In: Statistics and Probability
As you know, the risk of two assets is calculated by getting the square root of the equation:
sP2= w12s12+ w22s22+ 2w1w2r12s1s2
Explain what would happen if one of the two assets was a risk-free asset. In other words, what would be the risk of the combination of the two assets? You have to explain that in accordance with the above equation.
Given equation:
Sp2 =w12s12 + w22s22 + 2w1w2r12s1s2
So, here we have two assets - asset 1 and asset 2.
If one of the two assets was a risk-free asset, then it means that the standard deviation of risk-free asset is zero.
Let us assume that asset 2 is risk-free. So, it's risk =Standard deviation =s2 =0
So, the entire last term of the above equation, i.e., 2w1w2r12s1s2 becomes 0. Also the term, w22s22 becomes 0. The only term that remains is w12s12. It means the risk of the combination of the two assets would become the risk of one asset only. The portfolio risk is same as the risk of that asset which has the risk.
sp2 = w12s12 + w22s22 + 2w1w2r12s1s2
sp2 =w12s12 + w22(0)2 + 2w1w2r12s1(0)
(since, the risk-free asset's risk, i.e., standard deviation is s2 =0)
sp2 =w12s12 + 0 + 0 =w12s12
sp =
Thus, the portfolio risk is only the risk of the asset which is not risk-free (here, asset 1).