## Risk And Risk Premiums

Risk means uncertainty about future rates of return. We can quantify that uncertainty using probability distributions.

For example, suppose you are considering investing some of your money, now all invested in a bank account, in a stock market index fund. The price of a share in the fund is currently $100, and your time horizon is one year. You expect the cash dividend during the year to be $4, so your expected dividend yield (dividends earned per dollar invested) is 4%.

Your total holding-period return (HPR) will depend on the price you expect to prevail one year from now. Suppose your best guess is that it will be $110 per share. Then your capital gain will be $10 and your HPR will be 14%. The definition of the holding-period return in this context is capital gain income plus dividend income per dollar invested in the stock at the start of the period:

HpR = Ending price of a share - Beginning price + Cash dividend

Beginning price

In our case we have

This definition of the HPR assumes the dividend is paid at the end of the holding period. To the extent that dividends are received earlier, the HPR ignores reinvestment income between the receipt of the payment and the end of the holding period. Recall also that the percent return from dividends is called the dividend yield, and so the dividend yield plus the capital gains yield equals the HPR.

There is considerable uncertainty about the price of a share a year from now, however, so you cannot be sure about your eventual HPR. We can try to quantify our beliefs about the state of the economy and the stock market in terms of three possible scenarios with probabilities as presented in Table 5.1.

How can we evaluate this probability distribution? Throughout this book we will characterize probability distributions of rates of return in terms of their expected or mean return, E(r), and their standard deviation, ct. The expected rate of return is a probability-weighted average of the rates of return in each scenario. Calling p(s) the probability of each scenario and r(s) the HPR in each scenario, where scenarios are labeled or "indexed" by the variable s, we may write the expected return as

CHAPTER 5 History of Interest Rates and Risk Premiums 137

Table 5.1 Probability Distribution of HPR on the Stock Market

CHAPTER 5 History of Interest Rates and Risk Premiums 137

State of the |
Ending | ||

Economy |
Probability |
Price |
HPR |

Boom |
.25 |
$140 |
44% |

Normal growth |
.50 |
110 |
14 |

Recession |
.25 |
80 |
-16 |

Applying this formula to the data in Table 5.1, we find that the expected rate of return on the index fund is

E(r) = (.25 X 44%) + (.5 X 14%) + [.25 X (-16%)] = 14%

The standard deviation of the rate of return (ct) is a measure of risk. It is defined as the square root of the variance, which in turn is the expected value of the squared deviations from the expected return. The higher the volatility in outcomes, the higher will be the average value of these squared deviations. Therefore, variance and standard deviation measure the uncertainty of outcomes. Symbolically, ct 2 =£ p(s) [r(s) - E(r)]2 (5.2)

Therefore, in our example, ct2 = .25(44 - 14)2 + .5(14 - 14)2 + .25(-16 - 14)2 = 450

Clearly, what would trouble potential investors in the index fund is the downside risk of a -16% rate of return, not the upside potential of a 44% rate of return. The standard deviation of the rate of return does not distinguish between these two; it treats both simply as deviations from the mean. As long as the probability distribution is more or less symmetric about the mean, ct is an adequate measure of risk. In the special case where we can assume that the probability distribution is normal—represented by the well-known bell-shaped curve—E(r) and ct are perfectly adequate to characterize the distribution.

Getting back to the example, how much, if anything, should you invest in the index fund? First, you must ask how much of an expected reward is offered for the risk involved in investing money in stocks.

We measure the reward as the difference between the expected HPR on the index stock fund and the risk-free rate, that is, the rate you can earn by leaving money in risk-free assets such as T-bills, money market funds, or the bank. We call this difference the risk premium on common stocks. If the risk-free rate in the example is 6% per year, and the expected index fund return is 14%, then the risk premium on stocks is 8% per year. The difference in any particular period between the actual rate of return on a risky asset and the risk-free rate is called excess return. Therefore, the risk premium is the expected excess return.

The degree to which investors are willing to commit funds to stocks depends on risk aversion. Financial analysts generally assume investors are risk averse in the sense that, if the risk premium were zero, people would not be willing to invest any money in stocks. In theory, then, there must always be a positive risk premium on stocks in order to induce risk-averse investors to hold the existing supply of stocks instead of placing all their money in risk-free assets.

Although this sample scenario analysis illustrates the concepts behind the quantification of risk and return, you may still wonder how to get a more realistic estimate of E(r) and ct for common stocks and other types of securities. Here history has insights to offer.

