Risk Tolerance. Presented to the International Forum of Sovereign Wealth Funds

Transcription

1 Risk Tolerance Presented to the International Forum of Sovereign Wealth Funds Mark Kritzman Founding Partner, State Street Associates CEO, Windham Capital Management Faculty Member, MIT Source: A Practitioner s Guide to Asset Allocation. W. Kinlaw, M. Kritzman and D. Turkington. Wiley & Sons. Forthcoming May 2017.

2 Risk Tolerance Investors have two goals: to grow wealth and to avoid large losses along the way. These goals conflict with each other. The more a portfolio is structured to grow wealth, the more it is exposed to loss. Risk tolerance defines how investors balance these conflicting goals. 2

3 Agenda Risk tolerance in theory Risk tolerance in practice End-of-horizon exposure to loss Within-horizon exposure to loss Regimes Evidence Measuring financial turbulence Case study 3

4 Expected Return Risk Tolerance in Theory High Risk Tolerance Low Risk Tolerance Efficient Frontier Risk 4

5 Risk Tolerance in Practice In practice investors are unprepared to determine how many units of expected return to exchange for risk reduction. Instead, they map expected return and risk onto probability distributions. Based on these probability distributions, investors estimate the likelihood of generating a particular gain or experiencing a particular loss over different investment horizons. From this information, they choose a portfolio that best balances their goal to increase wealth and to avoid loss. A given asset mix implies a particular risk aversion coefficient whose inverse is risk tolerance. 5

6 Risk Tolerance in Practice E U = R S W S + R B W B λ(σ S 2 W S 2 + σ B 2 W B 2 + 2ρσ S W S σ B W B ) λ = ( R S W S R B W B )/( σ S 2 W S 2 + σ B 2 W B 2 + 2ρσ S W S σ B W B ) E U = expected utility R S = expected return of stocks W S = weighting in stocks R B = expected return of bonds W B = weighting in bonds λ = risk aversion coefficient σ S = standard deviation of stocks σ B = standard deviation of bonds ρ = correlation of stock and bond return 6

7 Risk Tolerance in Practice Expected return of stocks 10% Expected return of bonds 6% Standard deviation of stocks 20% Standard deviation of bonds 8% Correlation 30% Stock Bond Risk Risk Weight Weight Aversion Tolerance 90% 10% % 20% % 30% % 40% % 50% % 60% % 70% % 80% % 90%

8 End-of-Horizon Exposure to Loss Probability of Loss The end-of-horizon probability of loss is given by the equation: Pr end = N ln 1 + L μ ct σ c T Pr end equals the probability of loss at the end of the horizon N[ ] is the cumulative normal distribution function ln is the natural logarithm L equals the cumulative percentage loss in discrete units μ c equals the annualized expected return in continuous units T equals the number of years in the investment horizon σ c equals the annualized standard deviation of continuous returns 8

9 End-of-Horizon Exposure to Loss Value at Risk Value at risk and probability of loss are flip sides of the same coin. Value at risk is given by the equation: VaR = W (exp μ c T + σ c TN 1 p L 1) VaR equals value at risk μ c equals the annualized expected return in continuous units T equals the number of years in the investor s horizon level N 1 p L is the inverse cumulative normal distribution function evaluated at a given probability σ c equals the annualized standard deviation of continuous returns W equals initial wealth 9

10 Within-Horizon Exposure to Loss Within-Horizon Probability of Loss End-of-Horizon probability of loss and value at risk assume that we only observe our portfolio at the end of the investment. To account for losses that might occur prior to the conclusion of the investment horizon, we use a statistic called first passage time probability, which gives the probability that a portfolio will depreciate to a particular value over some horizon if it is monitored continuously. It is equal to: Pr within = N ln 1 + L μt σ T + N ln 1 + L + μt σ T (1 + L) 2μ σ 2 Pr within equals the probability of a within-horizon loss 10

