1. Measuring Risk and Return
Chapter 5

2. 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

3. 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

4. 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

5. 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

6. 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?

7. 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%?

8. 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

9. 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
Good
Poor
Crash

10. 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?

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

12. 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

13. 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

14. 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
20%
$100
$120
Taco Price
$1/taco
Tacos
100 Tacos
120 Tacos

15. 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
20%
$100
$120
Taco Price
$1/taco
$1.091/taco
Tacos
100 Tacos
110 Tacos

16. 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))

17. 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 ≈
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

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

19. Determination of the Equilibrium Real Rate of Interest
Government Increases Deficit

20. Bills and Inflation,
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.

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

22. 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

23. 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

24. 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

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

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

27. 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

28. 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

29. 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

30. 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)

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

32. 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:

33. 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?”

34. 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.

35. 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.

36. 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?

37. 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

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

39. 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

40. 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

41. 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

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

43. 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

44. 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

45. 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

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

47. 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

48. 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

49. 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

50. 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

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

52. 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?

53. 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

54. Historical Distribution of Monthly Returns

55. 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 ) + ( * 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%

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

57. 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

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

59. 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

61. 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%

62. Variance and Standard Deviation
State Prob. of State r in State
Excellent
Good
Poor
Crash
E(r) = 9.76%
Variance = ?
Standard Deviation = ?

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