Risk and Rates of Return
1. RATES OF RETURN: A REVIEW
A. Security return in $ terms, either stock or bonds, is a combination of
dividend or interest payments plus any capital gain or loss.
B. The annual percentage return on investment is:
Percentage Return = [capital gain (loss) + dividend or interest] / initial
share or bond price
For stocks, the return is the sum of the dividend yield and the capital
gains yield.
C. The above return is a nominal return, reflecting how much more money
one has at the end of the year.
D. The real rate of return is the nominal rate adjusted for the inflation rate in
the period or the additional purchasing power one has with the investment
return:
1 + real rate of return = 1 + nominal rate of return
1 + inflation rate
2. The Historical Record
A. The historical returns of Treasury bills, long-term Treasury bonds,
corporate bonds, and common stock are compared in Figure below.
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B. With Treasury bills’ average returns at the low end of the risk scale, a
maturity premium is added for long-term Treasury bond returns.
C. The return differentials between risk-free Treasury bills and corporate
bonds and common stock returns is a risk premium, or the added return
required by investors to invest in risky securities.
D. The riskier securities had wider fluctuations in their yearly returns over the
73 years studied.
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MEASURING RISK
A. Variation around a central tendency or mean may be presented visually by
constructing a histogram and studying the dispersion or spread of possible
outcomes.
B. Another method is calculating a measure of variation used as a proxy for
measuring risk, such as the variance or standard deviation. Risk relates
to the variability of future returns.
Variance and Standard Deviation
A. The variance statistic is the average value of squared deviations from the
mean. The standard deviation is the square root of the variance.
B. The greater the variance or standard deviation, the greater the dispersion,
volatility, or variability of returns, and the greater the risk. See class
example.
Measuring the Variation in Stock Returns
A. Calculating historical average investment returns and the variability of
those returns, the comparison of average returns and volatility indicates
that historical risk and return are directly related.
B. Higher risk is associated with higher average returns.
C. Investors will expect a higher rate of return, risk premium over the
Treasury bill rate, with higher standard deviation of returns.
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RISK AND DIVERSIFICATION
A. Our measures of variation apply to groupings of securities or portfolios as
well as single securities.
B. The variability or risk of a portfolio, or a market portfolio such as the S&P
500, is not the simple average of the individual stock variability. The
portfolio risk is less than the average risk of the individual securities.
Diversification
A. The reduced risk of the portfolio is caused by diversification effects of
spreading the portfolio across many investments.
B. Portfolio diversification works because prices of different stocks do not
move exactly together or are not perfectly positively correlated (r = +1).
Asset Risk vs. Portfolio Risk
A. While historical returns on individual securities are good proxies for
estimating future returns on individual securities, historical standard
deviations of returns are not good risk proxies for stocks held in a
portfolio.
B. Since stocks and other securities are usually held in portfolios, because of
the diversification effect, the relevant risk of a stock to be added to the
portfolio is the incremental risk of the added stock on the portfolio risk.
C. Incremental risk does not depend upon the added stock’s variability; it
depends upon how the stock affects the portfolio’s variability when it is
added to the portfolio.
Market Risk vs. Unique Risk
A. The diversification effect or the reduction in portfolio risk takes place with
the addition of added securities until about 20 or 30 are included in the
portfolio. Beyond that, the diversification effect of added securities is
minimal.
B. While diversification eliminates the unique risk of individual securities, it
cannot eliminate the market risk or systematic risk, the risks that affect
the entire stock market.
C. For a diversified portfolio, only the market risk matters. When one
discusses securities investment, it is assumed that the security is held in a
diversified portfolio and the relevant risk is market risk.
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MEASURING MARKET RISK (What’s Beta?)
Market Portfolio: Conceptually, the more stocks we add into a portfolio,
the more risk we can diversify away. The portfolio’s risk will keep
reducing until there is no more stock to add. This final portfolio is called
market portfolio that consists of all stocks available.
Firm specific or unique risk are averaged out or diversified away when
considering the market portfolio. Market portfolio only has market risk.
A broad market base, such as the S&P 500 Composite, is often used as a
proxy for the market portfolio.
Measuring Beta
Beta is the sensitivity of a stock’s (or portfolio’s) return to the return on the
market portfolio. It is expressed as the Greek letter, .
To be more specific, beta is the slope of the regression line of the
individual stock returns relative to the market portfolio returns.
A. Investors with diversified portfolios are not concerned about the specific or
unique risk of a stock, only the impact of the stock on the risk of the entire
portfolio.
B. Beta is the relevant risk to consider when a new stock is added to the
portfolio.
Examples:
Defensive stocks: Beta < 1.
If the variation in the stock return, given a 1 percent variation in the market
returns, is less than the market variation, the beta is less than 1 and the
stock is noted as a “defensive” stock.
Aggressive stocks: Beta >1 meaning?
Portfolio Beta
port = W 11 + W 22 + … + W nn. (W i is the weight of stock i in portfolio.)
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RISK AND RETURN (What’s CAPM?)
1)
The Rationale behind CAPM:
The most popular model to price risk is Capital Asset Pricing Model
(CAPM). The rationale behind this theory is that the required return of a
security is the risk-free rate (rrf) plus a risk premium that reflects only the
risk remaining after diversification (recall: market risk or systematic risk).
CAPM equation:
ri(p)= rrf + (rM-rrf) bi(p)
Explanation:
ri(p): is the required rate of return (compensation) for holding a security i or
a portfolio p.
rrf : nominal risk-free rate, usually we use T-bill rate.
rM: the market portfolio’s return, usually we use SP500 index’s return.
(rM-rrf): the market risk premium, same to all stocks.
bi(p): the sensitivity measure of a stock i (or a portfolio p)’s return to the
market risk premium.
(rM-rrf) bi(p): risk premium of stock i or portfolio p.
For example, if a stock is an “average stock”, then
bi=1, and
ri = rrf + (rM-rrf) bi = rrf + (rM-rrf) = rM.
This stock has the same return as the market portfolio does.
2)
Several Properties of CAPM:
(1) Under CAPM, only market risk of a security or portfolio matters. No
other risk is relevant or priced.
(2) bi is the measure of individual security risk in market portfolio. This
kind of risk is called market risk that cannot be diversified away. The
higher the bi is, the higher the risk an individual security brings into a
portfolio. Therefore, the market risk of a stock or portfolio is measured by
its beta coefficient.
For example:
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b=0: the security is risk free, i.e. T-securities.
b=0.5: the stock is only half as volatile, or risky, as the market portfolio.
b=1: the stock has the same risk as an average risk.
b=2: the stock is twice as risky as an average stock.
Note: If there are two securities, A and B, and b A are twice as much as
bB, then we may say that the risk premium for security A is twice as much
as security B. However, we cannot say that the required return for stock A
is twice as much as that of stock B. Why?
(3) Stocks with same variance (risk), may not have same beta value;
different stocks usually have different betas. Most betas are within the
range of 0 to 2.
3)
Does the CAPM Work?
A. The CAPM assumes well-diversified investors; market risk is the only
relevant risk.
B. The expected return on a portfolio is equal to the risk-free interest rate
plus the expected portfolio risk premium.
Both assumptions, unfortunately, cannot be held in reality.
4)
Using the CAPM to Estimate Expected Returns - Application
To estimate expected investor returns for a selected stock, 3 numbers are
needed:
(1) the risk-free rate of interest,
(2) the expected market risk premium, and
(3) the beta of the stock.
Because there are 4 unknowns in one equation, given any 3 numbers,
you should be able to calculate the last one.
Examples.
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