Andrei Simonov (HHS) Finance I – HHS MBA Risk and Return Andrei Simonov (HHS) Finance I – HHS MBA

Rates of Return: Single Period - P + D HPR = 1 1 P HPR = Holding Period Return P1 = Ending price P0 = Beginning price D1 = Dividend during period one

Rates of Return: Single Period Example Ending Price = 24 Beginning Price = 20 Dividend = 1 HPR = ( 24 - 20 + 1 )/ ( 20) = 25%

Averaging Multiperiod Returns: Arithmetic vs. Geometric 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%

Annual Holding Period Returns Geom. Arith. Stan. Series Mean% Mean% Dev.% Lg. Stk 10.51 12.49 20.30 Sm. Stk 12.19 18.29 39.28 LT Gov 5.23 5.53 8.18 T-Bills 3.80 3.85 3.25 Inflation 3.06 3.15 4.40

Annual Holding Period Risk Premiums and Real Returns Excess Real Series Returns% Returns% Lg. Stk 8.64 9.34 Sm. Stk 14.44 15.14 LT Gov 1.68 2.38 T-Bills --- 0.60 Inflation --- ---

Real vs. Nominal Rates Fisher effect: Approximation nominal rate = real rate + inflation premium R = r + i or r = R - i Example: r = 3%, i = 6% R = 9% = 3% + 6% or r = 9% - 6% = 3% Fisher effect: Exact r = (R - i) / (1 + i) = (9%-6%) / (1.06) = 2.83%

Averaging Multiperiod Returns: Dollar Weighted Returns Internal Rate of Return (IRR) Discount rate that equates present value of the future cash flows with the current investment amount Initial Investment is regarded as an cash outflow Ending value is considered as an inflow Additional investment is considered as a negative flow The IRR is an average rate of return over the period, considering changes in investment amount over time CF1 CF2 CF3 CFT = + + Inv0 +  + (1+IRR) (1+IRR)2 (1+IRR)3 (1+IRR)T

Dollar Weighted Average Net CFs 0 Y1 Y2 Y3 Y4 $ (mil) -1 -0.1 -0.5 0.8 1.0 Solving for IRR 1.0 = -.1/(1+r)1 + -.5/(1+r)2 + .8/(1+r)3 + 1.0/(1+r)4 r = .0417 or 4.17%

Quoting Conventions APR = annual percentage rate = (per-period rate) X (periods per year) Simple interest approach, ignoring compound interest EAR = effective annual rate = (1+ per-period rate)Periods per yr – 1 = (1+ (APR/n) )n – 1 Example: Annualize monthly return of 1% APR = 1% X 12 = 12% EAR = (1.01)12 - 1 = 12.68%

Characteristics of Probability Distributions Mean Most likely value (expected value) Variance or standard deviation (volatility) Degree of deviation from the mean value Skewness Degree of asymmetry in distribution Kurtosis Degree of fatness in tail area ...., etc. If a distribution is approximately normal, the distribution is described simply by characteristics 1) and 2)

Normal Distribution r = E(r) s.d. s.d. Symmetric distribution 68.3% of observations within one std. dev. 95.4% of observations within two std. dev., or 95% of observations within 1.65 std.dev. 99% of observations within 2.33 std.dev. “Fat tails” if more observations are away from the normal distance

Annual vs. Daily Volatility Annual volatility = daily volatility × (number of trading days per year)1/2 (Ex) S&P 500 index return’s annual std. dev. = 10.88% Then, what is the daily volatility? 0.1088 =  × (252)1/2   = 0.0068 (or 0.68%) Therefore, we would expect about 11.5 days a year when returns exceeded 1.36% in absolute value. Or, we would expect that returns would be between -0.68% and +0.68% in 174 days a year.

Histogram of S&P 500 index returns Sample period = 1,250 trading days Number of days with returns outside of 2 std. dev. = 65 days How many days would you observe returns exceeding 1.36% in absolute value in normal distribution?

Pros of Std. Dev. As a Measure of Risk Easier to calculate and implement. If return distribution is symmetric, the upside risk is the same as downside risk, and therefore standard deviation is a good measure of downside risk. if returns are normally distributed, standard deviation would be adequate in characterizing the risk.

Cons of Std. Dev. As a Measure of Risk Investors are concerned about downside risk. Standard deviation includes both the above-average returns (upside risk) and the below-average returns (downside risk). If returns are skewed, standard deviation is not the only relevant measure of risk. Holding expected return and standard deviation constant, investor would prefer positive skewed distribution.

Skewed Distribution Negative Positive r Left-skewed distribution Large negative returns possible Bad surprises are more likely to be extreme A risk-averse investor would prefer right-skewd distribution to left-skewd distribution Median Negative Positive r

Skewed Distribution Negative r Positive Right-skewed distribution Large positive returns possible Bad surprises are more likely to be small Median Negative r Positive

Measuring Mean: Scenario or Subjective Returns ( r ) = p s S p(s) = probability of a state r(s) = return if a state occurs 1 to s states

State Prob. of State rin State Numerical Example: Subjective or Scenario Distributions State Prob. of State rin State 1 .1 -.05 2 .2 .05 3 .4 .15 4 .2 .25 5 .1 .35 E(r) = (.1)(-.05) + (.2)(.05)...+ (.1)(.35) = .15

S Measuring Variance Subjective or Scenario Variance = p ( ) [ r - E )] 2 Standard deviation = [variance]1/2 Using Our Example: Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2...+ .1(.35-.15)2] Var= .01199 S.D.= [ .01199] 1/2 = .1095

Sample Statistics 1 Sample mean return where rt = rate of return on the stock for period t T = number of periods Sample variance of returns

