problems about effective beta

in a question, it calcualted the effective beta by multiply the beta to the %change in index + gain/loss in futures / the market return.

Why is it to multiply the beta here? where is it a need?

The question normally asks you to calculate the number of futures you need to reduce the B to 0. This is normally following a statement which says we want to reduce the correlation to market as there is an expected downturn in equities.

After we have hedged the portfolio by selling S&P futures we don’t actually reduce the B to 0 entirely (otherwise gains on our short futures would completely offset losses on our equity). We can therfore calculate the effective B i.e. what the B was for the hedged portfolio. We do this by comparing the decline in the market vs the decline in the portfolio. If our hedge was excellent, then the B will be close to nil (gains on short futures offsets losses on equity holdings). If our hedge was poor, then the B will be higher.

I get your point, as it have already asked to compute the effective beta, it means the beta is not 1.

so when we calcualating the gain/loss in the portfolio, is it a must to multiple the beta? and which beta is? the origianl portfolio beta or the target beta? thx~~

We calculate the B that was the target i.e we targeted a B of Nil, but the actually B was 0.87411 or something like that…

this is a nice way to compute effective B that I didn’t read before. Thanks

it will gave the same result as this conventional way:

Effective Beta= returns of the new position (including futures)/ returns of the benchmark (index)