Lecture 6: Value At Risk Part 2

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Lecture 6 Value At Risk Part 2
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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.

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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 ones 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!

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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 portfolios value across time
• From this stochastic process of the portfolios
  value we will estimate the Value At Risk for a
  given time horizon

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Our Method

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

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

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Portfolio Value Diffusion
Portfolio Value
Value At Risk At Time T
Expected Path For Portfolio
PV0
Portfolio Value Will Only Go Bellow this 2.5 of
the time
Time
T
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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

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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.

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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?
• 100e0.1 110.57

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• 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?
• Xe0.15110.57
• gtX 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 dont actually lose any money if we hold the
  IOU to maturity (excluding credit risk)

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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

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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
• PI.e-T.r
• The relationship better P and rT is non linear
• We will make the relationship between DP and Dr
  linear by using a first order Taylor
  approximation!

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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

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First Order Approximation
P
We estimate DP for a Dr using the straight line
approximation
Q
Dr
Complicated Function
DP
Linear Expansion About Point Q
r
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First Order Approximation Of Bond Price

• This makes the relationship between DP and Dr
  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.

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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 (DrT) has a mean of 0

• DrT Normally distributed with mean 0 and variance
  s2rT
• If we treat DrT as a stochastic variable then we
  can say that

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Value At Risk For A Bond

• Since DP 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 Dr

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Value At Risk For The Bond
DP
Because of the negative relationship between DP
and Dr we need to estimate the upper boundary of
Dr to estimate the VaR of DP
Dr
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VaR For The Bond

• Since the mean is zero the 2.5 upper boundary
  for the interest rate will simply be 1.96srT
• So the VaR on P, the value of the bond will be

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

• 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

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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 portfolios value in
  terms of the changes in the various interest rates

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VaR of a Bond Portfolio

• Let DPV be the change in the portfolios 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, DrT be the change in the change in
  the continuously compounded annualised interest
  rate for bonds of maturity T.

Or

• 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

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• 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

Var(DR1) Cov(DR1, DR2) Cov(DR1, DR3)
Cov(DR1, DR2) Var(DR2) Cov(DR2, DR3)
Cov(DR1, DR3) Cov(DR2, DR3) Var(DR3)
T1.w1
T2.w2
T3.w3
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 DR is always
  zero
• Note the scaled weights will not add to one

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

• Now

• So the 2.5 VaR

• The loss is normally expressed as a positive
  figure

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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

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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?

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Our Call Option Payoff Diagram
Profit
If Price of gold is greater than 105 then we
make more than the initial cost of 5
10 profit in 1 year!
100
105
Price of Gold In 1 Years Time
115
-5
If price of gold is less than 100 we throw the
option away
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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)

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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

• 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

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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

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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!

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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

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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

• 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

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

• Where Si is the price of the share the option is
  written on
• We know from our earlier discussion that

• So we can re-express this as

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• Finally expressing this in the familiar weight
  format

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

• 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

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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 dont 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!

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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).