Measuring Risk and Return Chapter 5

Learning Objectives Outline the key factors that influence interest rates Describe the Fisher effect and its influence on interest rates and inflation Calculate Risk and Return Measures Discuss the characteristics of the Normal distribution Understand the historic returns on risky portfolios

Rate of Return Basic Formula Keep these in mind ALL SEMESTER You will need these throughout the whole semester Single Period Rate of Return Perpetuity Growing Perpetuity

Single Period Rates of Return Holding Period Return (single period): P0 = Beginning price P1 = Ending price D1 = Dividend during period one CG1 = Capital Gains during period one

HPR Break Down We earn returns in one of two ways Capital Gains: Price Appreciation Capital Gains Yield is CG1/P0 Dividends Dividend Yield is D1/P0

Holding Period Return Questions The current price of a stock is $25, you expect the stock’s price at the end of the period to be $29.50 after paying a dividend of $0.50. What is the holding period return? What is the capital gain yield? What is the dividend yield?

Remember Stocks are Perpetuities What is the price of a stock with a $10 dividend if the discount rate is 10%? What is the price next year? What is the price of a stock with a $10 dividend next year if grow is 2% and the discount rate is 10%?

Expected Return HPR tells us what we earned, but investments are based on what we EXPECT to earn Expectations are based on the possible states of the world r(s) outcome if state occurs p(s) probability that the state occurs Expected return is the weighted average of the possible returns

Scenario Analysis: Possible States of Nature and Holding Period Returns In this set-up there are only 4 possible “states of nature” (investment outcomes). Each “state” is associated with: a probability of that state occurs, and the return on the investment if the state occurs State Prob. of State HPR in State Excellent .25 0.3100 Good .45 0.1400 Poor .25 -0.0675 Crash .05 -0.5200

Backwards Inducing Price We expect an investment is equally likely to payoff either $125,000; $75,000; or -$20,000 next year. If we demand a return of 20%, how much are we willing to pay?

Nominal v Real Nominal Dollar: Real Dollar: The dollar in your wallet, or bank account Real Dollar: Refers to purchasing power

Nominal v Real Dollar Example Hershey Nickel Bar Example In 1930 bar was 2 oz In 1968 bar was ¾ oz How much does it cost (nominal & real) to buy 2 oz of chocolate? 1930 1968 Real Nickel buys 2oz Nominal Nickel buys 0.75oz

Real and Nominal Rates of Return Nominal interest rate (“rn”) Growth rate of your money Real interest rate (“rr”) Growth rate of your purchasing power Inflation rate (“i”): The general decline in what a dollar can purchase

Taco World Tacos are the only good in our world Cost: $1/Taco We can invest $100 today and earn 20% Forgo tacos today for more tacos next year How many tacos can we buy next year? Nominal HPR? Real HPR? Year 0 Year 1 Invest @ 20% $100 $120 Taco Price $1/taco Tacos 100 Tacos 120 Tacos

Taco World Tacos are the only good in our world Cost $1/Taco: Inflation is 9.1% We can invest $100 today and earn 20% Forgo tacos today for more tacos next year How many tacos can we buy next year? Nominal HPR? Real HPR? Year 0 Year 1 Invest @ 20% $100 $120 Taco Price $1/taco $1.091/taco Tacos 100 Tacos 110 Tacos

Equilibrium Nominal Rate of Interest As the inflation rate increases, investors will demand higher nominal rates of return If E(i) denotes current expectations of inflation, then we get the Fisher Equation: (1+rn) = (1+rr) * (1+E(i))

Real vs. Nominal Rate Example If you invest $10,000 at a nominal rate of 12% APR, how much will you have in 30 years? How much will you have in real terms if the rate of inflation is 4% per year? What is your nominal RoR? Real RoR? N = 30, I/Y = 12, PV= 10,000, PMT = 0, FV= ???? FV = $299,599.22 How much will you have in real terms if the rate of inflation is 4% per year? 1+Real = 1.12/1.04 ≈1.07692307692 N = 30, I/Y = 7.069, PV= 10,000, PMT = 0, FV= ???? FV = $92,372.03 OR N = 30, I/Y = 4, PV=????, PMT = 0, FV= 299,599.22 PV = $92,372.03, this is the Future Real Dollar Value

Equilibrium Real Rate of Interest Real Rate Determined by: Supply Household savings Demand Business Investment Government actions Federal Reserve

Determination of the Equilibrium Real Rate of Interest Government Increases Deficit

Bills and Inflation, 1926-2012 Moderate inflation can offset most of the nominal gains on low-risk investments. A dollar invested in T-bills from 1926–2012 grew to $20.25, but with a real value of only $1.55.

