Financial Return and Risk Concepts Dr. Mariusz Dybał Institute of Economic Sciences mariusz.dybal@uwr.edu.pl Chapter 12 Financial Return and Risk Concepts © 2014 John Wiley and Sons

Chapter Outcomes Know how to compute arithmetic averages, variances, and standard deviations using return data for a single financial asset. Understand the sources of risk Know how to compute expected return and expected variance using scenario analysis. Know the historical rates of return and risk for different securities.

Chapter Outcomes Understand the concept of market efficiency and explain the three types of efficient markets. Explain how to calculate the expected return on a portfolio of securities. Understand how and why the combining of securities into portfolios reduces the overall or portfolio risk. Explain the difference between systematic and unsystematic risk. Understand the importance of ethics in investment-related positions.

Historical Return and Risk for a Single Asset Return: periodic income and price changes Dollar return = ending price – beginning price + income

Historical Return and Risk for a Single Asset Percentage return = Dollar return/beginning price Beginning price = $33.63 Ending price = $34.31 Dividend = $0.13 Dollar return = $34.31-33.63+0.13 =$0.81 Percentage return = $0.81/$33.63 = 0.024 or 2.4 percent

Historical Return and Risk for a Single Asset Can be daily, monthly or annual returns Risk: based on deviations over time around the average return

Returns for Two Stocks Over Time

Arithmetic Average Return Look backward to see how well we’ve done: Average Return (AR) = Sum of returns number of periods

An Example YEAR STOCK A STOCK B 1 6% 20% 2 12 30 3 8 10 4 –2 –10 1 6% 20% 2 12 30 3 8 10 4 –2 –10 5 18 50 6 6 20 Sum 48 120 Sum/6 = AR= 8% 20%

Measuring Risk Variance 2 = (Rt - AR)2 Deviation = Rt - AR Sum of Deviations (Rt - AR) = 0 To measure risk, we need something else…try squaring the deviations Variance 2 = (Rt - AR)2 n - 1

Since the returns are squared: (Rt - AR)2 The units are squared, too: Percent squared (%2) Dollars squared ($ 2) Hard to interpret!

Standard deviation The standard deviation () helps this problem by taking the square root of the variance:

A’s RETURN RETURN DIFFERENCE FROM DIFFERENCE YR THE AVERAGE SQUARED 1 6%– 8% = –2% (–2%)2 = 4%2 2 12 – 8 = 4 (4)2 = 16 3 8 – 8 = 0 (0)2 = 0 4 –2 – 8 = –10 (–10)2 = 100 5 18– 8 = 10 (10)2 = 100 6 6 – 8 = –2 (-2)2 = 4 Sum 224%2 Sum/(6 – 1) = Variance 44.8%2 Standard deviation = 44.8 = 6.7%

Using Average Return and Standard Deviation If the future will resemble the past and the periodic returns are normally distributed: 68% of the returns will fall between AR -  and AR +  95% of the returns will fall between AR - 2 and AR + 2 99% of the returns will fall between AR - 3 and AR + 3

For Asset A 68% of the returns between 1.3% and 14.7%

Which of these is riskier? Asset A Asset B Avg. Return 8% 20% Std. Deviation 6.7% 20%

Another view of risk: Coefficient of Variation = Standard deviation Average return It measures risk per unit of return

Which is riskier? Asset A Asset B Avg. Return 8% 20% Std. Deviation 6.7% 20% Coefficient of Variation 0.84 1.00

Where Does Risk Come From: Risk Sources in Income Statement Revenue Business Risk Purchasing Power Risk Exchange Rate Risk Less: Expenses Equals: Operating Income Less: Interest Expense Financial Risk Interest Rate Risk Equals: Earnings Before Taxes Less: Taxes Tax Risk Equals: Net Income

Measures of Expected Return and Risk Looking forward to estimate future performance Using historical data: ex-post Estimated or expected outcome: ex-ante

Steps to forecasting Return, Risk Develop possible future scenarios: growth, normal, recession Estimate returns in each scenario: growth: 20% normal: 10% recession: -5% Estimate the probability or likelihood of each scenario: growth: 0.30 normal: 0.40 recession: 0.30

