b_08.pdf

Bachelor Studies in Finance

Year 2, Spring 2012

Overview

1. What is Value at Risk

2. Duration and convexity

BANKING

Lecture 8

Market risk management

Ewa Kania, Department of Banking

1. Value at Risk

•

DEaR or Daily Earnings at Risk is defined as the estimated

potential loss of a portfolio’s value over a one-day unwind

period as a result of adverse moves in market conditions,

such as changes in interest rates, foreign exchange rates,

and market volatility.

Measurement of market risk can help a bank risk manager:

•

Provide information on the risk positions taken by

individual traders

•

DEaR is comprised of

(a)

•

Establish limit positions on each trader based on the

market risk of their portfolios

the value of the position

(b)

the price sensitivity of the assets to changes

in the risk factor, and

•

Help allocate resources to departments with lower market

risks and appropriate returns

•

Evaluate performance based on risks undertaken by traders

in determining optimal bonuses

(c)

the adverse move in the yield.

•

The product of the price sensitivity of the asset and the

adverse move in the yield provides the price volatility

component.

•

Help develop more efficient internal models so as to avoid

using standardized regulatory models

•

VaR quantifies the size and probability of a portfolio loss.

The portfolio can be the entire bank’s assets and liabilities.

In this case, VaR measures a loss to the bank’s net worth or

capital.

•

Value at Risk or VaR is the cumulative DEaRs over N days

and is given by the formula

VaR = DEaR × √ N

•

VaR is used to set risk-based capital requirements for large

international banks under the Basel II Capital Accord.

•

VaR is a more realistic measure if it requires a longer period

to unwind a position, that is, if markets are less liquid.

•

A VaR measurement approach can also be applied to the

portfolio position of a single trader or single type of asset

(e.g., FX or fixed-income).

•

The relationship according to the above formula assumes that

the yield changes are independent. This means that losses

incurred on one day are not related to the losses incurred the

next day.

•

For a given probability, p , and a given future investment

horizon, h days, VaR is defined as the loss in value that has

a probability p of being exceeded over the next h days,

assuming that the portfolio position is not changed over the

investment horizon.

•

However, recent studies have indicated that this is not the

case, but that shocks are autocorrelated in many markets over

long periods of time.

Cumulative Distribution Function of Portfolio Return

Probability

•

Example . Suppose that the loss in portfolio value that has a

one per cent probability of being exceeded over the next 10

days is estimated to be €1 million. Then, €1 million is the

portfolio’s VaR for p = 1%, and h =10 days.

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

Probability that % return < p

•

VaR depends on the firm’s or portfolio’s distribution of

value, which, in turn, depends on the firm’s or portfolio’s

assets, liabilities, and derivative positions.

•

We can graphically illustrate VaR using the probability

distribution for a portfolio’s % return.

•

Suppose that over some given time horizon, h days, a

particular portfolio’s % return is estimated to have the

following cumulative probability distribution function.

-0.5

-0.4

-0.3

-0.2

-0.1

0.1

0.2

0.3

0.4

0.5

-0.23

p , % return

•

We see that there is a 5% probability of the portfolio returning

less than –23%. If the portfolio’s initial value is €1 m, then

VaR( p =5%, h days) = 0.23·€1m = €230,000.

•

One approach to computing VaR is to assume that the

portfolio’s returns are normally distributed. Let the port f olio’s

random rate of return over a period of h days be .

•

Th is implies that there is a 5% probability of a return le ss than

and a 1% probability of a return less than

( )

rNr σ

−

1.65

−

2.33

•

VaR is usually calculated over a measurement horizon of a

small number of days. For this short horizon, a portfolio’s

standard deviation is typically much greater than its expected

retu rn . Hence, the practice is to ignore the expected return and

set

Probability density function

r =

5% of area

under curve

•

If this is done, then we have

VaR( p =5%, h days) =1.65σ(portfolio value)

VaR( p =1%, h days) =2.33σ(portfolio value)

1% of area

under curve

•

Example : a portfolio’s one-day standard deviation is 10%, and

its initial value is €1m, then

VaR( p =1%, h =1 day) = 2.33·0.10·1m = €233,000

Rate of return, r %

−

2.33

−

1.65

•

Example . A bank holds a €20m portfolio of syndicated loans

that would likely take 5 days to arrange for a sale. The daily

standard deviation of the portfolio’s value is 0.3%. Therefore

•

What should be the time horizon ( h days) over which to

calculate VaR? If a bank can measure its risk and change it

once a day, a one-day VaR is most useful. This would be

relevant when a bank’s portfolio consists of liquid securities

that can be bought or sold quickly.

