Frm Lecture3b

MODELLING RISK FACTORSDR NGUYEN THI THUY LINH

[email protected]

Agenda1. Measuring returns2. Time aggregation 3. Portfolio aggregation4. Moving average5. Exponentially weighted moving average (EWMA)6. Mapping a portfolio7. Variance Covariance approach8. Principal Component Analysis9. Comparisons of approaches

DR NGUYEN THI THUY LINH2

Measuring returnsThe relative rate of change in the spot price

Alternatively, we could construct the logarithm of the price ratio

This is also

Since ln(1+x) is close to x if x is small, Rt should be closed to rtprovided the return is small


Time aggregation To translate parameters over a given horizon to another horizon.

For example: we have data for daily return which we compute a daily volatility, we want to extend to a monthly volatility Returns can be easily aggregated when we use the log of the price ratio


Time aggregation (cont.)The expected return

The variance

Assuming returns are uncorrelated and have identical distributions across days, we have

More generally, define T as the number of steps. The multiple-period expected return and volatility are


Time aggregation (cont.)Assume now that the distribution is stable under addition (stays the same whether over one period or over multiple periods). The multiple-period VaR is

If returns are not independent, we may be able to characterized longer-term risks:

The variance of two-day returns

is the autocorrelation coefficient or the serial autocorrelation coefficient


Microsoft Example

We have a position worth $10 million in Microsoft shares The volatility of Microsoft is 2% per day (about 32% per year) We use N=10 and X=99

DR. NGUYEN THI THUY LINH7

Microsoft Example continued

The standard deviation of the change in the portfolio in 1 day is $200,000 The standard deviation of the change in 10 days is

200 000 10 456, $632,


Microsoft Example continued

We assume that the expected change in the value of the portfolio is zero (This is OK for short time periods) We assume that the change in the value of the portfolio is normally distributed Since N(2.33)=0.01, the VaR is

2 33 632 456 473 621. , $1, ,


Portfolio aggregation


Portfolio

Now consider a portfolio consisting of both Microsoft and AT&T Assume that the returns of AT&T and Microsoft are bivariate normal Suppose that the correlation between the returns is 0.3


AT&T Example

Consider a position of $5 million in AT&T The daily volatility of AT&T is 1% (approx. 16% per year) The S.D per 10 days is

The VaR is50 000 10 144, $158,

158 114 2 33 405, . $368,


S.D. of Portfolio

A standard result in statistics states that

In this case X = 200,000 and Y = 50,000 and = 0.3. The standard deviation of the change in the portfolio value in one day is therefore 220,227

YXYXYX 222


VaR for Portfolio

The 10-day 99% VaR for the portfolio is

The benefits of diversification are(1,473,621+368,405)1,622,657=$219,369

657,622,1$33.210220,227


Moving average



Value at Risk (VaR) Models

Risk

Maximum Potential Loss ... 1. ... with a predetermined confidence level2. ... within a given time horizon

VaR = Market Value x Sensitivity x Volatility

Three main approaches:1. Variance-covariance (parametric)2. Historical Simulations3. Monte Carlo Simulations


10 yrs Treasury BondMarket Value: 10 mlnHolding period: 1 month YTM volatility: 30 b.p. (0,30%)Worst case: 60 b.p.Modified Duration: 6

VaR = 10m x 6 x 0.6% = 360,000

The probability of losing more than 360,000 in

the next month, investing 10 mln in a 10 yrs

Treasury bond, is lower than 2.5%

VaR models: an example


VaR models: an exampleVaR = 10 mln x 6 x (2*0.3%) = 360,000 Euro

Market Value

(Mark to Market)

A proxy of the sensitivity of the bond price to

changes in its yield to

maturity (for a stock it

would be the beta)

An estimate of the future

variability of interest

rates (for a stock it would

be the volatility of the

equity market)

A scaling factor needed to obtain the

desired confidence level under the

assumption of a normal distribution

of market factors returns


Estimating Volatility of Market Factors Returns

Historical Volatility

Backward looking

Implied Volatility

Option prices: forward looking

Three main alternative criteria

Garch models (econometric)

Volatility changes over time autoregressive



1

)(1

2

n

RRn

ti

t

01/10/96 6,74% 01/10/97 6,87%01/11/96 -5,38% 01/11/97 -3,20%01/12/96 6,92% 01/12/97 4,05%01/01/97 0,89% 01/01/98 7,68%01/02/97 14,42% 01/02/98 11,27%01/03/97 -3,76% 01/03/98 4,84%01/04/97 -1,93% 01/04/98 20,14%01/05/97 5,34% 01/05/98 -7,65%01/06/97 -1,47% 01/06/98 1,86%01/07/97 10,66% 01/07/98 1,33%01/08/97 7,76% 01/08/98 3,07%01/09/97 -2,37% 01/09/98 -16,69%

