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Stulz Chapter 2 - Page 57 - Exercice 2


New Member

Could you please walk me through exercise 2 of the Stulz Book Chapter 2 page 57 ("Question and Exercises" after the review questions page)? "2. During 1997, the monthly... 4%"

I have searched the forum but it does not seem that this question was asked before. (too easy? embarassing...) ;-)

(I have asked Kaplan who took a week to answer and it is not very well explained at all...)

Thanks a lot for the GREAT website!


David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi Y.,

Stulz writes clunky stuff. You want the 2-asset portfolio variance but note how it simplifies when one of the assets (the riskfree rate) has no volatility .

2-asset variance:
variance (2-asset portfolio) = weight_1^2*variance(1) + weight_2^2*variance(2) + 2*weight_1*weight_2*volatility(1)*volatility(2)*covariance(1,2)

But if one of the assets is the risk-free asset, then its volatility = 0, reducing the general to:
variance (1 asset + risk free) = weight_1^2*variance(1) + weight_2^2*0 + 2*weight_1*weight_2*volatility(1)*0*covariance(1,2);
variance (asset + risk free) = weight_asset^2*variance(asset); or
volatility (asset + riskfree weight) = SQRT[weight_asset^2*variance(asset)] = weight_asset * volatility_asset.
It is a long way to show but the elegant conclusion is:
volatility (asset + riskfree weight) = weight_asset * volatility_asset.
... please note this is essentially similar to the straight capital market line (CML): an allocation between a risky portfolio + a riskless asset returns a portfolio which is a linear function. The only difference between this and the CML is this uses the S&P index for the risky asset and the CML uses the risky market portfolio itself as the risky "asset"

Here, the original portfolio has volatility
= w1*4.0%

When the S&P volatility increases to 4.5%, the portfolio volatility becomes:
= w2 * 4.5%

If same $100,000 is invested, we need volatilities to match, such that:
w1*4% = w2*4.5%, or
w2 = w1 * 4%/4.5%
And, this this is my strict answer to the question:
If vol goes from 4% to 4.5%, the Current Weight (w1) allocated to the S&P Index must be trimmed to w1*4/4.5
and i find the question to have no specific numeric answer, even if we assume (eg) a riskless rate of 3% or 4%.

Under these assumptions, as i read them, there are multiple answers.
I input into an XLS to check myself, see http://db.tt/nWWsvaJ
(Trying to figure if i've missed something, then i remind myself this is Stulz ....)

For example, if the initial portfolio is 50%/50%, then portfolio volatility = 50%*4% = 2%,
a loss of $5,000 (it's all monthly) is 2.5 standard deviations.

If risk vol goes to 4.5%, then we need to re-balance to 50%*4/4.5 = 44.44%, and
rebalanced portfolio vol = 44.44% * 4.5% = 2%
(given the same $100,000, equivalent volatility implies equivalent probabilities for all loss thresholds).

I don't think the current risk free rate level is needed. As written, then, i think there is no given starting allocation. I hope that helps!

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