Risk & Risk Premium If T-Bills and Google both have an expected return of 10%, where does the average person invest? Why?

Risk and Risk Premiums Risk Aversion: People generally dislike risk To induce people to take on risk they must be rewarded with higher returns Risk Premium: Difference between the expected RoR and the risk-free rate Excess Return: Difference between the actual RoR and the risk-free rate

Measuring Risk What is risk? There is no universally agreed-upon measure However, variance and standard deviation are both widely accepted measures of total risk

Statistics Review: Variance Variance (σ2) measures the dispersion of possible outcomes around expected return Standard deviation (σ) is the square root of variance Higher variance (std dev), implies a higher dispersion of possible outcomes More uncertainty

Different Variances E(r) Possible Returns Draw/ Compare two distributions E(r) Possible Returns

Variance and Standard Deviation Variance (VAR): Standard Deviation (STD):

Example You invest in a stock at the current price of $50. Your expectation regarding the price and the dividend in the following year depends upon how the economy performs: Compute the expected return and standard deviation of this investment Economy Probability Dividend Ending Price Strong 30% $2.00 $60.00 Normal 50% $1.00 $54.00 Weak 20% $0.50 $44.50

Using the Time Series of Historical Returns We cannot determine the “true” mean and variance of an investment because we don’t know all the possible scenarios Therefore we often estimate the mean and variance based on historical information

Using Historical Returns Each observation is a “scenario” We view each is equally likely of recurring If there are “n” observations then each scenario’s probability of occurring is 1/n The expected return is: Where p(s) = 1/n

When E(r) is less informative 1996 1997 1998 1999 20% -20% Does an investor really expect to earn a 1 year HPR of 0%? (0.25*0.2)+(0.25*-0.2)+(0.25*0.2)+(0.25*-0.2)

Geometric Average Return Used to compute the average compound return of an investment over multiple periods rg= geometric average rate of return

When E(r) is less informative 1996 1997 1998 1999 20% -20% Arithmetic Average is 0%? (0.25*0.2)+(0.25*-0.2)+(0.25*0.2)+(0.25*-0.2) Geometric Average is:

Arithmetic v Geometric Average RoR The arithmetic average rate of return answers the question, “what was the average of the yearly rates of return?” The geometric average rate of return answers the question, “What was the growth rate of your investment?”

More on Average Returns The geometric average will be less than the arithmetic average unless all the returns are equal. Arithmetic average is an overly optimistic estimate of future returns for long horizons. The geometric average is an overly pessimistic estimate of future returns for short horizons.

Forecasting Return To achieve a more accurate estimate of expected returns, one can use Blume’s formula: where, T is the forecast horizon and N is the number of years of historical data we are working with. T must be less than N.

Blume Example Over a 30-year period a stock had an arithmetic average of 15% and a geometric average of 11%. Using Blume’s formula what is the best estimate of the future annual returns over the next 5 years? 10 years?

Dollar Weighted Average Return The internal rate of return earned on an investment Gives an idea of what the investor actually earned Treat the investment like a corp capital budgeting problem and find the IRR

Dollar Weighted Example 2008 bought 100 shares @ $50 2009 return 10% buy another 50 shares, $2 div/sh 2010 shares sold 75 @ $51, $4 div/sh 2011 shares sold 75 @ $54 $4.5 div/sh What are the cash flows? Year Price Share Cash Flow 2008 50 100 2009 2010 51 75 2011 54

Example Continues Use the CF button to find the IRR 2nd CE/C Year Price Share Cash Flow 2008 50 100 -5,000.00 2009 55 -2,650.00 2010 51 75 4,125.00 2011 54 4,387.50 Use the CF button to find the IRR 2nd CE/C CF – Cashflow – Enter – Down Arrow Fill in all Cashflows IRR - CPT

Historical Risk Estimated Variance = expected value of squared deviations Estimated Variance is biased downward We are using the historical average instead of the actual expected return To eliminate the bias we modify the variance formula

Historical Risk Example What is the historical variance of the Index? The average return over the period is 6.4% 2010 2011 2012 2013 2014 8 9 5 4 6

Comparing Investments Investment A earned 20% Investment B earned 8% Which did better?