Standard Deviation = 7,77%

Historical Volatility: monthly changes of the Morgan Stanley Italian

equity index (10/96-10/98)



Most VaR models use historical volatility It is available for every market factor Implied vol. is itself derived from historical Which historical sample? Long (i.e. 1 year) high information content, does not reflect

current market conditions Short (1 month) poor information content Solution: long but more weight to recent data (exponentially

weighted moving average)

DR. NGUYEN THI THUY LINH 22

Example of simple moving averages

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Figure 3 The Echo Effect Problem

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Daily returns (right hand scale)23-days moving standard deviation (left hand scale)

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Exponentially weighted moving average (EWMA)



01 2

23

34

1

1 2 3 1

xt xt xt xtn xt n

n

...

...

0 1

1 11

i xt ii

Estimating Volatility of Market Factors ReturnsExponentially weighted moving average (EWMA)

= return of day t = decay factor (higher , higher persistence, lower decay)

tx

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Figure 4 An Example of Volatility Estimation Based Upon an Exponential Moving Average

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Daily returns (right hand scale)23-days simple moving standard deviation (left hand scale)23-days exp. weighted moving standard deviation (left hand scale)

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Figure 5 An Example of Historical Volatility Estimation Based Upon Different Decay Factors

S&P 500 equally-weighted index daily returnsMoving standard deviations based on different decay factors

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Daily returns (right hand scale)23-days exp. weighted moving standard deviation (l =0,94)23-days exp. weighted moving standard deviation (l =0,90)23-days exp. weighted moving standard deviation (l =0,99)

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Diversification & correlations

VaR must be estimated for every single position and forthe portfolio as a whole

This requires to aggregate positions together to get arisk measure for the portfolio

This can be done by:mapping each position to its market factors;estimating correlations between market factors returns;measuring portfolio risk through standard portfolio theory.



An example

CurrencyPosition (

mln)

Worst case

(1.65*std.dev.)VaR (Euro)

USD -50 0.92% 460.000

Yen 50 1.76% 880.000

Sum of VaRs: 1,340,000

821,74054.0880)460(2880460

2

22

$,$22

$

mmmm

VaRVaRVaRVaRVaR YenYenYenTot


If correl. /$-/Yen = 0.54



Three main issues1) A 2 positions portfolio VaR may be lower than the more risky position VaR natural hedge2) Correlations tend to shoot up when market shocks/crises occur day-to-day RM is different from stress-testing/crises mgmt3) A relatively simple portfolio has approx.ly 250 market factors large matrices computationally complex an assumption of independence between different types of market factors is often made

222EquityIRFXTot VaRVaRVaRVaR


Mapping Estimating VaR requires that each individual position gets associated to its relevant market factors

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MMapping stock portfolio Equity positions can be mapped to their stock index through their beta coefficient In this case beta represents a sensitivity coefficient to the return of the market index

Individual stock VaR

Portfolio VaR

jiii VMVaR

j

N

iiij VMVaR

1


MMapping of a stock portfolioExample

817,7326,207,0481

%99,

j

N

iiiP VMVaR

Stock A Stock B Stock C PortfolioMarket Value (EUR m) 10 15 20 45Beta 1,4 1,2 0,8Position in the Market Portfolio (EUR m) 14 18 16Volatility 15% 12% 10%Correlation with A 1 0,5 0,8Correlation with B 0,5 1 0Correlation with C 0,8 0 1

Mapping of equity positions


MMapping of a stock portfolioExample with individual stocks volatilities and correlations

589,9222 ,,,222

%99, CBCBCACABABACBAP VaRVaRVaRVaRVaRVaRVaRVaRVaRVaR

Stock A Stock B Stock C PortfolioMarket Value (EUR m) 10 15 20 45Beta 1,4 1,2 0,8Position in the Market Portfolio (EUR m) 14 18 16Volatility 15% 12% 10%Correlation with A 1 0,5 0,8Correlation with B 0,5 1 0Correlation with C 0,8 0 1

Mapping of equity positions

Stock A Stock B Stock C MappingVolatilities & Correlations

VaR(99%) 3.490 4.187 4.653 7.817 9.589

VaR of an equity portfolio


Mapping of a stock portfolio

Mapping to betas: assumption of no specific risk the systematic risk is adequately captured by a CAPM type model it only works for well diversified portfolios

Stock A Stock B Stock C MappingVolatilities & Correlations

VaR(99%) 3.490 4.187 4.653 7.817 9.589

VaR of an equity portfolio


Figure 6 Main Characteristics of the Parametric Approach

2. Portfolio: 3. Risk measures:

stocks

rates

commodities

fx

1. Risk factors:Are defined either as price changes (assetnormal) or as changesin market variables(delta normal) theirdistribution is thensupposed to benormal.

