Question

There are two assets following single-factor model, Asset 1 and Asset 2. Their model parameters are given as follow,
Asset   αi   βi
1 0.10   1.5
2 0.08   0.5
Assume E[f]=e1=e2=0.
a) Construct a zero-beta portfolio from these two risky assets.
b) Find the factor risk premium λ using the principle of no-arbitrage.
c) What is the meaning of factor risk premium in single-factor models?

Answer 1

a)

The beta of the portfolio is the weighted average of the individual betas

Let the weight of asset 1 be w, then

0 = w*1.5 + (1-w)*0.5

w = 0.5

Hence, the portfolio would consist of asset 1 with weight 50% and asset 2 with weight 50%

b)

c is a constant

For asset 1

0.10 = c+ 1.5*λ

For asset 2

0.08 = c+ 0.5*λ

Solving, we get

λ = 0.02

c)

Factor risk premium indicates the excess returns due a particular factor. This factor may be a equity risk factor, a country risk premium etc. Any factor in addition to the the market risk, is a factor. In a single factor model, there is one factor which demand an excess returns.