This post is a follow up to this post about choosing between mutually exclusive investments by simultaneously considering return and risk using the Sharpe ratio. This time, we will consider investing in two securities simultaneously and learn how to find the mix that leads to the best return for a given level of risk.
For "fairness," this time we will use the historical daily NAVPUs of BPI's equity and fixed income (bond) funds (which you can download here), which I copied from UITF.com.ph. I used the NAVPUs from the same period that we used in Part 1 (January 2 to August 24, 2012) so that you can compare the funds provided by the two banks in terms of expected return and risk, if you want to. Applying the procedure described in Part 1 will result in the following statistics for the BPI data (please note that the BDO counterparts of these BPI UITFs both have higher expected returns and lower risk for the same period, so make of that what you will). In this example, as with the BDO UITFs, the BPI fixed income fund is theoretically preferable than the equity fund because of the Sharpe ratio, despite having a lower expected return.
Equity fund
|
Fixed income fund
|
|
Expected return (daily)
|
0.0743%
|
0.0155%
|
Risk (standard deviation)
|
1.7260%
|
0.2814%
|
Sharpe ratio
|
0.04305
|
0.05524
|
But what if we can invest in both funds (i.e., BPI equity and fixed income funds). What mix of these two UITFs will give us the highest expected return for a given level of risk?
The Efficient Portfolio Frontier
Say X1 is the proportion (or percentage) of your "investable" funds or portfolio that is invested in the equity fund and X2 is the proportion that is invested in the fixed income fund. Additionally, say the equity fund and the fixed income fund have expected returns of R1 and R2, respectively, and standard deviations (risk) of S1 and S2, respectively. Then, the expected return of your portfolio, R, is just the weighted average of R1 and R2, or
R = X1(R1) + X2(R2)
And the variance (or standard deviation squared) of your portfolio, S^2, is given by
S^2 = (X1^2)(S1^2) + (X2^2)(S2^2) + 2(X1)(X2)(S12)
Where S12 is the covariance between the two funds (applying the COVAR function in Excel to our data, we get 0.3442%^2).
With X1 + X2 = 1 (100% of your portfolio is invested in the two funds), we can express X1 and X2 in terms of R, R1, and R2
X1 = (R - R2)/(R1 - R2)
X2 = (R - R1)/(R2 - R1)
And substitute these into the equation for S^2
S^2 = {[(R - R2)/(R1 - R2)]^2}(S1^2) + {[(R - R1)/(R2 - R1)]^2}(S2^2) + 2[(R - R2)(R1 - R)/(R1 - R2)^2](S12)
Which we can simplify into:
[(R1 - R2)^2]S^2 = (S1^2 + S2^2 - 2S12)R^2 + 2[(R1 + R2)S12 - R2S1^2 - R1S2^2]R + (R2^2)(S1^2) + (R1^2)(S2^2) - 2R1R2S12
or
(R1 - R2)S = sqrt {(S1^2 + S2^2 - 2S12)R^2 + 2[(R1 + R2)S12 - R2S1^2 - R1S2^2]R + (R2^2)(S1^2) + (R1^2)(S2^2) - 2R1R2S12}
Using the following values that we have already computed,
R1 = 0.0743
R2 = 0.0155
S1 = 1.726
S2 = 0.2814
S12 = 0.3442
We get the equation
0.0588S = sqrt (2.37R^2 - 0.04254R + 0.000362)
We can use the free application "Graph" to plot this equation (with R on the vertical axis and S on the horizontal axis) and get this hyperbolic (green) curve which is called the Efficient Portfolio Frontier:
Points on the Efficient Portfolio Frontier represent different combinations of our two assets (BPI's equity and fixed income UITFs). If we assume that investors don't like standard deviation (everything else equal), then they should only consider portfolios above and to the right of the "minimum variance portfolio" (at R = 0.008982 and S = 0.2226, which we get by differentiating S with respect to R and setting dS/dR to 0).
At this point we still face an infinite number of combinations of the two funds, and the choice of a particular mix ultimately rests on the investor's expectation for returns and attitude towards risk. In Part 3, I will show that if we include another factor in our analysis, we can come up with a truly optimal portfolio that all investors should choose.