# The Investment Background - UMass D Risk and Return: Past and Prologue Risk Aversion and Capital Allocation to Risk Assets HPR: Rate of return over a given investment period HPR Ending Price - Beginning Price Dividends Beginning Price Ending Price =

110 Beginning Price = 100 Dividend = 4 \$100 \$50 r1 t= 0 1 r1, r2: one-period HPR

\$100 r2 2 What is the average return of your investment per period? Arithmetic Average: rA = (r1+r2)/2 Geometric Average: rG = [(1+r1)(1+r2)]1/2 1

Arithmetic return: return earned in an average period over multiple period r1 r2 rn rA n It is the simple average return. It ignores compounding effect It represents the return of a typical (average) period

Provides a good forecast of future expected return Geometric return Average compound return per period 1/ n r 1 r 1

r 1 r 1 1 2 n Takes into account compounding effect G Provides an actual performance per year of the investment over the full sample period

Geometric returns <= arithmetic returns Quarter HPR

1 .10 2 .25 3 4 (.20) .25 What are the arithmetic and geometric return of this mutual fund?

Arithmetic ra = (r1 + r2 + r3 + ... rn) / n ra = (.10 + .25 - .20 + .25) / 4 = .10 or 10% Geometric rg = {[(1+r1) (1+r2) .... (1+rn)]} 1/n - 1 rg = {[(1.1) (1.25) (.8) (1.25)]} 1/4 - 1 = (1.5150) 1/4 -1 = .0829 = 8.29% Invest \$1 into 2 investments: one gives 10% per

year compounded annually, the other gives 10% compounded semi-annually. Which one gives higher return APR = annual percentage rate (periods in year) X (rate for period) EAR = effective annual rate ( 1+ rate for period)Periods per yr - 1 Example: monthly return of 1% APR = 1% X 12 = 12% EAR = (1.01)12 - 1 = 12.68% m

APR EAR 1 1 m Risk in finance: uncertainty related to outcomes of an investment The higher uncertainty, the riskier the investment.

How to measure risk and return in the future Probability distribution: list of all possible outcomes and probability associated with each outcome, and sum of all prob. = 1. For any distribution, the 2 most important characteristics Mean Standard deviation

s.d. s.d. r or E(r) n E r pi ri i 1 pi the probability of each scenario

ri the HPR in each scenario i indexation variable for scenarios Variance or standard deviation: 2 n pi (ri E (r )) i 1

2 2 Suppose your expectations regarding the stock market are as follows: State of the economy Scenario(s) Probability(p(s)) Boom 1 0.3 Normal Growth

2 0.4 Recession 3 0.3 Compute the mean and standard deviation of the HPR on stocks. E( r ) = 0.3*44 + 0.4*14+0.3*(-16)=14% Sigma^2=0.3*(44-14)^2+0.4*(14-14)^2 +0.3*(-16-14)^2=540 Sigma=23.24% HPR 44%

14% -16% 0.4 0.35 0.25 Probability Two variables with the same

mean. 0.3 0.2 0.15 What do we know about their dispersion? 0.1

0.05 0 -5 -4 -3 -2 -1

0 Outcomes 1 2 3 4

5 Data in the n-point time series are treated as realization of a particular scenario each with equal probability 1/n n r t 1 2

rt n n 1 2 rt r n 1 t 1

Year Ri(%) 1988 1989 1990 1991 1992 16.9

31.3 -3.2 30.7 7.7 Compute the mean and variance of this sample Series World Stk US Lg Stk US Sm Stk Wor Bonds LT Treas.

T-Bills Inflation Geom. Mean% 9.80 10.23 12.43 5.80 5.35 3.72 3.04

Arith. Mean% 11.32 12.19 18.14 6.17 5.64 3.77 3.13 Stan. Dev.% 18.05

20.14 36.93 9.05 8.06 3.11 4.27 Risk Premium 7.56 8.42 14.37 2.40

2.07 0 18% Small-Company Stocks Annual Return Average 16% 14% Large-Company Stocks

12% 10% 8% 6% T-Bonds 4% T-Bills 2%

0% 5% 10% 15% 20% 25% Annual Return Standard Deviation

30% 35% Figure 5.2 Rates of Return on Stocks, Bonds and Bills A large enough sample drawn from a normal distribution looks like a bell-shaped curve. Probability

