The following question illustrates the APT. Imagine that there are only two pervasive macroeconomic factors. Investments X$,$ Y$,$ and Z have the following sensitivities to these two factors:

Investment b$1$ b$2$

X $1\mathrm{.}750$

Y $1\mathrm{.}002\mathrm{.}00$

Z $2\mathrm{.}001\mathrm{.}00$

We assume that the expected risk premium is $4\mathrm{.}8\%$ on factor $1$ and $8\mathrm{.}8\%$ on factor $2.$ Treasury bills obviously offer zero risk premium.

According to the APT, what is the risk premium on each of the three stocks?

Suppose you buy $$260$ of X and $$65$ of Y and sell $$195$ of Z$.$ What is the sensitivity of your portfolio to each of the two factors? What is the expected risk premium?

Suppose you buy $$104$ of X and $$78$ of Y and sell $$52$ of Z$.$ What is the sensitivity of your portfolio to each of the two factors? What is the expected risk premium?

Finally, suppose you buy $$208$ of X and $$26$ of Y and sell $$104$ of Z$.$ What is your portfolio's sensitivity now to each of the two factors? And what is the expected risk premium?