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 KulasekaranResearch 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

TZTLCCC

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

LT

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

LT

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

Bo

nd

s

Stocks

Bo

nd

s

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 EquityUS Fixed

IncomeReal 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 EquityUS Fixed

IncomeReal 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%

NormalEnd-of-Horizon

TurbulentEnd-of-Horizon

NormalWithin-Horizon

TurbulentWithin-Horizon

Probability of 10% Loss5 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%

NormalEnd-of-Horizon

TurbulentEnd-of-Horizon

NormalWithin-Horizon

TurbulentWithin-Horizon

Value at Risk1% 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

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Tomorrow, September 15th at 11am

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with Lucas Turton

Thursday, October 20th

www.windhamlabs.com/webinars/


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Economy & Finance

Rethinking Exposure to Loss