(

)

VaR

days

⋅

003

⋅

221

371

• Suppose a portfolio consists of n different assets. Its standard

deviation depends on the standard deviations and correlations

of the individual assets composing the portfolio.

•Let ω i be the proportion of the portfolio’s total value that is

invested in asset i , and let σ i asset i ’s standard deviation of

return. Further, let ρ ij be the correlation between the returns

on asset i and asset j . Then the portfolio return’s variance is

•

However, if a bank holds a portfolio of illiquid assets that

cannot be sold quickly, a longer horizon would be relevant.

The bank should choose the VaR’s h to be the number of days

over which it could change its portfolio.

•

If the return on a portfolio is estimated to have a one-day

standard deviation of σ, then, assuming the portfolio’s

composition stays the same over h days, its h -day standard

deviation can be estimated as

= ∑∑

ωωσσρ

2. Duration and convexity

•

Example . A portfolio has three assets held in proportions

ω 1 = 0.2, ω 2 = 0.5, and ω 3 = 0.3. The assets’ h -day standard

deviations are σ 1 = 0.3, σ 2 = 0.2, and σ 3 = 0.4. Their

correlations are ρ 12 = 0.1, ρ 13 = 0.6, ρ 23 = –0.1. The portfolio’s

h -day return standard deviation is then

•

Interest rate risk : The risk incurred by a bank when the

maturities of its assets and liabilities are mismatched.

•

Occurs when the value of a bank’s assets and liabilities have

different sensitivities to market interest rates.

σ=

0.03544

0.188

•

A fixed-income security’s sensitivity to market interest rates is

measured by its duration : Securities with longer durations

have more sensitivity to market interest rates.

•

If the three-asset portfolio was initially worth €50m, then,

VaR( p =5%, h days) =1.65 · 0.188 · €50m = €15.5m

•

Implementing VaR for a portfolio of many different assets

requires estimates of each asset return’s standard deviation and

the correlations between all of the assets’ returns.

•

Interest rate risk occurs because the prices and reinvestment

income characteristics of long-term assets react to changes in

market interest rates differently from the prices and interest

expense characteristics of short-term liabilities.

•

RiskMetrics™ provides daily estimates of standard deviations

and correlations for different types of assets in many different

countries. Of course, an individual bank could compute these

estimates on its own using historical data.

•

Interest rate risk is the effect on prices (value) and interim cash

flows (interest coupon payment) caused by changes in the level

of interest rates during the life of the financial asset.

•

YOUR Bank has €20 million in cash and a €180 million loan

portfolio. The assets are funded with demand deposits of €18

million, a €162 million CD and €20 million in equity.

•

OUR Bank has the following balance sheet (€million):

Assets

Liabilities and Equity

•

The loan portfolio has a maturity of 2 years, earns interest at

the annual rate of 7% and is amortized monthly.

Cash

€20

Demand deposits €100

15-yr loan

€160

5-yr CD

€210

•

The bank pays 7% annual interest on the CD, but the interest

will not be paid until the CD matures at the end of 2 years.

What is the maturity gap?

MA = ML = 1.80 years → MGAP = 0 years

30-yr mortgages

€300

20-yr bonds

€120

Equity

€50

•

What is the maturity gap for OUR Bank? Is OUR Bank exposed

to an increase or decrease in interest rates?

MA = [0·20 + 15·160 + 30·300]/480 = 23.75 years

ML = [0·100 + 5·210 + 20·120]/430 = 8.02 years

MGAP = 23.75 – 8.02 = 15.73 years

•

It is tempting to conclude that the bank is immunized

because the maturity gap is zero. However, the cash flow

stream for the loan and the cash flow stream for the CD are

different because the loan amortizes monthly and the CD

pays annual interest on the CD.

•

OUR Bank is exposed to an increase in interest rates. If rates rise,

the value of assets will decrease more than the value of liabilities.