Expected Return E(R) =  pi . Ri E(R) = .3(20%) + .4(10%) +.3(-5%) = 8.5% Interpretation: 8.5% is the long-run average outcome if the current three scenarios could be replicated many, many times

Once E(R) is found, we can estimate risk measures: 2 =  pi[ Ri - E(R)] 2 = .3(20 - 8.5)2 + .4(10 - 8.5)2 + .3(-5 - 8.5)2 = 95.25 percent squared

Standard deviation:  = 95.25%2 = 9.76% Coefficient of Variation = 9.76/8.5 = 1.15

Do Investors Really do These Calculations? Market anticipation of Fed’s actions Identify consensus; where does our forecast differ? Simulation and Monte Carlo analysis

Historical Returns and Risk of Different Assets Two good investment rules to remember: Risk drives expected returns Developed capital markets, such as those in the U.S., are, to a large extent, efficient markets.

Efficient Markets What’s an efficient market? Operationally efficient versus informationally efficient Many investors/traders News occurs randomly Prices adjust quickly to news on average reflecting the impact of the news and market expectations

More…. After adjusting for risk differences, investors cannot consistently earn above-average returns Expected events don’t move prices; only unexpected events (“surprises”) move prices or events which differ from the market’s consensus

Price Reactions in Efficient/Inefficient Markets Overreaction(top) Efficient (mid) Underreaction (bottom) Good news event

Types of Efficient Markets Strong-form efficient market Semi-strong form efficient market Weak-form efficient market

Source: Author analysis of Morningstar Principia data

Implications of “Efficient Markets” Market price changes show corporate management the reception of announcements by the firm Investors: consider indexing rather than stock-picking Invest at your desired level of risk Diversify your investment portfolio

Portfolio Returns and Risk Portfolio: a combination of assets or investments

Expected Return on A Portfolio: E(Ri) = expected return on asset i wi = weight or proportion of asset i in the portfolio E(Rp) =  wi . E(Ri)

If E(RA) = 8% and E(RB) = 20% More conservative portfolio: E(Rp) = .75 (8%) + .25 (20%) = 11% More aggressive portfolio: E(Rp) = .25 (8%) + .75 (20%) = 17%

Possibilities of Portfolio “Magic” The risk of the portfolio may be less than the risk of its component assets

Merging 2 Assets into 1 Portfolio Two risky assets become a low-risk portfolio

The Role of Correlations Correlation: a measure of how returns of two assets move together over time Correlation > 0; the returns tend to move in the same direction Correlation < 0; the returns tend to move in opposite directions

Diversification If correlation between two assets (or between a portfolio and an asset) is low or negative, the resulting portfolio may have lower variance than either asset. Splitting funds among several investments reduces the affect of one asset’s poor performance on the overall portfolio

The Two Types of Risk Diversification shows there are two types of risk: Risk that can be diversified away (diversifiable or unsystematic risk) Risk that cannot be diversified away (undiversifiable or systematic or market risk)

Capital Asset Pricing Model Focuses on systematic or market risk An asset’s risk depend upon whether it makes the portfolio more or less risky The systematic risk of an asset determines its expected returns

The Market Portfolio Contains all assets--it represents the “market” The total risk of the market portfolio (its variance) is all systematic risk Unsystematic risk is diversified away

The Market Portfolio and Asset Risk We can measure an asset’s risk relative to the market portfolio Measure to see if the asset is more or less risky than the “market” More risky: asset’s returns are usually higher (lower) than the market’s when the market rises (falls) Less risky: asset’s returns fluctuate less than the market’s over time

Blue = market returns over time Red = asset returns over time 0% 0% Time Time More systematic risk than market Less systematic risk than market

Implications of the CAPM Expected return of an asset depends upon its systematic risk Systematic risk (beta ) is measured relative to the risk of the market portfolio

Beta example:  < 1 If an asset’s  is 0.5: the asset’s returns are half as variable, on average, as those of the market portfolio If the market changes in value by 10%, on average this assets changes value by 10% x 0.5 = 5%

