Rethinking Exposure to Loss

1© 2016 Windham Capital Management, LLC. All rights reserved.
Confidential. Not for redistribution.
September 2016 1
Rethinking Exposure to Loss
Cel KulasekaranCel KulasekaranCel KulasekaranCel Kulasekaran
Research Director

2© 2016 Windham Capital Management, LLC. All rights reserved.
Agenda
1. Conventional methods of evaluating exposure to loss
2. Within-horizon exposure to loss
3. Regime shifts and applications

3© 2016 Windham Capital Management, LLC
Probability of Loss and Value at Risk
Exposure to Loss

Introduction
■Probability of Loss
►The likelihood of a given loss at a specified horizon.
►Uses expected distribution of returns.
■Value at Risk
►Maximum loss that could occur at a given confidence level
over a specified horizon.
►Rearrangement of another risk measure, probability of loss.

Introduction
■Probability of Loss
1. (percentage loss – expected return) / standard deviation = standardized variable.
2. standardized variable is converted to probability of loss.
■Value at Risk
1. expected return – (standardized variable × standard deviation) = percentage loss.
2. percentage loss × portfolio value = value at risk.

Lognormality
■Asset returns are not normally distributed.
►Compounding causes positive cumulative returns to drift
further above the mean than negative cumulative returns.
►Asymmetry! Returns tend to be lognormally distributed.
►Logarithmic returns (continuous returns) are normally
distributed.

Conventional Probability of Loss








 −+
Φ=
T
TL
c
c
E
σ
µ)1ln(
Pr








 −+
Φ=
T
TL
c
c
E
σ
µ)1ln(
Pr
( )•Φ : inverse normal distribution function
L : cumulative percentage loss in periodic units
Cµ : annualized expected return in continuous units
Cσ : annualized standard deviation in continuous units
T : number of years in the horizon

Conventional Value at Risk

Conventional Value at Risk
TZTL CCC σµ −=
WeVaR CL
×−−= )1(

Conventional Value at Risk: Example
■ What is the maximum loss that could occur at the end of one year with 95%
confidence for a portfolio that has an expected return of 5.44% and a standard
deviation of 16.87%?
■ Using the inverse of the normal distribution function, we find that the
standardized variable (z-score) of 1.645 corresponds to a 95% confidence level
(or 5% probability).
■ Next, we take the mean and subtract the standard deviation multiplied by the z-
score.
%31.22)64449.187.16(44.5 −=×−

Conventional Value at Risk: Example

Common Methods to Evaluate Exposure to Loss
■Daily Value at Risk
►Insufficient, because it ignores what might occur cumulatively
over future days.
■End-of-horizon return distribution
►Inadequate, because it ignores what might occur before the
end of the horizon.
■Maximum drawdown
►Unsatisfactory, because the maximum drawdown might occur
from a higher value than the portfolio’s inception value.

Within-Horizon
Exposure to Loss

Within-Horizon Exposure to Loss
■ Investors should measure exposure to loss throughout their investment horizon.
■ Not just at its conclusion!
■ How?
►First-passage time statistic.
►Measure the probability of a first occurrence of an event within a finite horizon.

Within-Horizon Probability of Loss

Within-Horizon Probability of Loss








+
−+
Φ+




 −+
Φ=
2
2
)1(
)1ln()1ln(
Pr C
C
L
T
TL
T
TL
W
σ
µ
σ
µ
σ
µ

Within-Horizon Exposure to Loss
■ Second-term is always positive.
■ So, end-of-horizon probability of loss is always less than the within-horizon
probability of loss.
■ The conventional probability of loss diminishes with time.
■ Within-horizon probability of loss increases at a decreasing rate. It never
decreases!

Within-Horizon Value at Risk
■ No closed form solution.
■ Solve for L at a given probability W .
■ The monetary value of loss is determined by converting the solved percentage
loss into a dollar value by multiplying with the initial wealth.
WPr
WealthLVaRW ×−=








+
−+
Φ+




 −+
Φ=
2
2
)1(
)1ln()1ln(
Pr C
C
L
T
TL
T
TL
W
σ
µ
σ
µ
σ
µ

Normal vs. Turbulent
Regime Shifts

Using Regime Shifts to Stress Test Portfolios
■ Standard deviations and correlations are not always stable through time.
■ So, we should consider separating historical returns into those associated with
normal market conditions and those associated with market turbulence.
■ This allows us to estimate risk measures under each regime.
■ Stress test portfolios by measuring exposure to loss based on risk characteristics
that prevail during turbulent periods.

Turbulence Regime
■ Method to partition historical returns into two regimes: those associated with quiet periods and those
associated with periods of market turbulence.
■ Financial turbulence is defined as a condition in which asset prices behave in an uncharacteristic fashion
given their historical patterns of behavior, including: extreme price moves, decoupling of correlated assets and
convergence of uncorrelated assets.
■ Advantages of TI over an implied volatility measure such as the VIX Index:
► TI can be estimated for any set of assets.
► TI captures unusual interactions between assets, as well as unusual variance of returns.

