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hichmoul


Total Posts: 12
Joined: Mar 2010
 
Posted: 2017-11-12 00:52
I am confused.
Looking historically over a time period [t0,t1], and
Given 5 assets in a portfolio, the end-of-day price of each being the time series P1 P2 P3 P4 P5 that stretch daily from to to t1, and given the value of the portfolio is simply
Pp = P1+P2+P3+P4+P5.

Then LNRp, LNR1, LNR2, LNR3, LNR4, LNR5, the time series of logreturns derived from the portfolio price serie and each asset serie (as part of the portfolio)

Taking VARp, VAR1,...VAR5, the sample variances of those logreturns series, and COVAR(i,j), the sample covariances of pairs i#j of those logreturns.

1. Numerically, should there be a relationship between VARp, the VARi and COVAR(i,j) over the fixed time period [t0,t1]?
Question applies to both the unbiased ( /N-1) and the biased (/N) sample var/covar.
I am confused as to how map the expression here
https://en.wikipedia.org/wiki/Modern_portfolio_theory
to the "Realized" case I'm looking backwards.

2. Can one then express the Percentage contribution of each asset's variance to total portfolio variance, in a way they sum up to 100%

3. Can then 1 go a step further to talk about this contribution in terms of sqrt of var, ie sample volatility?

4. same question for down-side volatility (ie deviations only down of a threshold)

5. same question for drawdown?

hichmoul


Total Posts: 12
Joined: Mar 2010
 
Posted: 2017-11-14 21:33
I now have a slightly better understanding:
Simple(aka linear) returns (Price_t/Price_t-1 -1) aggregate across the 5 assets to make the up the simple returns on the portfolio. Obviously important to consider the varying weights of each asset in the portfolio. Weight-adjusted simple return series sum up to portfolio simple returns serie.
Unbiased sample variance from portfolio simple returns = sum of covariances of weight adjusted asset simple returns series.
That takes care of question 1 and 2.

Question 2.5 Is it possible, in the logreturn world, to express assets' variances contributions?

Question 3-5 remain

theevildog


Total Posts: 1
Joined: Oct 2012
 
Posted: 2017-11-16 17:53
Would think "marginal risk contribution" is the answer to all questions. Quick google search should give you the details you need.
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