey National shares is £2, the current Abbey National call option price is £1 and the price of Manchester United shares is £1.25

• If we want to estimate this equity/option portfolio as a straight equity portfolio then the weights will be as follows:

• Total Portfolio Value = (15 * 1) + (20 * 1.25) = £40• Weight in Abbey = (0.3 * 15 * 2) / (40) = 0.225 = 22.5%• Weight in Manu = (20 * 1.25) / (40) = 0.625 = 62.5%• Notice the sum of our weight don’t add up to 100%!• Using the weights of 22.5% and 62.5% we can go onto find out the Value at Risk of our

option/equity portfolio by treating it simply as a equity portfolio!

Errors with First Order Approximations

• First Order Approximations are good for estimating VaR and statistical properties over short time periods

• However as the time horizon increases they become increasingly inaccurate

• There are no statistical tricks that allow us to get a high degree of accuracy

• We need to move onto Monte Carlo Simulations (next week).