ConfidentialReportfor theCompanysC.E.O.



Risk factors are mapped to individualpositions based on virtual componentsand linear coefficients(deltas). Portfolio riskis estimated based on the correlation matrix

VaR is quicklygenerated as a multiple () of the standard deviation.

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Variazioni di valore del portafoglio (euro, valore centrale)

%

d

i

c

a

s

i


Mapping FX forwardA long position in a USD forward 6 month contract is equivalent to: A long position in USD spot A short deposit (liability) in EUR with maturity 6 m A long deposit (asset) in USD with maturity 6 m

titiSF

f

dt

1

1

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MMapping FX forwardExample: Buy USD 1m 6 months forward

FX and interest rates

099.990$5,002,01

000.000.1

USDI

119.118.12,1099.990 EURD1. Debt in EUR

2. Buy USD spot

3. USD investment

2,1099.990 spotUSD

EUR/USD Spot 1,206 m EUR interest rate 3,50%6 m USD interest rate 2,00%EUR/USD 6 m forward 1,209


MMapping FX forward

849.18483,0326,2%5,1119.118.16 EURVaR miEUR

259.16549.13490,0326,2%2,1099.9906 EURVaR miUSD

919.82099.69326,2%3099.990 EURVaRUSDspot

Market factor Volatility EUR/USD EUR 6 m IR USD 6 m IREUR/USD Spot 3% 1 -0,2 0,4EUR 6 m IR 1,50% -0,2 1 0,6USD 6m IR 1,20% 0,4 0,6 1

Correlation withVolatilities and correlations - Forward position market factors


MMapping FX forward

USDspotmiUSDUSDspotmiUSDUSDspotmiEURUSDspotmiEUR

iUSDiEURmiUSDmiEURUSDspotmiUSDmiEURmUSD VaRVaRVaRVaR

VaRVaRVaRVaRVaRVaR

,66,66

,6622

62

6

622

2

646.834,0919.82)259.16(2)2,0(919.82849.182

6,0)259.16(849.182919.82259.16849.18 222

Total VaR of the USD 6 m forward position


Mapping of a FRA An FRA is an agreement locking in the interest rate on an investment (or on a debt) running for a pre-determined A FRA is a notional contract no exchange of principal at the expiry date; the value of the contract (based on the difference between the pre-determined rate and the current spot rates) is settled in cash at the start of the FRA period. A FRA can be seen as an investment/debt taking place in the future: e.g. a 3m 1 m Euro FRA effective in 3 months time can be seen as an agreement binding a party to pay in three months time a sum of 1 million Euros to the other party, which undertakes to return it, three months later, increased by interest at the forward rate agreed upon

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MMapping of a FRA

1-11-2000 1-2-20011-8-2000

investment

1m

1,013m1m

mf

1.013m

Example: 1st August 2000, FRA rate 5.136% Investment from 1st November to 1st February 2001 with

delivery: 1,000,000 *(1 + 0.05136 * 92/360) = 1,013,125 Euros. Equivalent to:

a three-month debt with final principal and interest of one million Euros;

A six-month investment of the principal obtained from the transaction as per 1.


Variance-covariance approach

Assumptions and limits of the variance-covariance approach Normal distribution assumption of market factor

returns Stability of variance-covariance approach Assumption of serial indepence of market factor returns linear sensitivity of positions (linear payoff)

Markowitz Result for Variance oof Return on Portfolio

sinstrument th and th of returns between ncorrelatio is

portfolio in instrument th on return of variance is

portfolio in instrument th of weightis

Return Portfolio of Variance

2

1 1

ji

iiw

ww

ij

i

i

n

i

n

jjijiij


VaR Result for Variance of PPortfolio Value (i = wiP)

day per value portfolio the in change the of SD the is return)daily of SD (i.e., instrument th of volatilitydaily the is

P

i

n

ijiji

jiijiiP

n

i

n

jjijiijP

n

iii

i

xP

1

222

1 1

2

1

2


Covariance Matrix (vari = covii)

nnnn

n

n

n

C

varcovcovcov

covvarcovcov

covcovvarcov

covcovcovvar

321

333231

223221

113121

covij = ij i j where i and j are the SDs of the daily returns of variables i and j, and ij is the correlation between them


Alternative Expressions for P2

transpose its is and is element th whosevector column the is where

T

T

i

P

j

n

i

n

jiijP

i

C

2

1 1

2 cov


Alternatives for Handling IInterest Rates Duration approach: Linear relation between P and y but assumes parallel shifts) Cash flow mapping: Cash flows are mapped to standard maturities and variables are zero-coupon bond prices with the standard maturities Principal components analysis: 2 or 3 independent shifts with their own volatilities


