Class Business Upcoming Groupwork Fund Clip #1 Fund Clip #2

Quiz A client asks your advice about her investments. She has invested $50,000 in a Mosaic mutual fund and $30,000 in risk-free bonds. She asks you whether she should re-allocate her assets. You are analyzing the returns of the Mosaic fund and compare them with the S&P 500 ETF. You expect the Mosaic fund to have an expected return of 10% and a standard deviation of 25%. The S&P 500 ETF has an expected return of 8% and a standard deviation of 16%. The risk-free rate is currently 3%.

Quiz Question 1: What is the expected return and the standard deviation of her current portfolio? – The expected return is: E(r)=0.625(0.10) (0.03) = 7.4% – The standard deviation is: Stdev(r)=0.625*0.25 = 15.62%

Quiz Question 2: Assume that she can borrow and lend at an interest rate of 3%. Find a portfolio that dominates her current portfolio (same total risk but higher expected return)?

Quiz Find reward to variability ratios: RTV Mosaic= RTV S&P 500 = So S&P 500 ETF has a higher-sloped CAL. – It is nearer to the “tangency” portfolio than Mosaic

Quiz

Equation for S&P 500 CAL: Your friend’s current portfolio has a standard deviation of.1562 and an expected return of 7.4%. By dumping Mosaic and investing in the S&P 500 ETF she could have an expected return of

Quiz Figure out the dollar amount required. To find the dollar amounts invested in each security, first find fraction of investment equity needed in each asset: w = 0.98 (1-w) =.02 The total equity = $80,000, Invest 2% of equity in risk-free asset ($1600) Invest 98% of equity ($78,400) in the S&P 500 ETF. If we knew the covariation between the Mosaic Fund and the S&P ETF, we could possibly find an even better combination including all three!

Portfolio Allocation Example 1 2

Suppose tangency portfolio is – Risky Asset 1: 60% – Risky Asset 2: 40% Suppose you have $100 and only want – $50 in tangency portfolio – $50 in risk-free asset – Then you would have – __________ in asset 1 – __________ in asset 2 – __________ in risk-free asset $50 or 50% $30 or 30% $20 or 20%

Portfolio Allocation Example Suppose tangency portfolio is – Asset 1: 60% – Asset 2: 40% Suppose you want – 125% of investment equity in tangency portfolio – -25% in risk-free asset – What fraction of funds would you put in each asset? Asset 1: 75% Asset 2: 50% Risk free: -25%

Portfolio Allocation What about the case of many risky assets? The efficient frontier has the same shape Intuition is the same Expected Return Standard Deviation Individual Assets Tangency Portfolio risk free rate Best possible CAL

Portfolio Allocation How to solve for Tangency Portfolio? Easy to solve for using a computer Spreadsheets are a little more awkward, but it can be done. Upcoming Homework

Portfolio Allocation Three steps to determine the optimal portfolio: (A short version of steps given in example.) – 1) Compute and draw the efficient frontier – 2) Incorporate the risk free asset to find the best possible CAL – 3) Choose where you want to be on the CAL by borrowing or lending at the risk-free rate. Money managers solve (1) and (2) – Inputs = estimates of E[r p ] and Stdev[r p ] – Better inputs, better results you earn more $$. – Individuals determine (3)

Diversification and Many Risky Assets Two risky assets – Recall example when  = -1 We were able to eliminate all risk – When -1 <  <1 we were able to eliminate some risk

Diversification and Many Risky Assets What if we create portfolios of many risky assets? Decomposing realized returns: – E [ r it ]=  i – r it =  i +  i M t + e it – M t = random variable (mean 0) – Macro-wide shock – e t = random variable (mean 0) – Firm-specific shock unique to each firm – The actual, realized return is the sum of What we expect Macro Shock Firm-Specific Shock

Diversification and Many Risky Assets Consider an equally weighted portfolio of these returns. Suppose we have n assets in portfolio Weight in each asset is 1/n As n gets large The variance of e does not matter.

Diversification In general, as the number of assets increases in a portfolio, firm-specific shocks are washed out. n Unsystematic risk What happens if there are no macro-shocks?

Diversification and Many Risky Assets Since returns can be written as The variance of the return is Systematic Risk Unsystematic Risk