Useful Formulas for Variance

Sample Statistics 2 Sample covariance where r1,t = rate of return on the stock 1 for period t r2,t = rate of return on the stock 2 for period t T = number of periods Sample correlation

Correlation pattern 1 rAB = +1 rAB = -1

Correlation pattern 2

Asset Allocation Across Risky & Risk-Free Assets Possible to split investment funds between safe and risky assets Risk-free asset: T-bills (proxy) Risky asset: stocks (or a portfolio) Issues Examine risk-return tradeoff Demonstrate how different degrees of risk aversion will affect allocations between risky and risk-free assets

Expected Return on the Combined Portfolio Example: 1. Risk-free asset: rf = 7% srf = 0% E(rp) = 15% sp = 22% y = % in p (1-y) = % in rf + 2. Risky asset portfolio p:

Expected Return on the Combined Portfolio E(rc) = yE(rp) + (1 - y)rf rc = combined portfolio return For example, assume y = .75 E(rc) = y(0.15) + (1-y)(0.07) = 0.07+0.08y =0.13 (13%)

Variance on the Combined Portfolio = 0, then s p c = Since rf y s s This is because Var(rc) = Var(y rp + (1-y) rf ) = Var(y rp) + Var((1-y) rf) + 2Cov(y rp, (1-y) rf) = y2 sp2

Combinations Without Leverage ( y ≤ 1: No borrowing) = .75(.22) = .165 (16.5%) If y = .75, then = 1(.22) = .22 (22%) If y = 1 (all in risky assets), then = 0(.22) = .00 (0%) If y = 0 (all in risk-free asset), then s s s

sc E(rc) CAL(Capital allocation line) = Possible Combinations with no leverage (y ≤ 1) = Lending portfolio E(rp) = 15% P 13% rf = 7% F sc (y=0) 16.5% (y=0.75) sp=22% (y=1)

Combinations With Leverage ( y > 1: With borrowing) Borrow at the risk-free rate and invest in stocks (Ex) With 50% Leverage, E(rc) = (-.5) (.07) + (1.5) (.15) = .19 (19%) sc = (1.5) (.22) = .33 (33%)

sc E(rc) CAL with leverage (y>1) = Borrowing portfolio 19% P E(rp) = 15% E(rp) - rf = 8% ) Slope = 8/22 rf = 7% F (=Sharpe ratio) sc sp=22% (y=1) 33% (y=1.5)

Two-Risky-Asset Case Two risky-asset portfolio expected return, rp r1 = Expected return on Security 1 r2 = Expected return on Security 2 w1 = Proportion of funds in Security 1 w2 = Proportion of funds in Security 2

Two-Risky-Asset Portfolio Variance Var(rp) = Var(w1 r1 + w2 r2 ) = Var(w1r1) + Var(w2r2) + 2Cov(w1r1, w2r2) which can be simply rewritten as, sp2 = w12s12 + w22s22 + 2w1w2 Cov(r1,r2) s12 = Variance of Security 1 s22 = Variance of Security 2 Cov(r1,r2) = Covariance of returns for Security 1 and Security 2

Three-Security Portfolio rp = w1r1 + w2r2 + w3r3 s2p = w12s12 + w22s22 + w32s32 + 2w1w2 Cov(r1,r2) + 2w1w3Cov(r1,r3) + 2w2w3Cov(r2,r3)

In General, n-Security Portfolio rp = Weighted average of the n-securities’ returns sp2 = Own variance terms + all pair-wise covariances : N variances, N(N-1) covariances if wi = 1/N (i.e., equal-weighted) Converge to average covariance (Diversification effect)

Diversification Effect Diversifiable risk = Unsystematic risk = Idiosyncratic risk = Firm-specific risk Average covariance Nondiversifiable risk = Systematic risk = Market risk The risk of a random portfolio of stocks as a percentage of the risk of an individual security. That is, about 73% of the risk of an individual security can be eliminated by holding a random portfolio of stocks in U.S.

Diversification Benefits The combined portfolio of domestic and foreign stocks reduces risk substantially.

Risk Aversion and Asset Allocation Greater levels of risk aversion lead to larger proportions in the risk-free assets Lower levels of risk aversion lead to larger proportions in the risky asset portfolio Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations and larger investment in risky assets How to measure the risk aversion? Many kinds of models

One Way of Quantifying Risk Aversion ( ) rp - = s 2 E rf 0.5 A Assume p E(rp) = Expected return on portfolio p rf = the risk free rate 0.5 = Scale factor A x s2 = Proportional risk premium The larger A is, the larger return will be required for the risk

One Way of Quantifying Risk Aversion Rearranging the equation and solving for A, r - r E ( ) p f A = 2 0.5 σ p Many studies have concluded that investors’ average risk aversion is between 2 and 4

Finding an optimal weight, y* Assume a utility function: U(r) = E(r) – 0.5 A s2 Note that it has a trade-off between E(r) and s2 Using the previous example: U(ry) = (0.07+0.08y) – 0.5 A (0.222 y2) = f(y) Find y* that maximizes the utility function f´(y) = 0.08 – A (0.222) y = 0 f´´(y) = – A (0.222) < 0 (Optimality condition satisfies)  y* = 0.08 / [A (0.222)] = [ E(rp) – rf ] / [A s2 ] A more risk-averse investor will invest less in risk asset If risk premium is negative, a risk-averse investor will sell the asset short

Finding an optimal portfolio for y* E(rc) U(r) with A=4 CAL E(rp) = 15% P E(rc) = 10.28% rf = 7% F sc sp=22% (y=1) sp= 9.02%