Motivation Why the emphasis on volatility (SD) in Finance? Javier - PDF document

Risk Revisited (I): Downside Risk Javier Estrada ADFIN – Winter/2014 1. Motivation • A brief history of risk • Problems with the standard deviation 2. Measures of downside risk • The semideviation • Morningstar risk • Value at risk (VaR) • Downside beta Motivation  Why the emphasis on volatility (SD) in Finance? Javier Estrada  An arbitrary choice by Markowitz in the early ‘50s IESE Business • Partly motivated by limitations in computing power School  The rest is history Barcelona Spain • Sharpe and others focused on an asset within a diversified portfolio in the early ‘60s  This led to beta • Many other variables to assess risk have been proposed since then  Including variables designed to capture downside risk (Like the semideviation, highlighted by Markowitz)  Why so many variables?  Investors assess risk in many different ways  Different goals require different variables ADFIN Winter/2014 1

The Standard Deviation  If two assets X and Y have the same mean return, Javier Estrada investors would prefer the one with the lower SD IESE Business School Barcelona Spain Y X AM R  Hence investors would prefer asset X ADFIN Winter/2014 The Standard Deviation – Problems  Problems with the SD as a measure of risk Javier Estrada 1. Incomplete and misleading when the underlying IESE Business distribution is not normal (or not symmetric) School • This affects one of the main uses of the SD Barcelona Spain  Construction of confidence intervals 2. Deviations are measured with respect to the AM • Nothing wrong with that  But it makes more sense for symmetric distributions  And the AM is not a particularly useful benchmark 3. Deviations above and below the AM are treated in the same way • Is this the way you think about risk?  Investors usually associate risk with ‘bad’ outcomes ADFIN Winter/2014 2

The Standard Deviation – Problems  Problems with the SD as a measure of risk Javier Estrada 1. Incomplete and misleading when the underlying IESE Business distribution is not normal (or not symmetric) School  More often than not in practice, the distributions Barcelona Spain we deal with are not normal  Many distributions are skewed • All else equal, investors prefer positive skewness  Many distributions are leptokurtic (have fat tails) • All else equal, investors prefer thin tails  Only the normal distribution can be fully described by its mean and variance • For all other distributions, whatever the task, we need more information  Example: Forecasting the probability of returns ADFIN Winter/2014 The Standard Deviation – Problems  Problems with the SD as a measure of risk Javier Estrada 1. Incomplete and misleading when the underlying IESE Business distribution is not normal (or not symmetric) School  Two (of the many possible) examples Barcelona Spain  The following probabilities are valid exclusively under normality • P(AM–SD , AM+SD) ≈ 68% • P(AM–2∙SD , AM+2∙SD) ≈ 95% • P(AM–3∙SD , AM+3∙SD) > 99% Go  The following VaR calculation is valid exclusively under normality • VaR = AM + ( z c )⋅SD ADFIN Winter/2014 3

The Standard Deviation – Problems  Problems with the SD as a measure of risk Javier Estrada 2. Deviations are measured with respect to the AM IESE Business  Investors typically use other benchmarks School  These include … Barcelona Spain • A target return • The return of another asset • The risk‐free rate • Expected inflation • 0 …  And needless to say … • investors do not treat deviations above or below any chosen benchmark in the same way ADFIN Winter/2014 The Standard Deviation – Problems  Problems with the SD as a measure of risk Javier Estrada 3. Deviations above and below the AM are treated in IESE Business the same way School  Clearly, investors do not feel the same way about Barcelona Spain deviations above or below any benchmark  Deviations above the benchmark are welcomed • This is good volatility  Deviations below the benchmark are shunned • This is bad volatility  But remember, there is no such thing as good and bad volatility in the Markowitz framework • Volatility is bad, period Go ADFIN Winter/2014 4

