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

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

Feb 21, 2023 308 likes •447 views

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)

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  • 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

Risk Revisited (I):

Downside Risk

Javier Estrada ADFIN – Winter/2014

Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

Motivation

  • Why the emphasis on volatility (SD) in Finance?
  • An arbitrary choice by Markowitz in the early ‘50s
  • Partly motivated by limitations in computing power
  • The rest is history
  • 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
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Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

The Standard Deviation

  • If two assets X and Y have the same mean return,

investors would prefer the one with the lower SD

  • Hence investors would prefer asset X

R

AM Y X

Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

The Standard Deviation – Problems

  • Problems with the SD as a measure of risk
  • 1. Incomplete and misleading when the underlying

distribution is not normal (or not symmetric)

  • This affects one of the main uses of the SD

 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

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Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

The Standard Deviation – Problems

  • Problems with the SD as a measure of risk
  • 1. Incomplete and misleading when the underlying

distribution is not normal (or not symmetric)

  • More often than not in practice, the distributions

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

Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

The Standard Deviation – Problems

  • Problems with the SD as a measure of risk
  • 1. Incomplete and misleading when the underlying

distribution is not normal (or not symmetric)

  • Two (of the many possible) examples
  • 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 + (zc)⋅SD
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Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

The Standard Deviation – Problems

  • Problems with the SD as a measure of risk
  • 2. Deviations are measured with respect to the AM
  • Investors typically use other benchmarks
  • These include …
  • 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

Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

The Standard Deviation – Problems

  • Problems with the SD as a measure of risk
  • 3. Deviations above and below the AM are treated in

the same way

  • Clearly, investors do not feel the same way about

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

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Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

The Standard Deviation – Problems

  • Problems with the SD as a measure of risk
  • 3. Deviations above and below the AM are treated in

the same way

(Oracle)

Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

The Semideviation

  • Given any chosen benchmark B, the semideviation

with respect to B (ΣB) is given by

  • For the sake of comparison, recall that the SD is

given by

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Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

The Semideviation

  • Some attractive properties of the semideviation
  • 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 Bs
  • It is equally plausible and useful for symmetric and

skewed distributions

  • It is almost as easy to calculate as the SD

Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

The Semideviation

(Oracle)

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Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

Morningstar Risk

  • A proprietary measure of risk
  • This much is known
  • Takes into account return variability
  • Weights more heavily downside variability
  • 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 Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

Value at Risk (VaR)

  • VaR is the ‘worst’ expected outcome, over a given

time horizon (T), for a given confidence level (c)

  • Over the chosen time horizon, a loss larger than

VaR will occur with a (1–c)% probability

  • 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

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Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

Value at Risk (VaR)

  • Under normality, and exclusively under normality,

VaR can easily be calculated based on AM and SD

VaR = AM + (zc)⋅SD

1‐5%

AM

X

VaR  c = 95% ⟹ zc = –1.65  c = 99% ⟹ zc = –2.33

  • 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

Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

Value at Risk (VaR)

  • What is the impact of changing c?
  • If c = 99.% ⟹ zc = –2.33
  • Before (c = 95%)

 VaR = $5m + (–1.65)($9.2m) = –$10.2m

  • 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

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Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

Downside Beta

  • The relationship between beta and downside beta

is similar to that between the SD and the semideviation

  • Just as beta measures relative volatility
  • % 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

Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

Appendix

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Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

The Standard Deviation – Problems

Back Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

The Standard Deviation – Problems

Time

Y X

B R

  • 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

Good volatility Bad volatility

Back

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Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

Morningstar Risk

Go Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

Morningstar Risk

Back

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Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

Value at Risk (VaR)

Back