Handling Interest Rates: CCash Flow Mapping We choose as market variables bond prices with standard maturities (1mth, 3mth, 6mth, 1yr, 2yr, 5yr, 7yr, 10yr, 30yr) Suppose that the 5yr rate is 6% and the 7yr rate is 7% and we will receive a cash flow of $10,000 in 6.5 years. The volatilities per day of the 5yr and 7yr bonds are 0.50% and 0.58% respectively


Example continued

We interpolate between the 5yr rate of 6% and the 7yr rate of 7% to get a 6.5yr rate of 6.75% The PV of the $10,000 cash flow is

540,60675.1

000,105.6


Example continued

We interpolate between the 0.5% volatility for the 5yr bond price and the 0.58% volatility for the 7yr bond price to get 0.56% as the volatility for the 6.5yr bond We allocate of the PV to the 5yr bond and (1- ) of the PV to the 7yr bond


Example continued

Suppose that the correlation between movement in the 5yr and 7yr bond prices is 0.6 To match variances

This gives =0.074)1(58.05.06.02)1(58.05.056.0 22222


Example continuedThe value of 6,540 received in 6.5 years

in 5 years and by

in 7 years.This cash flow mapping preserves value and variance

484$074.0540,6

056,6$926.0540,6


Principal Components AAnalysis for Interest Rates


Principal Components AAnalysis for Interest Rates

The first factor is a roughly parallel shift (83.1% of variation explained) The second factor is a twist (10% of variation explained) The third factor is a bowing (2.8% of variation explained)


Using PCA to calculate VaR Example: Sensitivity of portfolio to rates ($m)

Sensitivity to first factor is from Table 18.3:10h0.32 + 4h0.35 8h0.36 7 h0.36 +2 h0.36 = 0.08Similarly sensitivity to second factor = 4.40

1 yr 2 yr 3 yr 4 yr 5 yr+10 +4 -8 -7 +2


Using PCA to calculate VaRcontinued

As an approximation

The f1 and f2 are independent The standard deviation of P (from Table 20.4) is

The 1 day 99% VaR is 26.66 h 2.33 = 62.12

21 40.408.0 ffP

66.2605.640.449.1708.0 2222


When Linear Model Can be UUsed Portfolio of stocks Portfolio of bonds Forward contract on foreign currency Interest-rate swap


The Linear Model and Options

Consider a portfolio of options dependent on a single stock price, S. Define

andSP

SSx


Linear Model and Options ccontinued

As an approximation

Similarly when there are many underlying market variables

where i is the delta of the portfolio with respect to the ith asset

xSSP

iiii xSP


Example

Consider an investment in options on Microsoft and AT&T. Suppose the stock prices are 120 and 30 respectively and the deltas of the portfolio with respect to the two stock prices are 1,000 and 20,000 respectively As an approximation

where x1 and x2 are the percentage changes in the two stock prices

21 000,2030000,1120 xxP


But the distribution of the daily rreturn on an option is not normal

The linear model fails to capture skewness in the probability distribution of the portfolio value.


Impact of gamma

Positive Gamma Negative Gamma


Translation of Asset Price Change tto Price Change for Long Call

Long Call

Asset Price


Translation of Asset Price Change tto Price Change for Short Call

Short Call

Asset Price


Quadratic Model

For a portfolio dependent on a single stock price it is approximately true that

this becomes

2)(2

1 SSP

22 )(2

1 xSxSP


Quadratic Model continued

With many market variables we get an expression of the form

where

This is not as easy to work with as the linear model

n

i

n

ijiijjiiii xxSSxSP

1 1 2

1

jiij

ii SS

PSP

2


Monte Carlo Simulation

To calculate VaR using M.C. simulation we Value portfolio today Sample once from the multivariate distributions of the xi Use the xi to determine market variables at end of one day Revalue the portfolio at the end of day


Monte Carlo Simulation

Calculate P Repeat many times to build up a probability distribution for P VaR is the appropriate fractile of the distribution times square root of N For example, with 1,000 trial the 1 percentile is the 10th worst case.


Speeding Up Monte Carlo

Use the quadratic approximation to calculate P


Comparison of Approaches

Model building approach assumes normal distributions for market variables. It tends to give poor results for low delta portfolios Historical simulation lets historical data determine distributions, but is computationally slower


Stress Testing

This involves testing how well a portfolio performs under extreme but plausible market moves Scenarios can be generated using

Historical data Analyses carried out by economics group Senior management


Back-Testing

Tests how well VaR estimates would have performed in the past We could ask the question: How often was the actual 10-day loss greater than the 99%/10 day VaR?


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