Two Assets Normal vs. Turbulent Example
Stocks
Bonds
Stocks
Bonds
Normal Periods Turbulent Periods

Risk (Normal vs. Turbulent Times)
Asset Classes Normal Turbulent
US Equity 16.33% 23.89%
Foreign Equity 18.85% 25.48%
US Fixed Income 4.14% 6.51%
Real Estate 21.68% 34.67%
Commodities 22.46% 35.28%
Cash 0.69% 0.81%
■ The table above shows standard deviation estimates for both normal and
turbulent regimes.
■ Volatility rises during times of turbulence.
■ Turbulent Threshold = 20%
■ Turbulent sub-sample = 50 months (occurs 17.67% of full-sample)

Correlations (Normal Times)
Normal US Equity Foreign Equity
US Fixed
Income
Real Estate Commodities
Foreign Equity 0.7136
US Fixed Income 0.1708 0.0988
Real Estate 0.5264 0.4530 0.1397
Commodities 0.1170 0.2217 -0.0481 0.1033
Cash 0.0833 -0.0209 0.1157 -0.0715 -0.0495
■ Asset class correlation coefficients for normal times
■ We define normal times with the full-sample asset class returns.

Correlations (Turbulent Times)
Turbulent US Equity Foreign Equity
US Fixed
Income
Real Estate Commodities
Foreign Equity 0.7126
US Fixed Income 0.2540 0.1754
Real Estate 0.6089 0.5050 0.2165
Commodities 0.1439 0.1772 0.0018 0.1319
Cash 0.3157 0.0870 0.3192 0.1712 0.1039
■ Asset class correlation coefficients during turbulent times

Stress Testing with Exposure to Loss
■ We can easily estimate the likelihood that a portfolio with a particular expected
return and standard deviation will experience a certain loss over a particular
horizon.
►Probability of Loss
■ Alternatively, we can also easily estimate the largest loss a portfolio might
experience given a certain level of confidence.
►Value at Risk
■ For normal periods, risk parameters are based on the entire sample of returns for
normal periods.
■ For the turbulent regime, risk parameters are based on the turbulent sub-sample
of returns.

Stress Testing with Exposure to Loss
■ Investors typically measure exposure to loss at the end of their investment
horizon.
►This ignores what may happen along the way; this is a dangerous oversight.
■ We introduce two additional risk measures to evaluate exposure to loss
throughout the investment horizon
►Within-horizon probability of loss and continuous value at risk
►Uses first-passage of time probability
■ This provides a more complete risk assessment.

Exposure to Loss (Probability of Loss)
■ End-of-horizon estimates drastically understate a portfolio’s vulnerability to losses
along the way.
■ The moderate investor has a little over 1% chance of losing 10% or more at the end
of five years.
■ But there is a 19% chance that the portfolio will depreciate by similar amounts some
point along the way.
■ This increases to 45% if we expect a turbulent period to prevail.
■ These are huge differences.
0%
10%
20%
30%
40%
50%
60%
Normal
End-of-Horizon
Turbulent
End-of-Horizon
Normal
Within-Horizon
Turbulent
Within-Horizon
Probability of 10% Loss
5 year Horizon
Conservative Moderate Aggressive

Exposure to Loss (Value at Risk)
■ Similarly, we can evaluate both conventional value at risk (end-of-horizon) and
continuous value at risk (within-horizon) for both a normal and turbulent regime.
■ Again, we can observe drastic differences in exposure to loss.
■ The worst outcome for a moderate investor given a 1% probability during normal
times is a 13% loss.
■ In comparison, the worst outcome in the interim period is a decline of at least 24%.
■ If a turbulent regime prevails, the worst outcome increases to a 41% loss.
0%
10%
20%
30%
40%
50%
60%
Normal
End-of-Horizon
Turbulent
End-of-Horizon
Normal
Within-Horizon
Turbulent
Within-Horizon
Value at Risk
1% Level, 5 year Horizon
Conservative Moderate Aggressive

Further Reading
■ Chow, G., E. Jacquier, M. Kritzman, and K. Lowry, “Optimal Portfolios in Good
Times and Bad,” Financial Analysts Journal, May / June 1999.
■ Kritzman, M., “Long Live Quantitative Models!,” CFA Magazine, July / Aug 2011.
■ Kritzman, M., The Portable Financial Analyst, Wiley Finance, 2003.
■ Kritzman, M., Puzzles of Finance, Wiley Finance, 2000.
■ Kritzman, M. and D. Rich, “The Mismeasurement of Risk,” Financial Analysts
Journal, May / June 2002.
■ Kritzman, M. K. Lowry, and A-S Vanroyen, “Risk, Regimes, and Overconfidence,”
The Journal of Derivatives, Spring 2001.
■ Setchall, S., et al., Optimizing Optimization: The Next Generation of Optimization
Applications and Theory, Chapter 4, Academic Press, Oct 2009.

Upcoming Windham Webinars
Windham Software Overview
Tomorrow, September 15th at 11am
Asset Allocation in a Low Interest Rate World
with Lucas Turton
Thursday, October 20th
www.windhamlabs.com/webinars/

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Rethinking Exposure to Loss

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