 > 1 If an asset’s  is 1.4: the asset’s returns are 40 percent more variable, on average, as those of the market portfolio If the market changes in value by 10%, on average this assets changes value by 10% x 1.4 = 14%

Sample Beta Values accessed December 2012 from http://finance. yahoo Firm Beta Caterpillar 1.86 Coca-Cola 0.38 General Electric 1.37 Delta Airlines 0.60 FirstEnergy 0.28

Learning Extension 12 Estimating Beta Beta is derived from the regression line: Ri = a + RMKT + e

Ways to estimate Beta Once data on asset and market returns are obtained for the same time period: use spreadsheet software statistical software financial/statistical calculator do calculations by hand

Sample Calculation Estimate of beta: n(RMKTRi) - (RMKT)(Ri) n RMKT2 - (RMKT)2

The sample calculation Estimate of beta = n(RMKTRi) - (RMKT )( Ri) n RMKT2 - (RMKT)2 6( 34.77) - (3.00)(0.20) 6(38.68) - (3.00)(3.00) = 0.93

Security Market Line CAPM states the expected return/risk tradeoff for an asset is given by the Security Market Line (SML): E(Ri)= RFR + [E(RMKT)- RFR]i

An Example E(Ri)= RFR + [E(RMKT)- RFR]i If T-bill rate = 4%, expected market return = 8%, and beta = 0.75: E(Rstock)= 4% + (8% - 4%)(0.75) = 7%

Portfolio beta The beta of a portfolio of assets is a weighted average of its component asset’s betas betaportfolio =  wi. betai

6. Find the real return on the following investments: 5. RCMP, Inc. shares rose 10 percent in value last year while the inflation rate was 3.5 percent. What was the real return on the stock? If an investor sold the stock after one year and paid taxes on the investment at a 15 percent tax rate what is the real after-tax return on the investment? The nominal return is 10% and the inflation rate is 3.5%. The real return on RCMP’s shares is 10%- 3.5% = 6.5%. Taking taxes into consideration, using a 15% tax rate the nominal after-tax return is 10% (1-0.15) = 8.5%. Subtracting the inflation rate, the real after-tax return is 8.5% - 3.5% = 5.0%. 6. Find the real return on the following investments:   The real return is computed as nominal return – inflation rate as follows: Stock A: 10% - 3% = 7% Stock B: 15% - 8% = 7% Stock C: -5% - 3% = -8% 7. Find the real return, nominal after-tax return, and real after-tax return on the following: Real return is nominal return minus the inflation rate: Stock X: 13.5% - 5% = 8.5% Stock Y: 8.7% - 4.7% = 4.0% Stock Z: 5.2% - 2.5% = 2.7% Nominal after-tax return is nominal return (1-tax rate): Stock X: 13.5% (1- 0.15) = 11.48% Stock Y: 8.7% (1- 0.25) = 6.53% Stock Z: 5.2% (1-0.28) = 3.74% The real after-tax return is the nominal after-tax return minus the inflation rate: Stock X: 11.48% - 5% = 6.48% Stock Y: 6.53% - 4.7% = 1.83% Stock Z: 3.74% - 2.5% = 1.24% Stock Nominal Return Inflation A 10% 3% B 15% 8% C -5% 2% Stock Nominal Return Inflation Tax Rate X 13.5% 5% 15% Y 8.7% 4.7% 25% Z 5.2% 2.5% 28%

