Managing equity portfolio beta with futures

Vol 5, Reading 36, Pg 353

This concept has continued to confuse me throughout, even at Level 3

The example in Exhibit 3 states “because the market went up by 4.4% and the overall gain was 5.04%, the effective beta of the portfolio was 0.0504/0.044 = 1.15”

why is beta often computed as = return on portfolio / return on market? according to this formula, if beta = 0, portfolio return =0.

but a zero beta portfolio should generate a risk free return.

according to the CAPM, should’nt beta = excess return on portfolio / excess return on market? (excess in relation to risk free rate)


portfolio return = 0 when compared against risk free rate.

ri = rf + beta * (rm-rf)

beta = 0 implies ri = rf

ri - rf / rm -rf = beta from CAPM

Market went up 4.4% which is rm - rf = 4.4%

Overall gain was 5.04% which is ri - rf = 5.04%

it is the excess return over the risk free rate they are talking about when they say “went up”…

cpk , thanks for clarifying.

never thought about it , but beta is really the ratio of excess return of portolio/stock over risk free rate to the excess return of benchmark return over risk free rate.

Again I think of AM and PM q’s and wonder if they give this problem AND a risk free rate , will the answer be different ?

I assume it will be different.

So if they do not give a risk free rate , and they mention loosely that the market went up , we can take it that the implied risk free rate is zero.

beta is never calculated as = % change in portfolio/%change is market. It just that you are trying to gauge the eventual impact on portfolio due to rounding issues of futures. is in it?

so if beta = 0, then as per CAPM ri=rf which cpk 123 has rightly mentioned above.

I think due to rounding issue related to futures contract they are just trying to tell you the effective beta is some what closer to our target.

Slight mismatch in betas (i.e. Trgt & effective) have also been discussed later.

thanks cpk.

as presented in the example, the 4.4% and 5.04% returns are absolute, not excess returns. there is no reference to risk free rate.

because the market went up by 4.4% and the overall gain was 5.04%

that definitely does not look like absolute to me… it is with reference to something.

stock portfolio beginning value = $38,500,000

stock portfolio ending value = $40,103,000

profit on stock futures = $338,937.50

stock portfolio return = $40,441,937.50/$38,500,000 -1 = 5.04%

this looks like absolute return to me.

and if 5.04 is being compared to 4.4, I am assuming 4.4 is absolute too (although the example doesnt explicitly state that) since a ratio of absolute to excess return wouldn’t make much sense.

5.04 % = portfolio total ( absolute ) return

4.4% = market total ( absolute ) return

beta = ( 5.04 - 0 ) / ( 4.4 - 0 ) ~ 1.145