Alternative Measures of Risk. The Optimal Risk Measure Desirable Properties for Risk Measure A risk measure maps the whole distribution of one dollar.

Alternative Measures of Risk

The Optimal Risk Measure

Desirable Properties for Risk Measure

A risk measure maps the whole distribution of one dollar returns, X, into one measure, (X).

Monotonic - if a portfolio has systematically lower returns

than another, it must have a greater risk.

if X1≤ X2, (X1) ≥ (X2)

Translation Invariance – adding a constant k to a portfolio should reduce its risk by k.

(X+k) = (X)-k

Desirable Properties for Risk Measure

Homogeneity – increasing the size of a portfolio by a factor b should scale its risk measure by the same factor:

(bX) = b(X)

Subaditivity – merging portfolios cannot increase risk:

(X1+X2) ≤ (X1)+ (X2)

The most common measures for risk are: Standard Deviation,

Semi-Standard Deviation and VaR – Value at Risk.

Standard Deviation

The standard deviation is defined as the square root of the expected squared deviation from the expected return:

The advantage of this measure is that it takes into account all

observations – any large negative value will increase its value.

The disadvantage is that any large positive value will also increase its value.

n

1i

2ii ))x(Ex(p)X(

Standard Deviation

The SD satisfies the Homogeneity and the Subadditivity properties:

However, the SD fails to satisfy the Monotonic and the Translation Invariance properties:

First, adding cash to a portfolio does not affect its value:

Second, a portfolio which has systematically lower returns

than another does not necessarily have a greater standard

deviation.

)X()kX(

)X(b)bX(

)X()X()XX( 2121

Standard Deviation

ProbabilityPortfolio A

Portfolio B

1/424

1/4510

1/4816

1/4612

Mean5.310.6

SD2.24.4

VaR – Value at Risk

Value at risk (VaR) is a summary measure of the downside risk, expressed in dollars.

Definition - VaR is the maximum loss over a target horizon

such that there is a low, predetermined probability that the

actual loss will be larger.

Example

Consider for instance a position of $4 billion short the yen, long the dollar, which takes a bet that the yen will fall in value against the dollar.

How much could this position loss over a day with a confidence level of 95%?

VAR – Value at Risk

We could use historical daily data on the yen/dollar rate and simulate a daily dollar return:

where Q is the current value of the position and S is the spot

rate in yen per dollar.

1t

1tt0t S

SSQ($)R

VAR – Value at Risk

For two hypothetical days S1= 112 and S2 =111.8. Therefore, the hypothetical return is:

Repeating this simulation over the whole sample (for instance, 10 years historical daily data) creates a time-series of returns.

Then, we construct a frequency distribution of daily returns.

M2.7$112

1128.111M000,4$($)R t

Distribution of Daily Returns

0

50

100

150

200

250

300

350

400

-160 -120 -80 -40 0 40 80 120 160

5% of observations

The maximum loss over one day is about $47M at 95% confidence level.

VaR Limitations

VaR does not describe the worst loss

VaR does not describe the distribution of the loss in the left

tail – it just indicates the probability of such value occurring.

0

50

100

150

200

250

300

350

400

-160 -120 -80 -40 0 40 80 120 160

5% of observations

VaR Limitations

VaR is measured with some error - Another sample period,

or a period different length, we lead to a different VaR number.

0

50

100

150

200

250

300

350

-160 -120 -80 -40 0 40 80 120 160

5%

8-Years Historical Daily data : VaR = -$74M

Alternative Measures of Risk

The risk manager can report a range of VaR numbers for

increasing confidence levels.

The Expected Tail Loss (ETL) – another concept is to

calculate the expected value of the loss conditional on the fact

that it greater than VaR.

where q is the number of observations that are lower than VaR.

q

1ii

i

q

1ii

p

Xp]VaRX|X[E

VaR Properties

VaR satisfies all the desirable properties for risk measure

excluding the subadditivity property - (X1+X2) ≤ (X1)+ (X2)

Numerical example

Consider an investment in a corporate bond with a face value of $100K, and default probability of 0.5%. Over the next period, we can either have no default, with payoff of $500, or default with loss of $100,000.

StateProbabilityPayoff

No default0.995$500

default0.005-$100,000

VaR Properties

Now, consider three identical positions and assume that the defaults are independent.

