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Transcript of Measuring Risk and Return Chapter 5. Learning Objectives Outline the key factors that influence...
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 SEMESTERYou will need these throughout the whole semester
Single Period Rate of Return Perpetuity Growing Perpetuity
4
Single Period Rates of Return
Invested
Change
P
DCG
P
DPPHPR$
$
0
11
0
101
P0 = Beginning price
P1 = Ending price
D1 = Dividend during period one
CG1 = Capital Gains during period one
Holding Period Return (single period):
5
HPR Break Down
We earn returns in one of two ways Capital Gains: Price Appreciation
Capital Gains Yield is CG1/P0
DividendsDividend Yield is D1/P0
P
D
P
CG
P
DCG
P
DPPHPR0
1
0
1
0
11
0
101
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%?What is the price next year?
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 worldr(s) outcome if state occursp(s) probability that the state occurs
Expected return is the weighted average of the possible returns
( ) ( ) ( )s
E r p s r s
9
State Prob. of State HPR in State Excellent .25 0.3100Good .45 0.1400Poor .25 -0.0675Crash .05 -0.5200
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
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: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 ozIn 1968 bar was ¾ oz
How much does it cost (nominal & real) to buy 2 oz of chocolate?
1930 1968
Real Nickel buys 2oz 2oz
Nominal Nickel buys 2oz 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
Invest @ 20% $100 $120
Taco Price $1/taco $1/taco
Tacos 100 Tacos 120 Tacos
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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
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?
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, 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.
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 returnsRisk Premium: Difference between the expected
RoR and the risk-free rateExcess Return: Difference between the actual RoR
and the risk-free rate
23
Measuring Risk What is risk? There is no universally agreed-upon
measureHowever, 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 outcomesMore uncertainty
25
Different Variances
Possible ReturnsE(r)
26
Variance (VAR):
Variance and Standard Deviation
22 ( ) ( ) ( )s
p s r s E r
2STD
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
n
ssrsprE
1)()()(
30
When E(r) is less informative
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)
1996 1997 1998 1999
20% -20% 20% -20%
31
Geometric Average Return Used to compute the average compound return
of an investment over multiple periods
rg= geometric average rate of return
1)]1(*..*)1(*)1[(1
21 nnrrrrg
32
When E(r) is less informative
Arithmetic Average is 0%?(0.25*0.2)+(0.25*-0.2)+(0.25*0.2)+(0.25*-0.2)
Geometric Average is:
1996 1997 1998 1999
20% -20% 20% -20%
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:
AverageArithmeticN
TNverageGeometricA
N
TTR
11
1)(
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?
AverageArithmeticN
TNverageGeometricA
N
TTR
11
1)(
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 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 Flow2008 50 1002009 502010 51 752011 54 75
39
Example Continues
Use the CF button to find the IRR 2nd CE/C CF – Cashflow – Enter – Down Arrow Fill in all Cashflows IRR - CPT
Year Price Share Cash Flow2008 50 100 -5,000.002009 55 50 -2,650.002010 51 75 4,125.002011 54 75 4,387.50
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
2
1
_2 1
ˆ
n
s
rsrn
2
1
_2
1
1ˆ
n
j
rsrn
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 investmentA better risk return trade off
h𝑆 𝑎𝑟𝑝𝑒𝑅𝑎𝑡𝑖𝑜=𝑅𝑖𝑠𝑘𝑝𝑟𝑒𝑚𝑖𝑢𝑚
𝑆𝑡𝑑 𝐷𝑒𝑣𝑜𝑓 𝑒𝑥𝑐𝑒𝑠𝑠𝑟𝑒𝑡𝑢𝑟𝑛𝑠
45
Comparing Investments
Higher Return due to performanceInvestment A earned 20% - SR 3Investment B earned 8% - SR 1
Higher Return due to riskInvestment A earned 20% - SR 1Investment B earned 8% - SR 3
46
Risk Return TradeoffR
etur
ns
Risk
T-Bills
T-Bonds
LT Corp Bonds
Large Cap Stock
ST Corp Bonds
Small Cap Stock
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 discussedCredit Cards, LoansFound using simple interest
Effective Annual Rate (EAR): Return an investment actually makes over a yearFound using compound interest
EAR = {1+ (APR/n)}n – 1n 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% perWeek
Month
50
High Math Investment Foundation Underlying most of the class is the idea that
returns are normally distributedThis assumption is central to investment theory
and practice Implications
If security returns are normal then so are portfolioStandard deviation and the mean completely
describe the distributionStandard 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 returnsr 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 ) + (-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%
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 riskSharpe ratio is not a complete measure of
portfolio performanceNeed to consider skewness and kurtosis
60
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 DeviationState Prob. of State r in State Excellent .25 0.3100Good .45 0.1400Poor .25 -0.0675Crash .05 -0.5200E(r) = 9.76%Variance = ?Standard Deviation = ?
63
Historical Risk Example
Expected RoR
Variance =
Standard Deviation =
1996 1997 1998 1999 2000
20% 15% -5% 5% 10%