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 X 1 ≤ X 2,  (X 1 ) ≥  (X 2 ) 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:  (X 1 +X 2 ) ≤  (X 1 )+  (X 2 ) 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.

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.

Standard Deviation Portfolio B Portfolio A Probability 421/4 1051/4 1681/4 1261/ Mean SD

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.

VAR – Value at Risk For two hypothetical days S 1 = 112 and S 2 = 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.

Distribution of Daily Returns 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. 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. 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.

VaR Properties VaR satisfies all the desirable properties for risk measure excluding the subadditivity property -  (X 1 +X 2 ) ≤  (X 1 )+  (X 2 ) 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. PayoffProbabilityState $ No default -$100, default

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 PayoffProbabilityState $ *0.995*0.995= No default -$99,0003*0.005*0.995*0.995= default -$-199,5003*0.005*0.005*0.995= default -$300, *0.005*0.005= default

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 ModelMapping Distribution of risk factors Exposures VaR 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 x – the dollar exposure to risk factor f  f – the movement in risk factors

Risk Factors Instrument s 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: The vector of the exposures to risk factors The risk factors' covariance matrix

Covariance Matrix

Delta-Normal Method The portfolio’s VAR is directly obtained from the standard normal deviate  that corresponds to the confidence level, c: 95% %

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:

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:

Step 5: From this we can compute hypothetical changes in portfolio values: 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