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RISK AND RETURN: PAST AND PROLOGUE
Rates of Return
Holding-period return (HPR)—the rate of return that is earned on an investment over a particular period of time.
Ending value Beginning value + Cash received HPR Beginning value
The equation provides the ex post (historical) return for an investment. If the investment is a stock, the return can be computed as follows:
1 0 1 1 0 1 Stock 0 0 0
(P P ) D (P P ) D (^) Capital gains Dividend HPR P P P yield^ yield
where P 1 is the per share value of the stock at the end of the investment period, P 0 is the per share value of the stock at the beginning of the investment period, and D 1 is the total cash dividend paid per share during the investment period. Although the investment period over which HPR is measured can be any length (e.g., one month, one quarter, one year, five years, and so forth), often HPR is computed for one year.
Measuring average returns over multiple periods—generally investors want to know the annual (or other period) rate of return that is earned during an investment period that covers a number of years (periods). Following are the methods used to determine the average return per period: o Arithmetic average (simple average)—computation of the arithmetic average does not consider the effects of interest/return compounding.
1 2 n a
Arithmetic r^ r^ r average return r n
where rt = the return that was earned in Period t and n is the total number of periods (months, quarters, years, etc.). o Geometric average (time-weighted return)—considers compounding.
n ^1 n g 1 2 n 1 2 n
Geometric (^) r (1 r )(1 r ) (1 r ) (1 r )(1 r ) (1 r ) average return
o Dollar-weighted average (internal rate of return, IRR)—considers variations in cash flows throughout the investment period.
1 2 n 1 2 n
Dollar weighted CF^ CF^ CF Initial cash flow average return (IRR) (^) (1 IRR) (1 IRR) (1 IRR)
where the Initial cash flow represents the amount originally invested and CFt = the cash flow that occurs in Period t; CFt can be positive (e.g., when a dividend is received and it is not reinvested in the stock) or it can be negative (e.g., when more funds are invested in the stock); CFn is the terminal value of the investment, which is the total value of the investment at the
end of the investment period.
Annualized returns
o Annual percentage rate, or APR—represents the simple, or non-compounded, interest earned during the year. APR = Rate per period x Number of periods in the year o Effective annual rate, or EAR—represents the compounded rate of return earned during the year.
APR ^1 n EAR 1 1. n
where n is the number of periods in the year (e.g., n =12 if monthly returns are used)
Inflation and the Real Rate of Interest
Inflation affects the purchasing power associated with the return that is earned on an investment: In general, R Real rate + i Real rate R – i where R is the nominal rate of return and i is the inflation rate during the investment period. More exactly, (1 + R) = (1 + Real rate)(1 + i) Real rate = (1 + R)/(1 + i) – 1 = (R – i)/(1 + i)
The equilibrium nominal rate of interest; Fisher equation:
R Real rate + E(i) E(i) is the expected inflation rate
Risk and Risk Premiums
Risk
o Definition—variability (both upside and downside) of returns; more than one possible outcome o Risk exists if there is a possibility the actual outcome could be something other than expected—the outcome could be better than expected or worse than expected.
Risk attitudes
o aversion—an investor requires (demand) higher returns (greater rewards) for taking greater risk o seeker—an investor requires lower returns for taking greater risk (will pay to take risk) o neutral—risk doesn't matter—that is, doesn’t care about risk; an investor considers return only; If all investors were considered truly risk neutral, eventually market mechanisms would cause all financial assets to yield the same expected return—the risk-free rate.
Asset Allocation across Risky and Risk-Free Portfolios
Asset allocation—proportion of total funds invested in different types of securities.
o Capital allocation to risky assets—percentage of total funds invested in securities that are considered risky; also the proportion of “risky funds” that is invested in each type of risky security. o Complete portfolio—combination of risky assets and “risk-free” assets
Risk-free asset—an investment with a guaranteed outcome; Treasury inflation-protected securities (TIPS), CDs, and other money market generally are used as proxies for risk-free assets.
Capital allocation line (CAL)—assume that you are considering investing funds in two types of investments: (1) a risk-free asset and (2) a portfolio of risky assets. The relationship between return and risk for various combinations (allocations) of these two assets might be depicted as follows:
Proportion/Weights: Portfolio Risk-Free Risky Expected return Standard Deviation A 0.5 0.5 8.5% = 0.5(5%) + 0.5(12%) 9% = 0.5(18%) P 0.0 1.0 12.0% = 0.0(5%) + 1.0(12%) 18% = 1.0(18%) Q* – 0.5 1.5 15.5% = -0.5(5%) + 1.5(12%) 27% = 1.5(18%)
- To form this portfolio, 50 percent of the total funds invested will be borrowed at the risk-free rate of return (i.e., the borrowed funds would cost 5 percent). The borrowed funds will be invested in the risky assets. o Following is information about the riskiness of the three portfolios shown in the CAL graph:
Portfolio Expected Return Risk Premium σ Sharpe Ratio A 8.5% 3.5% = 8.5% – 5.0% 9.0% 0.39 = 3.5%/9.0% P 12.0% 7.0% = 12.0% – 5.0% 18.0% 0.39 = 7.0%/18.0% Q 15.5% 10.5% = 15.5% – 5.0% 27.0% 0.39 = 10.5%/27.0%
σp = 18%
σ (risk)
E(r)
E(rp) = 12.0%
rf = 5.0%
wp = 100%
wp = 50% wf = 50%
σA = 9%
E(rA) = 8.5% (^) RPP = 7%
wp = 150% wf = -50%
σB = 27%
E(rB) = 15.5%
CAL
Slope = 7/18 = 0. = Sharpe ratio
A
P
Q
o What would be your preferred allocation of funds in the situation described here? To answer this question, let’s first suppose that you are comfortable investing in a portfolio that promises an expected risk premium [RP = E(r) – rf] equal to 16 percent and a variance of returns, σ^2 , equal to 0.04 (σ = 20%). In this case, your price of risk is A = 4.0 as described earlier. Based on this information and the information given in the CAL, you can use the following equation to determine the proportion of your total funds that should be invested in the risky asset based on your risk preferences:
f (^2) p
2
[E(r) r ) Pr oportion to invest Available risk premium to variance ratio in risky asset (^) Required risk premium to variance ratio A
[0.12 0.05)
^
Thus, you should invest 54 percent of your funds in the risky asset and 46 percent (= 1 – 0.54) of your funds in the risk-free asset. This portfolio would yield the following results: Expected return = E(r) = 0.46(5%) + 0.54(12%) = 8.78% Standard deviation = σ = 0.54(18%) = 9.72%