The probability that a yearly return will fall within 20.2 percent of the mean of 12.2 percent will be approximately 2/3. 3 48.3% 2 28.1% 1

7.9% 0 12.2% 68.26% 95.44% 99.74% + 1 32.5% + 2 52.7%

+ 3 72.9% Return on large company common stocks The 20.14% standard deviation we found for large stock returns from 1926 through 2005 can now be interpreted in the

following way: if stock returns are approximately normally distributed, the probability that a yearly return will fall within 20.14 percent of the mean of 12.2% will be approximately 2/3. Risk aversion: higher risk requires higher return, risk averse investors are rational investors Risk-free rate: the rate you can earn by leaving the money in risk-free assets such as T-bills. Risk premium (=Risky return Risk-free return) It is the reward for investor for taking risk involved in

investing risky asset rather than risk-free asset. The Risk Premium is the added return (over and above the riskfree rate) resulting from bearing risk. Historically, stock is riskier than bond, bond is riskier than bill Return of stock > bond > bill More risk averse, put more money on bond

Less risk averse, put more money on stock This decision is asset allocation John Bogle, chairman of the Vanguard Group of Investment Companies The most fundamental decision of investing is the allocation of your assets: how much should you own in stock, how much in bond, how much in cash reserves. That decision accounts for an astonishing 94% difference in total returns achieved by institutionally managed pension funds. ... There is no reason to believe that the same relationship does not hold true for individual investors. The complete portfolio is composed of: The risk-free asset: Risk can be reduced by allocating

more to the risk-free asset The risky portfolio: Composition of risky portfolio does not change This is called Two-Fund Separation Theorem. The proportions depend on your risk aversion. Total portfolio value = \$300,000 Risk-free value = 90,000 Risky (Vanguard & Fidelity) = 210,000

Vanguard (V) = 54% Fidelity (F) = 46% Vanguard 113,400/300,000 = 0.378 Fidelity Portfolio P 96,600/300,000 =

0.322 210,000/300,000 = 0.700 Risk-Free Assets F Portfolio C 90,000/300,000 = 300,000/300,000 =

0.300 1.000 Only the government can issue defaultfree bonds Guaranteed real rate only if the duration of the bond is identical to the investors desire holding period T-bills viewed as the risk-free asset Less sensitive to interest rate fluctuations

Its possible to split investment funds between safe and risky assets. Risk free asset: proxy; T-bills Risky asset: stock (or a portfolio) Example: Let the expected return on the risky portfolio, E(rP), be 15%, the return on the risk-free asset, rf, be 7%. What is the return on the complete portfolio if all of the funds are invested in the risk-free asset? What is the risk premium? 7% 0 What is the return on the portfolio if all of the funds are

invested in the risky portfolio? 15% 8% Example: Let the expected return on the risky portfolio, E(rP), be 15%, the return on the risk-free asset, rf, be 7%. What is the return on the complete portfolio if 50% of the funds are invested in the risky portfolio and 50% in the risk-free asset? What is the risk premium? 0.5*15%+0.5*7%=11% 4% In general:

E rC yE (rp ) (1 y )rf y fraction of funds invested in the risky asset E rC rf risk premium on the complete portfolio E rP rf risk premium on the risky asset C2 y 2 P2 1 y 2 r2 2 y 1 y Pr P r f f f

C y P if r 0 f where c - standard deviation of the complete portfolio P - standard deviation of the risky portfolio rf - standard deviation of the risk-free rate y - weight of the complete portfolio invested in the risky asset Example: Let the standard deviation on the risky portfolio, P, be 22%. What is the standard deviation of the complete portfolio if 50% of the funds are invested in the risky

portfolio and 50% in the risk-free asset? 22%*0.5=11% We know that given a risky asset (p) and a risk-free asset, the expected return and standard deviation of any complete portfolio (c) satisfy the following relationship: E (rc ) r f y ( E (rp ) rf ) c y p Where y is the fraction of the portfolio invested in the risky asset

E (rc ) rf c E (rc ) rf E (rp ) rf p E (rp ) rf

p c for every complete portfolio c Risk Tolerance and Asset Allocation: More risk averse - closer to point F Less risk averse - closer to P S

E rP r f P S is the increase in expected return per unit of additional standard deviation S is the reward-to-variability ratio or Sharpe Ratio Example: Let the expected return on the risky portfolio, E(rP), be 15%, the return on the risk-free asset, rf, be 7% and the standard deviation on the risky portfolio, P, be 22%. What is the slope of the CAL for the complete portfolio?

