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Standardized or Unitized returns are so obvious they are rarely described explicitly in financial literature.
Unitizing Returns can often prove a better solution when simulating multivariate distributions,
computing parametric VaR and extracting marginal measures such as incremental or marginal
Value-at-Risk or to couple univariate and multivariate distributions with
a copula !
A c++ dll demonstrating unitized returns is available here.
The online correlation and volatility calculator is also based on this approach.
(See Sigma, One-Rho or
Combined/Benchmarked Returns returns modules).
The conventional approach to modeling multivariate
distributions usually proceeds by computing conventional variance and covariance
and perhaps subsequently standard deviations and correlations, and then decomposing the matrix either through Cholesky decomposition or Singular Value Decomposition.
However an alternative approach, which consists in unitizing returns, can yield, in some
cases, much better results with increased computer efficiency.
The idea behind this technique is very simple:
Rather
than compute returns from raw (or converted) prices, we scale or "standardize
" returns to the Normal Distribution so that the mean equals zero
and the standard deviation equals one, thus the term unitized. (or normalized)
This approach requires subtracting the return at time t by the mean of the
returns and then scaling this mean adjusted return by the return's
volatility.
Some
(Obvious)
properties:
1)
The transpose of the Unitized Matrix of returns multiplied by the Unitized Matrix of Returns gives
the correlation matrix.
(with or without exponential moving average)
2)
This has substantial advantages when generating multivariate
normal distributions with mean zero
and variance one .
Indeed, we can take a dot product
with a vector of independent mean zero
unit one random numbers (a
vector of equal length to the normalized returns r) and obtain the exact same distribution we
would have obtained had we:
- 1) Computed the correlation matrix from the returns.
- 2) Decomposed the matrix.
Either through Cholesky or Spectral
decomposition technology
- 3) Multiplied the independent random number by the factorized matrix.
3)
In computational terms, the
gain in efficiency must be balanced by the storage cost of keeping the returns cached
in memory.
4)
VaR then reduce s to a simple
multiplication requiring m(n+1) multiplies instead of n(n+1) with n number
of series and m observations.
5)
The returns are scaled to the standard normal distribution N(0,1)
with mean zero and variance one, which means we can weight and aggregate (to
compute country /industry weighted indices),
subtract
(to compute relative
Value-at-Risk), multiply by normal random variable and still respect
normality.
6) Incremental and Marginal VaR are obtained by simple multiplication.
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