Calculating a Portfolio's Expected Return

There are two key __properties of expected values__ which are very useful. The first property of expected value is that if you multiply a random variable, $$X_i$$, by a constant, $$w_i$$, the expected value is given by: $$\displaystyle E(w_iX_i) = w_iE(X)$$. For example, suppose a weight of 30% multiplied by a bunch of observations leads to an expected value of 2.5. If you then double the value of each observation, what happens to the expected value?

In general, if you have $$i=1$$ to $$n$$, $$w_i$$ constants and $$R_i$$ random variables, then $$E(w_1R_1 + w_2R_2 + ... + w_nR_n)$$ > $$= w_1E(R_1) + w_2E(R_2) + ... + w_nE(R_n)$$ So maybe $$R_1$$ is the return on stocks, $$R_2$$ the return on bonds, $$R_3$$_ the return on money market accounts, and $$w_1$$, $$w_2$$, and $$w_3$$ denote the respective weights on each portfolio. Then the expected return on the portfolio is $$\displaystyle E(R) = w_1E(R_1) + w_2E(R_2) + w_3E(R_3) $$ Use as many assets as you'd like.

Suppose the current market values of Mr. Nader's investments are: Money market fund: EUR 22,000 Blue chip stocks: EUR 52,000 Stock mutual fund: EUR 84,000 Bond fund: EUR 42,000 Mr. Nader expects to earn a return of 2.4% on money market funds, 6.8% on blue chip stocks, 8.1% on stock mutual fund, and 3.2% on his bond fund. What is the expected return on his portfolio?

Incorrect. Consider the calculations step by step. First, you need to calculate the weights on each of the asset classes in Mr. Nader's portfolio. The total market value of Mr. Nader's investments is EUR 200,000. So, 11% of his investments are in money market funds, 26% in blue chip stocks, 42% in stock mutual funds, and 21% in bond funds. Use those percentages to derive the weights needed.

Correct. The first step is to calculate the weights on each of Mr. Nader's investments in his portfolio, dividing each investment value by the EUR 200,000 invested in total: | Asset Class | Value Invested (EUR) | Weight | Expected Return | |---|---|---|---| | Money Market | 22,000 | 11% | 2.4% | | Blue Chip Stocks | 52,000 | 26% | 6.8% | | Stock Mutual Fund | 84,000 | 42% | 8.1% | | Bond Fund | 42,000 | 21% | 3.2% | Then the second property of expected value allows you to calculate the expected return: $$ E(R) = 0.11 \times 2.4 \% + 0.26 \times 6.8 \% + 0.42 \times 8.1\% + 0.21 \times 3.2 \% $$ > $$= 6.106 \% $$.

Incorrect. This is just the unweighted average. You need to consider how much Mr. Nader has invested in each asset class, calculate weights that sum to 1.0, and then apply one of the properties of expected value.

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The second property of expected value can be stated as: Let $$w_i$$ be any constant and $$X_i$$ be a random variable. Then: $$E(w_1X_1 + w_2X_2 + ... + w_nX_n) $$ > $$= w_1E(X_1) + w_2E(X_2) + ... + w_nE(X_n)$$ This is where it gets interesting as far as calculating the expected return on a portfolio, which is essentially a weighted sum of returns on a number of assets. For example, imagine a bond portfolio that returns 5%, and a stock portfolio that returns 8%. If you invested part of your money in each of these two, what sort of expected return would be possible?

No, that would be the case if you multiplied each observation by 1.3 rather than 2.

Not quite; this is a combination of the weight and the factor of 2, but just consider a simple example where every observation doubles.

Yes, that's the idea. Double the observations, you double the expected value. The weight can be applied to each individual observation or the sum at the end; either way.

No, that would only be the case if all of your money was invested in the bond portfolio.

Exactly! An expected return of 5% would require a 100% investment in bonds, and a 9% return isn't possible with these two. But an expected 7% return is possible by investing 2/3 of your money in stocks: $$\displaystyle \frac{1}{3} \times 5 \% + \frac{2}{3} \times 8 \% = 7 \% $$

No, it wouldn't be possible to get a return this high with these two offerings.

It increases by 30%

It increases by 60%

It increases by 100%

5.125%

5.333%

6.106%

5%

7%

9%

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