The Standard Deviation – Problems  Problems with the SD as a measure of risk Javier Estrada 3. Deviations above and below the AM are treated in IESE Business the same way School Barcelona Spain (Oracle) ADFIN Winter/2014 The Semideviation  Given any chosen benchmark B , the semideviation Javier Estrada with respect to B ( Σ B ) is given by IESE Business School Barcelona Spain  For the sake of comparison, recall that the SD is given by ADFIN Winter/2014 5

The Semideviation Javier Estrada IESE Business School  Some attractive properties of the semideviation Barcelona Spain  It accommodates any chosen benchmark • It does not restrict the benchmark to the AM  It gives weight only to deviations below B • Risk is defined as volatility below the benchmark  Given the asset, it may differ across investors • Plausibly, different investors can have different B s  It is equally plausible and useful for symmetric and skewed distributions  It is almost as easy to calculate as the SD ADFIN Winter/2014 The Semideviation Javier Estrada (Oracle) IESE Business School Barcelona Spain ADFIN Winter/2014 6

Morningstar Risk  A proprietary measure of risk Javier Estrada  This much is known IESE Business • Takes into account return variability School • Weights more heavily downside variability Barcelona Spain • Relative measure within each Morningstar category  Low (Bottom 10%)  Below average (Next 22.5%)  Average (Middle 35%)  Above average (Next 22.5%)  High (Top 10%) Go ADFIN Winter/2014 Value at Risk (VaR)  VaR is the ‘worst’ expected outcome, over a given Javier Estrada time horizon ( T ), for a given confidence level ( c ) IESE Business  Over the chosen time horizon, a loss larger than School VaR will occur with a (1– c )% probability Barcelona Spain  VaR was introduced by JP Morgan in 1994  Several financial disasters at the time (Barings, Daiwa, Orange County, …) and subsequent calls for regulation increased VaR’s popularity  The Basle Committee for Banking Supervision recommended that capital requirements for banks be based on their daily VaR ADFIN Winter/2014 7

Value at Risk (VaR)  Under normality, and exclusively under normality , Javier Estrada VaR can easily be calculated based on AM and SD IESE Business School VaR VaR = AM + ( z c ) ⋅ SD Barcelona Spain  c = 95% ⟹ z c = –1.65 1 ‐ 5%  c = 99% ⟹ z c = –2.33 X AM  Consider …  a (normal) distribution of daily profits with … • AM = $5m • SD = $9.2m  a confidence level of 95% VaR = $5m +(–1.65)($9.2m) = –$10.2m ADFIN Winter/2014 Value at Risk (VaR)  What is the impact of changing c ? Javier Estrada  If c = 99.% ⟹ z c = –2.33 IESE Business • Before ( c = 95%) School  VaR = $5m + (–1.65)($9.2m) = –$10.2m Barcelona Spain • Now ( c = 99%)  VaR = $5m + (–2.33)($9.2m) = –$16.4m  This company expects … • to make $5m on the average day • to lose $10.2m one out of every 20 days • to lose $16.4m one out of every 100 days  But remember, these calculations are as accurate as is the assumption of normality Go ADFIN Winter/2014 8

Downside Beta  The relationship between beta and downside beta Javier Estrada is similar to that between the SD and the IESE Business semideviation School  Just as beta measures relative volatility Barcelona Spain • % change in an asset given a 1% change in the market  Downside beta measures relative downside volatility • % fall in an asset given a 1% fall in the market  Also similar to the semideviation  Falls are calculated with respect to a benchmark • The benchmark is chosen by the investor  The upside has no weight in the calculation • It is irrelevant how much an asset rises when the market rises ADFIN Winter/2014 Appendix Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014 9

The Standard Deviation – Problems Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014 Back The Standard Deviation – Problems Javier Estrada R IESE Business Good School volatility Barcelona X Spain B Y Bad volatility Time  Asset X has a much higher SD than asset Y  But it is good volatility ( above the benchmark)  Asset Y has very little volatility, but it is all bad ADFIN Winter/2014 Back 10

Morningstar Risk Javier Estrada IESE Business School Barcelona Spain Go ADFIN Winter/2014 Morningstar Risk Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014 Back 11

Value at Risk (VaR) Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014 Back 12