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) Beta is measurement of volatility or systematic risk .A zero Beta indicates that the security's Return is independent of the market return.Zero Beta portfolio can be constructed asfollows

Beta of portfolio is average beta of individual assets

LET THE WEIGHT BE W

Therefore Step1

0=W1.5 + 0.5(1-W)

W=0.5

HENCE WHEN INVESTED EQUALLY IN BOTH THE ASSETS BETA WOULD BE ZERO.

ASSET1=50%

ASSET 2=50%

b) According To CAPM E(ri) = rf + βi * (E(rM) - rf)

rf + βi * λ

(λ is known as lambda)

where average risk premium i.e Lambda is (E(rM) - rf)

therefore forming two simultaneous Equation

0.1=rf+1.5*(E(rM) - rf)

0.08=rf+0.5*(E(rM) - rf)

solving the above we get value of λ =0.02

C)The market risk premium is the difference between the expected return on a market portfolio and the risk-free rate.IN case of a single factor model such premium or excess return is assumed to be due to the factor which is responsible.It could be any factor like market index,inflation etc.