138 PART I Introduction

Small |
Large |
Long-Term |
Intermediate- | |||

Year |
Stocks |
Stocks |
T-Bonds |
Term T-Bonds |
T-Bills |
Inflation |

1926 |
-8.91 |
12.21 |
4.54 |
4.96 |
3.19 |
-1.12 |

1927 |
32.23 |
35.99 |
8.11 |
3.34 |
3.12 |
-2.26 |

1928 |
45.02 |
39.29 |
-0.93 |
0.96 |
3.21 |
-1.16 |

1929 |
-50.81 |
-7.66 |
4.41 |
5.89 |
4.74 |
0.58 |

1930 |
-45.69 |
-25.90 |
6.22 |
5.51 |
2.35 |
-6.40 |

1931 |
-49.17 |
-45.56 |
-5.31 |
-5.81 |
0.96 |
-9.32 |

1932 |
10.95 |
-9.14 |
11.89 |
8.44 |
1.16 |
-10.27 |

1933 |
187.82 |
54.56 |
1.03 |
0.35 |
0.07 |
0.76 |

1934 |
25.13 |
-2.32 |
10.15 |
9.00 |
0.60 |
1.52 |

1935 |
68.44 |
45.67 |
4.98 |
7.01 |
-1.59 |
2.99 |

1936 |
84.47 |
33.55 |
6.52 |
3.77 |
- 0.95 |
1.45 |

1937 |
-52.71 |
-36.03 |
0.43 |
1.56 |
0.35 |
2.86 |

1938 |
24.69 |
29.42 |
5.25 |
5.64 |
0.09 |
-2.78 |

1939 |
-0.10 |
-1.06 |
5.90 |
4.52 |
0.02 |
0.00 |

1940 |
-11.81 |
-9.65 |
6.54 |
2.03 |
0.00 |
0.71 |

1941 |
-13.08 |
-11.20 |
0.99 |
-0.59 |
0.06 |
9.93 |

1942 |
51.01 |
20.80 |
5.39 |
1.81 |
0.26 |
9.03 |

1943 |
99.79 |
26.54 |
4.87 |
2.78 |
0.35 |
2.96 |

1944 |
60.53 |
20.96 |
3.59 |
1.98 |
- 0.07 |
2.30 |

1945 |
82.24 |
36.11 |
6.84 |
3.60 |
0.33 |
2.25 |

1946 |
-12.80 |
-9.26 |
0.15 |
0.69 |
0.37 |
1 8.13 |

1947 |
-3.09 |
4.88 |
-1.19 |
0.32 |
0.50 |
8.84 |

1948 |
-6.15 |
5.29 |
3.07 |
2.21 |
0.81 |
2.99 |

1949 |
21.56 |
18.24 |
6.03 |
2.22 |
1.10 |
- 2.07 |

1950 |
45.48 |
32.68 |
-0.96 |
0.25 |
1.20 |
5.93 |

1951 |
9.41 |
23.47 |
-1.95 |
0.36 |
1.49 |
6.00 |

1952 |
6.36 |
18.91 |
1.93 |
1.63 |
1.66 |
0.75 |

1953 |
-5.68 |
-1.74 |
3.83 |
3.63 |
1.82 |
0.75 |

1954 |
65.13 |
52.55 |
4.88 |
1.73 |
0.86 |
- 0.74 |

1955 |
21.84 |
31.44 |
-1.34 |
-0.52 |
1 .57 |
0.37 |

1956 |
3.82 |
6.45 |
-5.12 |
-0.90 |
2.46 |
2.99 |

1957 |
-15.03 |
-11.14 |
9.46 |
7.84 |
3.14 |
2.90 |

1958 |
70.63 |
43.78 |
-3.71 |
-1.29 |
1.54 |
1.76 |

1959 |
17.82 |
12.95 |
-3.55 |
-1.26 |
2.95 |
1.73 |

1960 |
-5.16 |
0.19 |
13.78 |
11.98 |
2.66 |
1.36 |

1961 |
30.48 |
27.63 |
0.19 |
2.23 |
2.13 |
0.67 |

1962 |
-16.41 |
-8.79 |
6.81 |
7.38 |
2.72 |
1.33 |

1963 |
12.20 |
22.63 |
-0.49 |
1.79 |
3.12 |
1.64 |

1964 |
18.75 |
16.67 |
4.51 |
4.45 |
3.54 |
0.97 |

1965 |
37.67 |
12.50 |
-0.27 |
1.27 |
3.94 |
1.92 |

1966 |
-8.08 |
-10.25 |
3.70 |
5.14 |
4.77 |
3.46 |

1967 |
103.39 |
24.11 |
-7.41 |
0.16 |
4.24 |
3.04 |

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