The VaR numbers add up to $1500

Thus, there is a higher than 1% probability of some default – with a confidence level of 98.5% the VaR value is -$99K

StateProbabilityPayoff

No default0.995*0.995*0.995=0.985075$1500

1 default3*0.005*0.995*0.995=0.01485-$99,000

2 default3*0.005*0.005*0.995=0.000075-$-199,500

3 default0.005*0.005*0.005=0.0000001-$300,000

VaR Parameters

To measure VaR, we need to defined two parameters: the confidence level and the horizon

Confidence Level

The higher the confidence level, CL, the greater the VaR measure.

As CL increases, the number of unlikely losses increases.

The choice of the CL depends on the use of VaR:

If the VaR is being used for benchmark measure – the consistency of the CL across trading desks or across time.

If the VaR is being used to decide how much capital to set aside to avoid bankruptcy – a high CL is advisable.

VaR Parameters

Horizon

The longer the horizon, the greater the VaR measure.

The choice of the horizon depends on the used of VaR:

If the VaR is being used for benchmark measure – the

horizon should be relatively short – the period for the

portfolio major rebalancing.

If the VaR is being used to decide how much capital to

set aside to avoid bankruptcy – a long horizon is advisable

– institutions would like to have enough time for corrective

actions.

Element of VaR System

Risk Factors Portfolio

Historical Data

Portfolio positions

Model Mapping

Distribution of risk factors

ExposuresVaR

method

VaR

Element of VaR System

Portfolio Positions – The assumption is that the positions are

constant over the horizon.

The Risk Factors represent a subset of all market variables

that related to the current portfolio’s positions.

There are many securities available, but a much more

restricted set of useful risk factors.

For each national market – one stock market, bond and

currency factor explain 50% of the variance of all assets.

There are three Method: Delta-Normal, Historical

Simulation and Monte Carlo Simulation

VaR Methods

VaR methods can be classified as either analytical - using

closed-form or local valuation solution, or simulation –

historical or Monte Carlo.

Mapping Approach – replacing the instruments by position on a limited number of risk factors

1ti

N

1ii1tp fxR

,,

x – the dollar exposure to risk factor f

f – the movement in risk factors

Risk Factors

Instruments

1 2 3 54

1 2 3

Risk Aggregation

Delta-Normal (Risk-Matrix) Methods

Delta-Normal method assumes that the portfolio exposure is

linear and that the risk factors are jointly normally distributed.

As the portfolio is a linear combination of normal variable, it

is itself normally distributed:

XXR p2 '

X The vector of the exposures to risk factors

The risk factors' covariance matrix

2.0

5.0

4.0

2,3

1,3

1,2

ρ

ρ

ρ

01.0σ

018.0σ

014.0σ

3

2

1

2

2

2

01.001.0018.02.001.0014.05.0

01.0018.02.0018.0018.0014.04.0

01.0014.05.0018.0014.04.0014.0

Covariance Matrix

Delta-Normal Method

The portfolio’s VAR is directly obtained from the standard

normal deviate that corresponds to the confidence level, c:

pRασVaR

95%

0-1.645-2.325

99%

Historical Simulation Method

The HS method assumes that the historical movements

represent the distribution of the future possible movement.

Step 1: The current portfolio value is a function of the current

risk factors.

Step 2: We sample the factor movements from the historical

distribution:

tN,t2,t1, f....f,fftP

iN,i2,i1,h

i f....f,fff

Historical Simulation Method

Step 3: From this we can construct hypothetical factor values,

starting from current time values

Step 4: We use this to construct a hypothetical value of the

current portfolio, under the new scenario:

iN,tN,i2,t2,i1,t1,h

i ff,....ff,fff

hiN,

hi2,

hi1, f,....f,ffh

iP

Step 5: From this we can compute hypothetical changes in

portfolio values:

thi

hi PPR

0

50

100

150

200

250

300

350

-160 -120 -80 -40 0 40 80 120 160

5%

Monte Carlo Simulation Method

The Monte Carlo simulation method is similar to the

historical simulation, except that the movements in risk

factors are generated by drawing random number from some

distribution.

It requires to make assumptions about the stochastic

process:

A Movement model

A Joint distribution of the risk factors