In this chapter…
Discuss how asset return distributions tend to deviate from the normal distribution.
Explain potential reasons for the existence of fat tails in a return distribution and discuss the implications fat tails have on analysis of return distributions.
Distinguish between conditional and unconditional distributions.
Discuss the implications regime switching has on quantifying volatility.
Explain the various approaches for estimating VaR.
Compare, contrast and calculate parametric and non-parametric approaches for estimating conditional volatility, including: Historic simulation
Compare, contrast and calculate parametric and non-parametric approaches for estimating conditional volatility, including: Historical standard deviation
Compare, contrast and calculate parametric and non-parametric approaches for estimating conditional volatility, including: Exponential smoothing
Compare, contrast and calculate parametric and non-parametric approaches for estimating conditional volatility, including: GARCH approach
Compare, contrast and calculate parametric and non-parametric approaches for estimating conditional volatility, including: Multivariate density estimation
Compare, contrast and calculate parametric and non-parametric approaches for estimating conditional volatility, including: Hybrid methods
Explain the process of return aggregation in the context of volatility forecasting methods.
Explain how implied volatility can be used to predict future volatility and discuss its advantages and disadvantages.
Explain the implications of mean reversion in returns and return volatility for forecasting VaR over long time horizons.
Discuss the effects non-synchronous data has on estimating correlation and describe approaches that mitigate the impact of non-synchronous data on risk estimates.
Discuss the use of backtesting for comparing VaR results using different volatility estimation approaches and the desirable attributes of VaR estimates.
Key terms
Risk varies over time. Models often assume a normal (Gaussian) distribution (“normality”) with constant volatility from period to period. But actual returns are non-normal and volatility varies over time (volatility is “time-varying” or “non-constant”). Therefore, it is hard to use parametric approaches to random returns; in technical terms, it is hard to find robust “distributional assumptions for stochastic asset returns”
Conditional parameter (e.g., conditional volatility): a parameter such as variance that depends on (is conditional on) circumstances or prior information. A conditional
parameter, by definition, changes over time.
Persistence: In EWMA, the lambda parameter (λ). In GARCH (1,1), the sum of the alpha (α) and beta () parameters. High persistence implies slow decay toward to the long-run average variance.
Autoregressive: Recursive. A parameter (today’s variance) is a function of itself (yesterday’s variance).
Heteroskedastic: Variance changes over time (homoskedastic = constant variance).
Leptokurtosis: a fat-tailed distribution where relatively more observations are near the middle and in the “fat tails (kurtosis > 3)
How to Estimate Volatility
Take two steps to compute historical (not implied) volatility:
1. Compute the series of periodic (e.g., daily) returns,
2. Choose a weighting scheme (to translate a series into a single metric)
Compute the series of periodic returns (e.g., 1 period = 1 day)
Assume that one period equals one day. You can either compute the “continuously compounded daily return” or the “simple percentage change.” If Si-1 is yesterday’s price and Si is today’s price,
Continuously compounded return:
The simple percentage return is given by:
1
Linda Allen contrasts three periodic returns (i.e., continuously compounded, simple percentage change, and absolute level change). She argues continuously compounded must be used when computing VAR because it is “time consistent” (except for
interest-Choose a weighting scheme
The series can be either un-weighted (each return is equally weighted) or weighted. A weighted scheme puts more weight on recent returns because they tend to be more relevant.
The “standard” un-weighted (or equally weighted) scheme
The un-weighted (which is really equally-weighted) variance is a “standard” historical variance.
In this case, the variance is given by:
2 2
most recent m observations
the mean/average of all daily returns ( )
n
For practical purposes, the above equation is often simplified with the following assumptions:
The average daily return of zero is assumed to be zero:
u 0
The denominator (m-1) is replaced with m
This produces a simplified version of the standard (un-weighted) variance:
2 2
This simplified version replaces (m-1) with (m) in the denominator. (m-1) produces an
“unbiased” estimator and (m) produces a “maximum likelihood” estimator.
The weighted scheme (a better approach, generally)
The standard approach gives no weight (or equal weight) to each return. But for forecasting purposes, it is better to give greater weight to more recent data. A generic model for this approach is given by a weighted moving average:
2 2
The alpha () parameters are simply weights; the sum of the alpha () parameters must equal one because they are weights.We can now add another factor to the model: the long-run average variance rate. The idea here is that the variance is “mean regressing:” think of it the variance as having a “gravitational pull” toward its long-run average. We add another term to the equation above, in order to capture the run average variance. The added term is the weighted long-run variance:
2 2 1 m
n L i n i
i
V u
The added term is gamma (the weighting) multiplied by () the long-run variance because the variance is a weighted factor.
-formatted ARCH (m) model:
2 2
1 m
n i n i
i
u
Stochastic behavior of returns
Risk measurement (VaR) concerns the tail of a distribution, where losses occur. We want to impose a mathematical curve (a “distributional assumption”) on asset returns so we can estimate losses. The parametric approach uses parameters (i.e., a formula with parameters) to make a distributional assumption but actual returns rarely conform to the distribution curve. A parametric distribution plots a curve (e.g., the normal bell-shaped curve) that approximates a range of outcomes but actual returns are not so well-behaved: they rarely “cooperate.”
Value at Risk (VaR) – 2 asset, relative vs. absolute
Know how to compute two-asset portfolio variance & scale portfolio volatility to derive VaR:
Inputs
Trading days /year 252 Initial portfolio value (W) $100 VaR Time horizon (days)
(h) 10
Autocorrelation (h-1, h) 0.25 If independent, = 0. Mean reverting returns = negative
Outputs Exp portfolio return (per
year) 18.5%
Scaling factor 15.78 Don’t need to know this, used for AR(1) Std deviation (h),
Autocorrelation 3.12% Standard deviation if auto-correlation.
Normal deviate (critical z
value) 1.64 Normal deviate
Expected future value 100.73
Relative VaR, i.i.d $4.08 Doesn’t include the mean return Absolute VaR, i.i.d $3.35 Includes return; i.e., loss from zero
Relative VaR, AR(1) $5.12 The corresponding VaRs, if autocorrelation incorporated.
Note VaR is higher!
Absolute VaR, AR(1) $4.39