S = (15%-7%)/22% = 8/22 So far, we only consider 0<=y<=1, that means we use only our own money. Can y > 1? Borrow money or use leverage Example: budget = 300,000. Borrow additional 150,000 at the risk-free rate and invest all money into risky portfolio y = 450,000/150,000 = 1.5 1-y = -0.5 Negative sign means short position. Instead of earning risk-free rate as before, now have to pay riskfree rate

E rC yE ( rp ) (1 y ) r f 1.5 *15 ( 0.5) * 7 19% c 1.5 * 22 0.33 S E rP r f P 19 7 0.36

.33 The slope = 0.36 means the portfolio c is still in the CAL but on the right hand side of portfolio P Example: Let the expected return on the risky portfolio, E(rP), be 15%, the return on the risk-free asset, rf, be 7%, the borrowing rate, rB, be 9% and the standard deviation on the risky portfolio, P, be 22%. Suppose the budget = 300,000. Borrow additional 150,000 at the borrowing rate and invest all money into risky portfolio What is the slope of the CAL for the complete portfolio for points where y > 1, y = 1.5; E(Rc) = 1.5(15) + (-0.5)*9 = 18%

c 1.5 * 22 0.33 Slope = (0.18-0.09)/0.33 = 0.27 Note: For y 1, the slope is as indicated above if the lending rate is rf. SPECIAL CASE OF CAL (I.e., P=MKT) The line provided by one-month T-bills and a broad index of common stocks (e.g. S&P500) Consequence of a passive investment strategy based on stocks and T-bills

E(r) P3? E(Rm) = 12% M P1? S=0.45 rf = 3% P2?

F 0 20% 48 Risk Preference Risk averse

Require compensation for taking risk Risk neutral No requirement of risk premium Risk loving Pay to take risk Utility Values: A is risk aversion parameter U E (r ) 0.5 A 2 FIN 8330 Lecture 7 10/04/07

49 1 2 U E (r ) A 2 Where U = utility E ( r ) = expected return on the asset or portfolio A = coefficient of risk aversion = variance of returns

Greater levels of risk aversion lead to larger proportions of the risk free rate. Lower levels of risk aversion lead to larger proportions of the portfolio of risky assets. Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations. 55

Solve the maximization problem: Max U E (r ) 0.5 A 2 rf y ( E ( R p ) rf ) 0.5 A( y p ) 2 Two approaches: 1. Try different y 2. Use calculus:

Solution: y* dU E R r Ay 2 0 p f p dy E ( R p ) rf A p2 56

If A = 4, rf = 7%, E(Rp) = 15%, p 22% y* E ( R p ) rf A p2 .15 .07

.41 2 4 .22 58 If CAL is from 1-month T-bills and a broad index of common stocks, then CAL is also called Capital Market Line (CML) Why passive strategy: (1) strategies are costly; (2) market is competitive. Mutual fund separation theorem: capital should be invested in the (same optimal) risky portfolio and riskfree asset.

FIN 8330 Lecture 7 10/04/07 66 Passive strategy involves a decision that avoids any direct or indirect security analysis

Supply and demand forces may make such a strategy a reasonable choice for many investors A natural candidate for a passively held risky asset would be a well-diversified portfolio of common stocks

Because a passive strategy requires devoting no resources to acquiring information on any individual stock or group we must follow a neutral diversification strategy Definition of Returns: HPR, APR and AER. Risk and expected return Shifting funds between the risky portfolio to the risk-free asset reduces risk Examples for determining the return on the risk-free asset Examples of the risky portfolio (asset) Capital allocation line (CAL)

All combinations of the risky and risk-free asset Slope is the reward-to-variability ratio Risk aversion determines position on the capital allocation line