The standard deviation of the annual return is therefore
Investment Opportunities
Suppose we choose to characterize every investment opportunity by its expected return and standard deviation of return. We can plot available risky investments on a chart such as Figure 1.1 where the horizontal axis is the standard deviation of the return and the vertical axis is the expected return.
FIGURE 1.1 Alternative Risky Investments
Once we have identified the expected return and the standard deviation of the return for individual investments, it is natural to think about what happens when we combine investments to form a portfolio. Consider two investments with returns R1 and R2. The return from putting a proportion w1 of our money in the first investment and a proportion w2 = 1 − w1 in the second investment is
The portfolio expected return is
(1.1)
where μ1 is the expected return from the first investment and μ2 is the expected return from the second investment. The standard deviation of the portfolio return is given by
(1.2)
where σ1 and σ2 are the standard deviations of R1 and R2 and ρ is the coefficient of correlation between the two.
Suppose that μ1 is 10 % per annum and σ1 is 16 % per annum, while μ2 is 15 % per annum and σ2 is 24 % per annum. Suppose also that the coefficient of correlation, ρ, between the returns is 0.2 or 20 %. Table 1.2 shows the values of μP and σP for a number of different values of w1 and w2. The calculations show that by putting part of your money in the first investment and part in the second investment a wide range of risk-return combinations can be achieved. These are plotted in Figure 1.2.
TABLE 1.2 Expected Return, μP, and Standard Deviation of Return, σP, from a Portfolio Consisting of Two Investments
The expected returns from the investments are 10 % and 15 %; the standard deviation of the returns are 16 % and 24 %; and the correlation between returns is 0.2.
FIGURE 1.2 Alternative Risk-Return Combinations from Two Investments (as Calculated in Table 1.2)
Most investors are risk-averse. They want to increase expected return while reducing the standard deviation of return. This means that they want to move as far as they can in a “northwest” direction in Figures 1.1 and 1.2. Figure 1.2 shows that forming a portfolio of the two investments we have been considering helps them do this. For example, by putting 60 % in the first investment and 40 % in the second, a portfolio with an expected return of 12 % and a standard deviation of return equal to 14.87 % is obtained. This is an improvement over the risk-return trade-off for the first investment. (The expected return is 2 % higher and the standard deviation of the return is 1.13 % lower.)
1.2 THE EFFICIENT FRONTIER
Let us now bring a third investment into our analysis. The third investment can be combined with any combination of the first two investments to produce new risk-return combinations. This enables us to move further in the northwest direction. We can then add a fourth investment. This can be combined with any combination of the first three investments to produce yet more investment opportunities. As we continue this process, considering every possible portfolio of the available risky investments, we obtain what is known as an efficient frontier. This represents the limit of how far we can move in a northwest direction and is illustrated in Figure 1.3. There is no investment that dominates a point on the efficient frontier in the sense that it has both a higher expected return and a lower standard deviation of return. The area southeast of the efficient frontier represents the set of all investments that are possible. For any point in this area that is not on the efficient frontier, there is a point on the efficient frontier that has a higher expected return and lower standard deviation of return.
FIGURE 1.3 Efficient Frontier Obtainable from Risky Investments
In Figure 1.3 we have considered only risky investments. What does the efficient frontier of all possible investments look like? Specifically, what happens when we include the risk-free investment? Suppose that the risk-free investment yields a return of RF. In Figure 1.4 we have denoted the risk-free investment by point F and drawn a tangent from point F to the efficient frontier of risky investments that was developed in Figure 1.3. M is the point of tangency. As we will now show, the line FJ is our new efficient frontier.
FIGURE 1.4 The Efficient Frontier of All Investments
Point I is achieved by investing a percentage βI of available funds in portfolio M and the rest in a risk-free investment. Point J is achieved by borrowing βJ − 1 of available funds at the risk-free rate and investing everything in portfolio M.
Consider what happens when we form an investment I by putting βI of the funds we have available for investment in the risky portfolio, M, and 1 − βI in the risk-free investment F (0 < βI < 1). From equation (1.1) the expected return from the investment, E(RI), is given by
and from equation (1.2), because the risk-free investment has zero standard deviation, the return RI has standard deviation
where σM is the standard deviation of return for portfolio M. This risk-return combination corresponds to the point labeled I in Figure 1.4. From the perspective of both expected return and standard deviation of return, point I is βI of the way from F to M.
All points on the line FM can be obtained by choosing a suitable combination of the investment represented by point F and the investment represented by point M. The points on this line dominate all the points on the previous efficient frontier because they give a better risk-return combination. The straight line FM is therefore part of the new efficient frontier.
If we make the simplifying assumption that we can borrow at the risk-free rate of RF as well as invest at that rate, we can create investments that are on the continuation of FM beyond M. Suppose, for example, that we want to create the investment represented by the