Finance 3310

Lecture 11

Rates of return & Risk

1. Rates of return

2. Statistics

3. Capital market history

4. Efficient Markets Hypothesis

5. Risk Premiums - Equity

6. Risk Premiums - Bonds

 

Overview: We have now finished applying time value of money analysis to the valuation of bonds, stocks, and capital budgeting projects. In all these examples we always took the 'discount rate' (R) as given. When valuing bonds we referred to this rate as the 'required return on bonds.' When valuing stocks we called this rate the 'required return on equity' and in the context of capital budgeting, we called this rate the 'cost of capital.' We now want to discuss where 'required returns' come from. Previously in class we said that required returns were a function of risk. Before being more specific, it will be useful to look at capital market history and how risk and return have been related historically.

1. Returns

By this time everyone should be very familiar with, if not tired of, the following formula:

1 + R = FV/PV

In the context of an investment, PV represents what we paid for the investment (stock price or bond price) and FV represents the dollar return. If we are talking about bonds, then FV represents the interest you receive at the end of the period plus the value of the bond at the end of the period. If we are discussing stocks, then FV represents the dividend you receive at the end of the period plus the value of the stock at the end of the period. (See T12.3)

R represents the (nominal) rate of return.

To put rates of return in perspective, it will now be useful to examine some historical average rates of return. (Note: These are arithematic averages, not geometric averages. Our formula for time value of money calculates geometric averages.) Before reviewing this capital market history it might be helpful to review a couple of basic statistics.

2. Statistics: Recall that a variable may either be stochastic (random) or deterministic (known). Stochastic variables are governed by probability distributions. These distributions may be either discrete or continuous. Because we do not know the price of a stock or bond next period, FV is random and therefore rates of return are random. Because the variables are random we cannot talk about their true values, but rather about what we expect the values to be. Expectations are simply weighted averages of probabilities and something else. Two particular expectations are useful. The mean is the expected value of the variable itself. The variance is the expected deviation from the mean squared. If we are using a sample to estimate the true, but unknown, mean and variance of a distribution then:

= and

The standard deviation is the square root of the variance. We will describe historical returns below using the average and standard deviation of returns. Standard deviation is one measure of risk. Recall that we defined risk to be the chance that actual returns turn out to be much different than what was expected. The greater the standard deviation, the greater this chance.

3. Capital Market History:

T12.5 - T12.15 contain historical returns from the period 1926 - 1996. These transparencies contain much information about historicl returns. They indicate the value in 1996 of $1 invested in 1926. They indicate returns by year for different financial assets. They also indicate the average risk premiums. The risk premium is the difference between the asset's return and the return on Treasury Bills. The nominal return on Treasury Bills is risk-free because we know both PV and FV. (We assume the government will not default so that there is no uncertainty about FV.) Below are the historical returns from 1926 - 1996:

Asset Return Premium Stand. Dev.
Small Stocks 17.7% 13.9% 34.1%
Common Stocks 12.7% 8.9% 20.3%
L-T Corp. Bonds 6.0% 2.2% 8.7%
L-T. Govt. Bonds 5.4% 1.6% 9.2%
U.S. Treas. Bills 3.8% 0 3.3%
Inflation 3.2%   4.5%

 

What 'lessons' do the above data tell us?

As an example, look at small stocks. Small stocks have the highest risk as measured by standard deviation, but also have the highest risk premium. Bonds have much lower risk premiums and lower risk.

 

4. Efficient Market Hypothesis

There are many different ways to characterize an 'efficient' market. Among these are:

• The market is always in equilibrium (or moving there)

==> supply = demand

==> expected returns = required returns

• There are no apparent bargains (some people beat the market some of the time)

• Security prices reflect available information

A statement about how much information is reflected in stock prices. First of all, we should discuss what we mean by 'information' and 'reflected in price.' Information is anything of fundamental importance to an investor. Would knowledge of the something in question influence your investment decision about an asset? If I sell thousands of shares of stock because I want to buy a new house, would that sale influence the average investor? Contrast that sale with a sale of thousands of shares because I think the company is about to be sued for product liability.

To say the information is reflected in price is to say that someone has traded based on the information, and that price has adjusted to that trade. If I have good news about a company and go out and buy a lot of shares, then the price should rise. Once the price has risen it 'reflects' my information. Market efficiency implies that the market reacts quickly and in the correct direction to new information.

There are three 'forms' of the hypothesis that relate to the set of information that is reflected in price.

The Efficient Market Hypothesis doesn't say the market is right, only unbiased.

Summary: An efficient market is one in which prices fully reflect available information. This hypothesis has been tested more than any other in the social sciences. Efficiency is a matter of degree. Tests indicate that prices do not reflect insider information, but that most other informaation is reflected in price. Note that not all information will be reflected in price even in an efficient market if there are transaction costs. If I have information that is 'worth' $0.50 per share, but it is going to cost me $0.60 per share to trade, then I will not trade and my information will not be reflected in price.

