Foundations of Finance
Lecture 13-14: Risk and Return III
In the previous lecture we found out that the most efficient portfolio is the tangent portfolio, which
has the highest sharpe ratio. This lecture is mainly about finding out if a given portfolio is a tangent
portfolio or not.
Portfolio Improvement: Beta and the Required Return
Consider an arbitrary portfolio P and assume that further investment i is invested by borrowing
money from the risk free market. This will have 2 consequences:
Expected Return: because we are taking out money from the risk free market and investing
in a more riskier investment, the expected return will increase be i’s excess return, E[Ri]-rf
Volatility: risk of investment i that is common with the portfolio’s risk will be added (since
rest of the risk of investment i will be diversified). Therefore the incremental risk would be
SD(Ri)*Corr(Ri,Rp).
Another way through which we could have increased our risk would have been investing directly in
portfolio P. in that case, P’s Sharpe ratio is:
E[RP ] rf
SD(RP )
We can ow just compare the sharpe ratio of portfolio P to the incremental sharpe ratio due to
investment i.
E [Ri ] rf E[RP ] rf
SD(Ri ) Corr (Ri ,RP ) SD(RP )
Incremental Sharpe Ratio Sharpe Ratio
of investment i of portfolio P
Rearranging this equation, we would get:
For further interpretation, lets combine the correlation and volatility terms to define the beta of
investment i with portfolio p.
SD(Ri ) Corr (Ri ,RP ) Cov(Ri ,RP )
iP
SD(RP ) Var (RP )
Beta measure the sensitivity of the investment i to fluctuations in portfolio p. That is, for 1% change
in the portfolio’s return, investment i’s return is expected to change by β% due to the common risks.
Therefore investment i will increase the sharpe ratio of the portfolio if its actual return is higher than
E Ri rf iP (E[ RP ] rf )
required return for i ri
Lecture 13-14: Risk and Return III
In the previous lecture we found out that the most efficient portfolio is the tangent portfolio, which
has the highest sharpe ratio. This lecture is mainly about finding out if a given portfolio is a tangent
portfolio or not.
Portfolio Improvement: Beta and the Required Return
Consider an arbitrary portfolio P and assume that further investment i is invested by borrowing
money from the risk free market. This will have 2 consequences:
Expected Return: because we are taking out money from the risk free market and investing
in a more riskier investment, the expected return will increase be i’s excess return, E[Ri]-rf
Volatility: risk of investment i that is common with the portfolio’s risk will be added (since
rest of the risk of investment i will be diversified). Therefore the incremental risk would be
SD(Ri)*Corr(Ri,Rp).
Another way through which we could have increased our risk would have been investing directly in
portfolio P. in that case, P’s Sharpe ratio is:
E[RP ] rf
SD(RP )
We can ow just compare the sharpe ratio of portfolio P to the incremental sharpe ratio due to
investment i.
E [Ri ] rf E[RP ] rf
SD(Ri ) Corr (Ri ,RP ) SD(RP )
Incremental Sharpe Ratio Sharpe Ratio
of investment i of portfolio P
Rearranging this equation, we would get:
For further interpretation, lets combine the correlation and volatility terms to define the beta of
investment i with portfolio p.
SD(Ri ) Corr (Ri ,RP ) Cov(Ri ,RP )
iP
SD(RP ) Var (RP )
Beta measure the sensitivity of the investment i to fluctuations in portfolio p. That is, for 1% change
in the portfolio’s return, investment i’s return is expected to change by β% due to the common risks.
Therefore investment i will increase the sharpe ratio of the portfolio if its actual return is higher than
E Ri rf iP (E[ RP ] rf )
required return for i ri