a) What is Uma’s expected return forecast for Wallnut stock? 9. Using the information below, compute the percentage returns for the following securities: 12. Ima’s sister, Uma, has completed her own analysis of the economy and Wallnut’s stocks. Uma used recession, constant growth and inflation scenarios but with different probabilities and expected stock returns. Uma believes the probability of recession is quite high, at 60 percent and that in a recession Wallnut’s stock return will -20 percent. Uma believes the scenarios of constant growth and inflation are equally likely and that Wallnut’s returns will be 15 percent in the constant growth scenario and 10 percent under the inflation scenario. a) What is Uma’s expected return forecast for Wallnut stock? b) What is the standard deviation of the forecast? c) If Wallnut’s current price is $20 a share and is expected to pay a dividend of $0.80 a share next year, what price does Uma expect Wallnut to sell for in one year? With the probability of recession set at 60 percent, the probability of not having a recession is 1-0.60 or 0.40. As the probability of the constant growth and inflation scenarios are equally likely, there probabilities are 0.40/2 or 0.20 (20%) each. Using these probabilities we have: c) The return is computed as the (change in price + income)/beginning price. If the expected return is -7.00% (or -0.07 in decimal form) we have: (Expected price - $20) + 0.80 = -0.07 $20 or (Expected price - $20) + 0.80 = -0.07 ($20) = -$1.40 = Expected price - $19.20 = -$1.40 Solving, we see the expected price = $17.80.   Price today Price one year ago Dividends received Interest received Dollar Return= change in price + income Percentage Return=Dollar return/initial price a) RoadRunner stock $20.05 $18.67 $0.50 $1.88 10.07% b)Wiley Coyote stock $33.42 45.79 $1.10 -$11.27 -24.61% c)Acme long-term bonds $1,015.38 $991.78 $100.00 $123.60 12.46% d) Acme short-term bonds $996.63 $989.84 $45.75 $52.54 5.31% e) Xlingshot stock $5.43 $3.45 $0.02 $2.00 57.97% Scenario Probability Wallnut return Recession 60% -20% Constant growth 20% 15% Inflation 10% a) Expected return -7.00% = 60% (-20%) + 20%(15%) + 20% (10%) Variance 256.00% = 60% (-20%-(-7%))^2 + 20%(15%-(-7%))^2 + 20% (10%-(-7%))^2 b) Standard Deviation 16.00%

15. Below is annual stock return data on Hollenbeck Corp and Luzzi Edit, Inc. Year Hollenbeck Luzzi Edit 2010 10% -3% 2011 15% 0% 2012 -10% 15% 2013 5% 10%   a. What is the average return, variance, and standard deviation for each stock? Average return = (Sum of returns)/n Hollenbeck Corp: 20/4 = 5.0% Luzzi Edit: 22/4 = 5.5% Variance = ∑ (Ri – Average)2/ (n – 1) Hollenbeck Corp = 350/(4 – 1) = 116.67%2 Luzzi Edit= 213/(4 – 1) = 71.0%2 Standard deviation Hollenbeck Corp = √116.67= 10.80% Luzzi Edit = √ 71.0= 8.43% b. What is the expected portfolio return on a portfolio comprised of i. 25% Hollenbeck Corp and 75% Luzzi Edit? ii. 50% Hollenbeck Corp and 50% Luzzi Edit? iii. 75% Hollenbeck Corp and 25% Luzzi Edit?   E(portfolio return) = .25(5%) + .75(5.5%) = 5.375% E(portfolio return) = .5(5%) + .5(5.5%) = 5.25% E(portfolio return) = .75(5%) + .25(5.5%) = 5.125% c. Without doing any calculations, would you expect the correlation between the returns on Hollenbeck Corp and Luzzi Edit's stock to be positive, negative, or zero? Why? Probably negative, as the changes in returns from year-to-year moved in opposite directions in two of the three years (2010-2011: return rose for both Hollenbeck Corp and Luzzi Edit; 2011-2012: return fell for Hollenbeck Corp, rose for Luzzi Edit; 2012-2013: return rose for Hollenbeck Corp, fell for Luzzi Edit).

Case study #12 16. Below is annual stock return data on AAB Company and YYZ, Inc. Year AAB YYZ 2009 0% 5% 2010 5% 10% 2011 10% 15% 2012 15% 20% 2013 -10% -20%   a. What is the average return, variance, and standard deviation for each stock? b. What is the expected portfolio return on a portfolio comprised of i. 25% AAB and 75% YYZ? ii. 50% AAB and 50% YYZ? iii. 75% AAB and 25% YYZ?   c. Without doing any calculations, would you expect the correlation between the returns on AAB and YYZ's stock to be positive, negative, or zero? Why?