Expected Returns and Risk:

Earlier we said that because future values are unknown (stochastic), returns are also uncertain. When we are dealing with uncertainty we have to talk in terms of 'expected' returns. All expectations are just weighted averages of probabilities and something else. (As discussed above, the mean is the expected value of the variable itself; the variance is the expected deviation from the mean squared.)

We have thus far discussed individual assets. Most investors actually own portfolios of assets. Just as we can discuss the expected return and variance of an individual asset, we can discuss the expected return and variance of a portfolio. The expected return for a portfolio is simply a weighted average of all the individual expected returns. More specifically,

 

Rp = Summation (Wi x Ri) , where:

 

Wi = % of portfolio invested in asset i

Ri = expected return on asset i

 

Example: Suppose you have the following portfolio:

 

Asset Amount Invested Wi Expected Return Wi x Ri
X $40,000 40% 12% 4.8%
Y 35,000 35% 10% 3.5%
Z 25,000 25% 20% 5.0%
Portfolio $100,000 100%   13.3%

Thus the portfolio expected return is 13.3%. The portfolio variance and standard deviation can be calculated in the same way as for individual assets if we know all the possible portfolio returns and probabilities.

Equity Risk:

We now want to discuss risk as it relates to an investment in stocks (equity). Consider buying an individual stock. Your actual return on the stock will consist of two parts: what you expected and what was unexpected. The unexpected part is where the uncertainty (risk) is.

More formally we can write:

Re = E(R) + Surprise

The 'surprise' is the unexpected part. (Otherwise it wouldn't be a surprise would it?) The surprise or unexpected part of your return arises from two sources. One source is movement in the overall market. Another source involves events that are specific to the company in question. We call this first source of risk systematic risk. The second part is called unsystematic risk. Se we can write:

Surprise = systematic part + nonsystematic part

Systematic - uncertainty from changes in the overall market; also called 'market' risk.

Unsystematic - firm specific, unique surprise. For example, invention of a new product by a company; a lawsuit against a company, or some event that affects only one company. (or at most a few)

 

Investors cannot do anything about systematic or market risk, other than decide not to invest in stocks. Unsystematic risk can, however, be virtually eliminated by holding a diversfied portfolio. The unsystematic part is like a random variable with an expected value of 0. Any one 'realization' (ex: one stock) may be positive or negative, but if we have enough realizations (ex: several stocks) then the positive and negative will tend to average out. (This has to do with the law of large numbers from statistics.) The bottome line is that investors can eliminate unystematic risk from the their portfolio by holding several stocks, that is, by diversifying. (by not 'putting all their eggs in one basket') See Figure 13.1 in your text, which shows how your portfolio standard deviation declines as you add more stocks, until you are left with nondiversifiable risk. (market risk)

Since investors can eliminate unsystematic risk on their own, they should not expect to be rewarded for it.

Systematic risk is market risk, or that risk which cannot be eliminated (GNP, inflation, R, etc.). Since investors cannot eliminate this risk, they are rewarded for it. The extent of the the reward depends on how sensitive to overall market movements a particular stock is. So we need some way to measure how sensitive a security's returns are to overall market movements. This measure is called beta (B). Beta measures the risk of a security relative to the market. Some facts about beta:

We can now write down the expected return for a stock as a function of risk as follows:

 

E(Ri) = Rf + Bi[E(Rm) - Rf]

where:

Ri = expected return on security i

Rf = risk-free rate

Bi = beta of security i

Rm = expected return on the market

The above model is known as the Capital Asset Pricing Model. In words, the expected (required) return on a stock is equal to the risk-free rate (Rf) plus a risk premium - Bx(Rm - Rf). B measures the sensitivity to the overall market. Note that Rf is the nominal risk-free rate, that is it includes the real rate plus expected inflation.

 

Example:

 

Suppose Rf = 7% ; B=1.2, E(Rm) = 15%; ----- What is the expected return on the stock?

E(Ri) = 7% + 1.2(15% - 7%) = 16.6%

 

Finally, we can also apply the capital asset pricing model to portfolios by substituting the portfolio beta into the above equation. The beta of a portfolio is a weighted average of all the individual betas.


Bonds:

We have already discussed risk with respect to bonds. A brief review of these risks follows:

Default risk - is the risk that the issuer defaults on some or all of the bond's promised payments.

Liquidity risk - is the risk that the bond cannot be liquidated quickly w/o loss of value.

Maturity risk - longer term bonds have longer durations & greater interest rate risk. Also, investors may require a higher return to induce them to invest longer term.

In general, bonds without default risk face interest rate risk, a measure of which is duration. This risk not rewarded because investors can change the duration to eliminate it.


Summary:

Rates of return consist of the following components:

 

R = r + I + RP

Where:

r = real rate

I = expected inflation

RP = risk premium

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