New York University



A Guide to the Interactive Risk/Return Trade-offOmbretta Pettinato and William L. Silber The Set-Up and Some Insights1) This discussion is a guide to the initial screen you will see when opening the accompanying Excel file on your computer. There are 2 risky assets, stock A (a mean return of 13% and a standard deviation of 6.4% in B4 and B5) and stock B (a mean return of 24% and a standard deviation of 13.6% in C4 and C5).2) Cell B6 contains the correlation coefficient between the two assets and allows you to change the value (any value between -1 and 1). In the initial set-up the correlation coefficienti is 1 but changing the value shows the effects on the Risk/Return Trade-off in the graph below the table.3) Columns B and C (lines 9-29) contain various combinations of the two securities. There are 21 possible portfolios shown in the table (out of an infinite number of theoretically small steps), beginning with a portfolio containing only the riskier asset, stock B (with a mean return of 24% and a standard deviation of 13.6%). All the other portfolios are constructed by changing the fraction allocated to each risky asset (their weights), namely by increasing the amount invested in the less risky asset (stock A), ending with a portofolio entirely composed of stock A (with a mean return of 13% and a standard deviation of 6.4%). 4) Column E (lines 9-29) shows the expected return of each two-asset portfolio given by the formula:Portfolio return = XA RA + XB RBFor example, consider the portfolio composed by XA= .5 (Cell B19) and XB = .5 (C19). We know from the formula that the portfolio return is:R = .5(.13)+.5(.24) = .185 (see Cell E19)5) Column F (lines 9-29) shows the standard deviation of each two-asset portfolio given by the formula:The standard deviation of the half-and-half portfolio is therefore equal to:σ = ((0.52 * .0642)+ (0.52 * .1362)+(2*0.5*0.5*1*.064*.136))1/2= 10% (see Cell F19)6) Column H (lines 9-29) shows the standard deviation of each two-asset portfolio given by the simple weighted average of the underlying standard deviation of the two assets. Recall that when ρ is equal to 1, the variance of a portfolio becomes a perfect square and therefore the standard deviation of the portfolio will be (only in that case) equal to the weighted average of the underlying standard deviations (just like the mean). In our example, the standard deviation of the portfolio is exactly half way between the two:σ = (0.5*.064 + 0.5*.136)= 10% (Cell H19)7) Finally, Column I (lines 9-29) is simply the difference between Column F and Column H. You can see that as long as ρ = 1, the σ of each portfolio is exactly equal to the weighted average of σA and σB so the difference between the two (Column I) is always zero.8) The graph Risk/Return Trade-off displayed in the bottom of the table plots all the combinations of risk (horizontal axis ) and return (vertical axis) that are available to an investor based on the steps of the table. Recall that the shape of the risk return trade-off depends on the correlation of returns (ρ) among the securities.Lessons on Correlations 9) When two assets are perfectly positively correlated (=1), increasing the amount of money invested in the riskier asset (moving from the bottom to the top of the table), you get a proportional increase of return as well as risk. In fact, the graph of the risk-return trade off looks like a straight line, therefore there are no gains from diversification (showed by the zeros in Column I).10) By changing the correlation coefficient in Cell B6, you can see the change in the risk-return trade-off graph, and also how the values reported in Column F diverge from the values computed in Column H. Namely, the difference between the two becomes always negative (Column I) suggesting that the standard deviation of a portfolio is always less than the simple average of the underlying standard deviations (σA and σB) when ρ < 1. Column I is labelled the Power of Covariance because it shows that as long as the two assets are not perfectly correlated, you always gain from diversification by adding a little bit of the riskier security. The nice thing about diversification is that it always produces gains to a portfolio in the form of increased return that exceeds the cost in terms of increased standard deviation except when the correlation of returns is equal to one. 11) Another way of thinking about this is to see that changing the correlation coefficient between the two assets does not change the portfolio return but a lower the correlation coefficient always produces a higher (absolute) difference between columns F and H and therefore you get more gains from diversification.12) Changing the correlation coefficient in Cell B6 changes the shape of the risk return trade-off. For a correlation equal to 1 the trade-off lies on a straight line and as the correlation is lowered below 1 the trade-off becomes bowed to the left and eventually bends backward (it happens for any value of ρ < σ1 /σ2 ). The backward bend means that it is possible to lower the risk and at the same time increase the expected return of your portfolio. Note that for a correlation coefficient equal to zero the curve is always backward bending.14) Finally, the standard deviation never never exceeds a weighted average of the two underlying standard deviations. The reason is that ρ can never be greater than one. Thus the largest possible standard deviation is a weighted average of σ1 and σ2.Extension15) It is interesting to observe what happens if the correlation between the two assets is perfectly negative (equal to -1). Recall that in this case the variance and the standard deviation of the portfolio become the following: σ2(Portfolio) = (X1 σ1 - X2 σ2)2 and σ (Portfolio) = |X1 σ1 - X2 σ2|. By setting that equation equal to zero and solving it for X1 you get the specific allocation to asset 1 and asset 2 which makes the risk equal to zero (and at the same time you get more return). In our example, the allocations for XA and XB that produce zero risk are equal to 68% and 32% respectively. Putting a value of -1 in Cell B6 and putting 68% and 32% into B22 and C22 and recalculating shows the risk-return graph touching the vertical axis for that allocation, confirming that the standard deviation is zero. ................
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