11 Within-Horizon Exposure to Loss Within-Horizon Value at Risk Whereas value at risk measured conventionally gives the worst outcome at a chosen probability at the end of an investment horizon, within-horizon value at risk gives the worst outcome from inception to any time throughout an investment horizon. It is not possible to solve for within-horizon value at risk analytically. Instead, we set the within-horizon probability of loss equation equal to the chosen confidence level and solve iteratively for L. Within-horizon value at risk equals L multiplied by initial wealth. These two measures of within-horizon exposure to loss bring us closer to the real world because they recognize that investors care about drawdowns that might occur throughout the investment horizon. 11

12 Regimes Thus far we have assumed implicitly that returns come from a single distribution. It is more likely that there are distinct risk regimes such as a turbulent regime and a calm regime. We detect a turbulent regime by observing whether or not returns across a set of asset classes behave in an uncharacteristic fashion, given their historical pattern of behavior. One or more asset class returns, for example, may be unusually high or low, or two asset classes that are highly positively correlated may move in the opposite direction. 12

13 Regimes Empirical Evidence There is persuasive evidence showing that returns to risk are substantially lower when markets are turbulent than when they are calm. This is to be expected, because when markets are turbulent investors become fearful and retreat to safe asset classes, thus driving down the prices of risky asset classes. Conditional Annualized Returns to Risky Assets (January 1976 to December 2015) 10% Most Turbulent Months Other 90% U.S. Equities -5.5% 13.7% Foreign Developed Market Equities -10.0% 13.1% Emerging Market Equities -43.0% 20.4% Commodities -12.5% 8.2% 13

14 Regimes Measuring Turbulence This description of turbulence is captured by a statistic known as the Mahalanobis distance. It is used to determine the contrast in different sets of data. In the case of returns, it captures differences in magnitude and differences in interactions, which can be thought of respectively as volatility and correlation surprise. Turbulence t = 1 N x t μ Σ 1 (x t μ) x t equals a set of returns for a given period, μ equals the historical average of those returns, and Σ is the historical covariance matrix of those returns. The term x t μ captures extreme price moves. By multiplying this term by the inverse of the covariance matrix, we capture the interaction of the returns. 14

15 Foreign Equity Returns Regimes Visualizing Turbulence Scatter Plot of Hypothetical Returns: U.S. and Foreign Equities 25% 20% 15% 10% 5% 0% -5% -10% -15% -20% -25% -25% -20% -15% -10% -5% 0% 5% 10% 15% 20% 25% U.S. Equity Returns 15

16 Regimes Exposure to Loss We suggest that investors measure probability of loss and value at risk not based on the entire sample of returns, but rather on the returns that prevailed during the turbulent subsamples, when losses occur more commonly. This distinction is especially important if investors care about losses that might occur throughout their investment horizon, and not only at its conclusion. 16

17 Case Study Hypothetical Portfolio Treasury Bonds, 14.30% Emerging Market Equities, 9.10% U.S. Corporate Bonds, 22% Expected Return: 7.5% Standard Deviation: 10.8% Foreign Developed Equities, 23.20% Commodities, 5.90% U.S. Equities, 25.50% 17

18 Value at Risk (%) Value at Risk (%) Case Study Exposure to Loss Based on Pre-Crisis Data ( ) Probability of 35.9% or Greater Loss (Five-Year Period) Value at Risk (Five-Year Period, 1%) 5 End-of-Horizon 0 Within-Horizon Full Sample 1.4 Turbulent Sample Crisis Loss = 35.9% End-of-Horizon Within-Horizon Full Sample Turbulent Sample 18

19 The Bottom Line Investors seek to grow wealth and to avoid large losses along the way. These goals conflict with each other. Risk tolerance balances these conflicting goals. Investors under-estimate their portfolios exposure to loss, because they focus on the distribution of returns at the end of the investment horizon and disregard losses that might occur along the way. Moreover, investors base their estimates of exposure to loss on full-sample standard deviations which obscure episodes of higher risk that prevail during turbulent periods. 19