Question

You estimate that a passive portfolio invested to mimic the S&P 500 stock index yields an expected rate of return of 10% with a standard deviation of 19%. Assume you manage a risky portfolio with an expected rate of return of 12% and a standard deviation of 24%. The T-bill rate is 5%. Your client would like to switch an 80% allocation in your portfolio to an 80% allocation in the stock market index. How much would your client have to earn in your portfolio to earn the same return? Convert your answer to a percentage and round to one decimal places

Answer 1

Existing Situation:
Passive Portfolio which is mimicking the S&P 500 Index gives 10% return.
Passive Portfolio's Risk (Standard Deviation, σ) is 19%

Active Portfolio (risky portfolio) gives returns of 12%  
Active Portfolio's Risk (Standard Deviation, σ) is 24%, which is certainly higher due to higher risk.

Standard Deviation is the measure of risk or measure of market volatility. It can be understood as the number of times the stock or portfolio value has deviated from the mean value .
Higher the risk taking ability, greater are the chances for high returns yielding.

Changed Situation: 80% of the Funds in Passive Portfolio
Weight/ Allocation to Passive Portfolio i.e. Stock Market Index = 80%
Weight/ Allocation to Active Portfolio, thus would be = 100% - 80%
= 20%

Active/ Risky Portfolio Return = 12%
T-Bill rate or Risk free rate = 5%
Rreturns on Stock Index = 10%

Portfolio Return = {(Weight of Active Portfolio * E(r) of Active Portfolio)} + {( Weight of Passive Portfolio * E(r) of Passive Portfolio)}

If the client needs the same return as before when his funds were in actively managed portfolio i.e. 12%, Let us assume our actively managed new fund gives us "X" returns

Then, putting all the value in the formula of calculating portfolio return, we get,
12% = {(20% * X%)} + {(80% * 10%)
0.12 = ( 0.20 * X% )+ (0.8 * 0.10)
0.12 = (0.20 * X%) + 0.08
0.12 - 0.08 = 0.20 * X%
0.04 = 0.20 * X%
0.04/0.20 = X%
0.2 = X%
0.2 = X/100
X= 0.2*100
X = 20%

Therefore, the active portfolio needs to give atleast 20% of returns for the client to earn the returns of 12% as earlier from their entire portfolio.