Risk contribution (2008 question 37)

[ajsa]

Hi David,

Could you explain the Risk Contribution here? How is it calculated? it seems to be different from the RC of Ong..

Also why increasing an asset's position will decrease its Marginal Return/ Marginal Risk? (like asset 1 and 4)

Thanks!

37. You are given the following information about a portfolio and are asked to make a recommendation
about how to reallocate the portfolio to improve the risk/return tradeoff.
Asset Expected Standard Current Covariance of Marginal Marginal Marginal Return/ Risk
Return Deviation Weight portfolio and Return Risk Marginal Risk Contribution
asset returns
Asset 1 7.10% 17.00% 38.70% 1.43% 3.10% 13.99% 22.17% 5.41%
Asset 2 8.00% 40.60% 6.20% 2.44% 4.00% 23.93% 16.71% 1.48%
Asset 3 6.70% 44.80% 5.50% 2.39% 2.70% 23.39% 11.55% 1.29%
Asset 4 6.90% 21.40% 14.60% 1.41% 2.90% 13.86% 20.92% 2.02%
Risk Free 4.00% 0.00% 35.00% 0.00%
Which of the following the recommendations will improve the risk/return tradeoff of the portfolio?
a. Increase the allocations to assets 1 and 3 and decrease the allocations to assets 2 and 4.
b. Increase the allocations to assets 1 and 2 and decrease the allocations to assets 3 and 4.
c. Increase the allocations to assets 2 and 3 and decrease the allocations to assets 1 and 4.
d. Increase the allocations to assets 1 and 4 and decrease the allocations to assets 2 and 3.
Answer: d
a. Incorrect. Asset 3 should be decreased since it has the lowest marginal return-to-marginal risk ratio.
b. Incorrect. Asset 4 should be increased since it has the highest marginal return-to-marginal risk ratio.
c. Incorrect. Asset 4 should be increased since it has the highest marginal return-to-marginal risk ratio.
d. Correct. A portfolio optimizing the risk-reward tradeoff has the property that the ratio of the
marginal return to marginal risk of each asset is equal. Therefore, this option is the only recommendation
that will move the ratios in the right direction.
Reference:
Philippe Jorion, Value at Risk: The New Benchmark for Managing Financial Risk, 3rd ed. (New York:
McGraw-Hill, 2007)., Chapter 17 – VaR and Risk Budgeting in Investment Management

 

[David Harper CFA FRM]

Hi asja,

Hmmm...don't recognize this, surprised hasn't come up ... as you know, question is just asking you to look at the marginal return/risk.
But, in a literal sense, I do agree, this particular type of RC is not (to my knowledge) covered except very indirectly by Jorion Ch 7.

Because this is superficially similar to Ong's RC: both are first derivatives: change in portfolio risk/change in asset risk
(only Ong doesn't assume normality, whereas this is mean variance)

so the RC here is equal to marginal VaR but where the deviate = 1.0; i.e., d_Portfolio Vol/d_Asset
marginal VaR = cov(portfolio, asset) / portfolio volatilty * deviate; but under low confidence (84%), deviate = 1.0,
and this RC = cov(portfolio, asset) / portfolio volatilty ....

Re: why increasing an asset’s position will decrease its Marginal Return/ Marginal Risk? (like asset 1 and 4)
because the assets are part of the portfolio! as they increase in weight, like their portfolio betas tend toward 1.0, their RC tends toward portfolio volatility

David

 

[ajsa]

Re: why increasing an asset’s position will decrease its Marginal Return/ Marginal Risk? (like asset 1 and 4)
because the assets are part of the portfolio! as they increase in weight, like their portfolio betas tend toward 1.0, their RC tends toward portfolio volatility

David​

Hi David,

Sorry for being slow.. but how is it related to Marginal Return/ Marginal Risk? maybe you need to explain what are Marginal Return and Marginal Risk? I am afraid I do not have a clear understanding of them

Thanks.

 

[David Harper CFA FRM]

Hi asja,

i can't vouche for the precision of the question (you know how it is with these samples, often times they aren't rigorous)...just eyeballing it, it appear to me they defined marginal return simply as *excess return* (i.e., - RF rate).

then, as i said above, marginal risk is probably: covariance(asset, portfolio) / portfolio (volatilty)
port vol isn't given but looks like it is constant at ~ 10%; i.e., for each, cov() / marginal risk looks like ~10%

so the key is: as the weight increases to the asset, the asset occupies more of the portfolio, and the covariance (asset, portfolio) increases...what is the limit of this? when the asset is the portfolio (!), and we have covariance (asset, itself) = variance(asset), such that the marginal risk = variance (Asset) / portfolio (volatility), so that's why i said marginal risk will tend toward portfolio volatilty

...more simply, as asset increases, cov(asset, portfolio that includes asset) increases such that MRisk increases and MR/MRisk decreases

David