CFAI textbook, Example 18, Page 44


Example 18, point #3 asks to calculate the correlation between private equity and small-mid cap equity. The formula given in the response is as following:

correlation = (ß1 * ß2 * variance market) / (standard deviation1 * standard deviation2)

I may be suffering from amnesia, but I don’t remember seeing this formula before. Does anyone know how it is derived?

Thanks guys!

I don’t know how it was derived, and it’s wrong.

Given particular betas for securities 1 and 2, a particular variance of the market’s returns, and particular standard deviations of returns for securities 1 and 2, there isn’t a (unique, specific) correlation of returns for securities 1 and 2; there is a range of possible correlations.

that is called the Market model.

Pg 47 - CME chapter

with one factor, the covariance between any two assets in a one-beta model (such as the ICAPM) is equal to the product of each asset’s beta with respect to the market times the variance of the market. The needed betas can be calculated as

and covariance / (std dev 1 * std dev 2 ) = correlation.

I remember seeing the market model, but I don’t know how they justify the calculation of the covariance (or correlation) between security 1 and security 2. As I wrote above, there is a range of possible correlations that will work. A simple model in Excel using rand() will show you that the formula doesn’t work.

Unless I’m missing something fundamental from the model formulation, that equation just doesn’t make any sense.

Still, CPK’s correct: it’s in there.

Ok, I guess I’ll just have to memorize some formulas without knowing much background info. I’ve done it before, but it’s sticks better when I know the logic behind formulas. Thanks!

I remember vaguely that this was a Level I formula using a CAPM model ( 1 factor)

well corr= Cov(x,y)/SDx* SDy in the market model cov(xy)= beta x*beta y *var(M) Put them together and you get the answer in the text

The covariance is a very broad term . It ranges from +infinity to - infinity . It is standardised by dividing by the product of standard deviation of the two securities . This standardised figure is called correlation of the two securities . It is calculated by the above formula given. It varies from +1 to -1. So it can be used in statistical analysis and measurement.

A correlation of +1 means both the assets move in the same direction . A correlation of -1 means they move in the opposite direction. a correlation of 0 means they there is no linear relationship between two securities.

This is an algebraic manipulation beta = sum (market i-mean)(xi-mean)/var(M) . Decompose the two betas and you end up with the var(M) on top cancelling out one of the denominators and the two (market i -mean)s combining to cancel out the other var(M) in the denomiator leaving (xi-mean)(yi-mean) = covariance.