Comparing Investments We cannot simply compare returns when we compare investments. WHY? FYI: How are mutual funds advertised? To fairly compare investments we need to examine both the return earned and the risk involved → Sharpe Ratio

Sharpe Ratio: Reward to Volatility A measure of risk adjusted performance Is higher return due to good performance or more risk? Higher Sharpe Ratios → a more efficient investment A better risk return trade off 𝑆ℎ𝑎𝑟𝑝𝑒 𝑅𝑎𝑡𝑖𝑜= 𝑅𝑖𝑠𝑘 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 𝑆𝑡𝑑 𝐷𝑒𝑣 𝑜𝑓 𝑒𝑥𝑐𝑒𝑠𝑠 𝑟𝑒𝑡𝑢𝑟𝑛𝑠

Comparing Investments Higher Return due to performance Investment A earned 20% - SR 3 Investment B earned 8% - SR 1 Higher Return due to risk Investment A earned 20% - SR 1 Investment B earned 8% - SR 3

Risk Return Tradeoff Small Cap Stock Returns Large Cap Stock LT Corp Bonds ST Corp Bonds T-Bonds T-Bills Risk

Given $100,000 to invest: What is the expected return and standard deviation of each of the investment opportunities? What is the expected risk premium in dollars of investing in equities versus risk-free T-bills? Invest in: Probability Return ($) Equities .6 $50,000 .4 -$30,000 T-Bills 1.0 $5,000

Annualizing Returns Annual Percentage Rate (APR): This is the return commonly discussed Credit Cards, Loans Found using simple interest Effective Annual Rate (EAR): Return an investment actually makes over a year Found using compound interest EAR = {1+ (APR/n)}n – 1 n is the number of compounding periods per year

Examples: APR & EAR What is the EAR of a 4% APR that compounds: Semi-annually? Quarterly? Monthly? What is the APR and EAR of 0.5% per Week Month

High Math Investment Foundation Underlying most of the class is the idea that returns are normally distributed This assumption is central to investment theory and practice Implications If security returns are normal then so are portfolio Standard deviation and the mean completely describe the distribution Standard deviation is an appropriate measure of risk Fortunately, returns appear to normal

The Normal Distribution Mean = 10%, SD = 20%

Normal Distribution Example A security with normally distributed returns has an annual expected return of 18% and standard deviation of 23%. What is the probability of getting a return between -28% and 64% in any one year?

Measuring “Surprise” Standard Deviation Score: How much of a surprise an observed return sr i = [r i – E(r i)] / σi Return surprise divided by standard deviation

Historical Distribution of Monthly Returns

How Much Could I loss? Value at Risk (VaR): What is the worse loss that an investment will suffer, given a probability (often 5%) VaR = E(r ) + (-1.64485* σ) VaR at 5% with normal return distribution What is my value at risk on an investment with an expected return on 12%, and a standard deviation of 5%

Non-Normal Distributions When distributions are non-normal we need to consider more than mean and variance

Distribution Characteristics Mean Most likely outcome Variance or standard deviation The spread of possible outcomes Skewness How asymmetrical is the distribution Kurtosis Flat or “Peakie” * If a distribution is approximately normal, the distribution is described by the mean and standard deviation

Normal and Skewed Distributions Mean = 6%, SD = 17%

Normality and Risk Measures What if excess returns are not normally distributed? Standard deviation is no longer a complete measure of risk Sharpe ratio is not a complete measure of portfolio performance Need to consider skewness and kurtosis

Expected Return Example Amy has just purchased 1,000 shares of GE. She expects that the return over the next year will depend on the state of the economy. Given her expectations what is her expected return? State Probability Expected Return Boom 10% 35% Normal 70% 12% Bust 20% -18%

Variance and Standard Deviation State Prob. of State r in State Excellent .25 0.3100 Good .45 0.1400 Poor .25 -0.0675 Crash .05 -0.5200 E(r) = 9.76% Variance = ? Standard Deviation = ?

Historical Risk Example 1996 1997 1998 1999 2000 20% 15% -5% 5% 10% Expected RoR Variance = Standard Deviation =