•

Thus, any change in interest rates will affect the earning

power of the loan more than the interest cost of the CD.

What determines price sensitivity to changes in interest rates?

•

Immunization is a strategy that involves a fixed-income

portfolio. Its objective is to determine the smallest initial

portfolio value that will have a future value sufficient to pay a

given liability at a specific future date.

•

The longer the duration , the greater the price impact of any

given interest rate shock.

•

Duration is the weighted-average time to maturity on an

investment; duration is the investment’s interest elasticity:

measures the change in price for any given change in interest

rates.

•

If we use duration to immunize a portfolio, the change in net

worth for a given change in interest rates is given by the

following equation:

•

Duration D equals time to maturity M for pure discount

instruments only; duration of floating rate instrument = time to

first roll date; for all other instruments: D < M

∆

[

]

∆

−

⋅

where

•

Duration decreases as:

– Coupon payments increase

– Time to maturity decreases

– Yields increase

•

Immunizing the equity from changes in interest rates requires

that the leverage-adjusted duration gap be 0.

•

Thus, ( D A – D L k ) = 0 ⇒ D A = D L k

• This equation can be rewritten to provide a practical

application:

The three factors are important in determining ∆ E :

• D A – D L k or the leveraged adjusted duration gap. The larger

this gap, the more exposed is the bank to changes in

interest rates.

• A or the size of the bank. The larger is A , the larger is the

exposure to interest rate changes.

• ∆ R /(1+ R ) or interest rate shocks. The larger is the shock,

the larger is the exposure.

∆

⎡

⎤

∆

−

⎣

⎦

• In other words, if duration is known, then the change in the

price of a bond due to small changes in interest rates, R , can be

estimated using the above formula.

• Example . Calculate the duration of a 2-year, €1,000 bond that

pays an annual coupon of 10 per cent and trades at a yield of

14 per cent. What is the expected change in the price of the

bond if interest rates decline by 0.50 per cent (50 basis points)?

•

The economic interpretation : D is a measure of the

percentage change in price of a bond for a given percentage

change in yield to maturity (interest elasticity).

⎡

⎤

PVIF

CF·PVIF

CF·PVIF·t

∑ =

Convexity

(

)

⋅

(

)

⎣

⎦

⋅

(

)

(

)

100

0.8772

87.72

1100

0.7695

846.41

1692.83

∆

−

Price

Price =

934.13

1780.55

∆

−

⎡

∆

−

∆

⎤

1780.548

⎣

⎦

duration(

)

≅

P +

934.1336

∆

−

005

P –

∆

−

934

SF (scaling factor) = 10 8

new

price(

)

934

941

The actual price using conventional bond price discounting is

€941.99. The difference of €0.05 is due to convexity, which was

not considered in this solution. Note a linear relationship between

∆ P and – D .

Yield

R-0,01% R R+0,01%

∆

⎧

⎫

∆

−

⋅

∆

⋅

Example . Estimate the convexity for a 7-yr, 10 per cent

annual coupon bond which trades at YTM of 8 per cent and

has face value of €1,000.

Change of market value at ±1 basis point:

PV(7.99%;7; –100)+PV(7.99%;7;; –1000) = €1,104.68

PV(8%;7; –100)+PV(8%;7;; –1000) = €1,104.13

PV(8.01%;7; –100)+PV(8.01%;7;; –1000) = €1,103.57

7.99 per cent → 0.55643682 (capital gain)

8.01 per cent → –0.55606169 (capital loss)

What is the effect of a 1 per cent increase in interest rates?

•

Using present values, the percentage change is

(1,050.33 – 1,104.13)/1,104.13= –4.8724%

•

Using the duration/convexity formula:

–5.44·0.01/1.08 + 0.5·34·0.01 2 = –4.8679%

•

Adding convexity adds more precision. Duration alone

would have given the answer of –5.0379%

•

The duration measure is a linear approximation of a

non-linear function. If there are large changes in R , the

approximation is much less accurate.

55643682

−

55606169

•

All fixed-income securities are convex. Convexity is

desirable, but greater convexity causes larger errors in

the duration-based estimate of price changes.

00000034

104

CX = 10 8 ⋅0.00000034=34

•

Assets are more convex than liabilities.

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