chapter 5.pptx risk and return
TRANSCRIPT
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Risk and Return: Past andPrologue
Chapter 5
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Real and Nominal Rates of Interest
Nominal interest rate:Growth rate of your money
Real interest rate:
Growth rate of your purchasing power
(1 + rnom)= (1 + rreal) x (1 + i)
rnom rreal +i
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Real and Nominal Rates of Interest
Fisher Equation:
rnom= rreal+ E(i)
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Rates of Return
Holding-period return (HPR):
Example: Suppose an investor wants to invest in astock-index fund, which sells for $100 per sharetoday. Suppose also that this fund is expected to
pay a $4 cash dividend and sell for $110 one yearlater. What is the expected HPR? Capital gainsyield? Dividend yield?
t
ttt
P
DPPHPR 11
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Rates of Return
Measuring returns over multiple periods:
Arithmetic average:
Geometric average: Single per-period return that givesthe same cumulative performance as the sequence ofactual returns.
Dollar weighted return: The IRR of an investment.
n
ssr
n 1)(
1
1)1)...(1)(1( /121 n
nrrrg
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Rates of Return
Example: Consider a fund that starts with $1 millionunder management at the beginning of the year. Thisfund receives both additional funds to invest andredemptions from existing shareholders as given below.
Find the arithmetic and geometric averages, and thedollar-weighted return.
1st
Quarter
2nd
Quarter
3rd
Quarter
4th
Quarter
AUM at the start of the
quarter
1.0 1.2 2.0 0.8
HPR (%) 10.0 25.0 (20.0) 25.0
Net Inflow ($ million) 0.1 0.5 (0.8) 0.0
AUM at the end of the
quarter
1.2 2.0 0.8 1.0
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xpec e e urns an an arDeviation
Scenario analysis: What HPRs are possible and howlikely are they?
Expected return:
Variance (Var):
Standard Deviation (Std):
( ) ( ) ( )s
E r p s r s
2
2 ( ) ( ) ( )s
p s r s E r
2STD
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Expected Returns and StandardDeviation
Example: Suppose that, over the next year, there arefour possible scenarios with their probabilities andHPRs. Find the expected return and the standarddeviation of returns.
Scenario Probability HPR (%)
SevereRecession
0.05 -37
MildRecession
0.25 -11
NormalGrowth
0.40 14
Boom 0.30 30
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The Normal Distribution
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The Normal Distribution
Suppose that riis normally distributed with expectedreturn E(ri) and standard deviation i.
Then, the standardized return: =()
is normally
distributed with a mean of zero and a standard deviation of1. Therefore, is a standard normal variable.
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Risk
Value at Risk (VaR):Attempts to answer the followingquestion:How many dollars can I expect to lose on my portfolio in a
given time period at a given level of probability?
The typical probability used is 5%. We need to know what HPR corresponds to a 5% probability.
If returns are normally distributed then we can use a standard normaltable or Excel to determine how many standard deviations below themean represents a 5% probability:
From Excel: =Norminv (0.05,0,1) = -1.64485 standard deviations From the standard deviation we can find the
corresponding
level of the portfolio return:
VaR = E[r] + -1.64485
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Risk
Example: A $500,000 stock portfolio has an annualexpected return of 12% and a standard deviationof 35%. What is the portfolio VaR at a 5%probability level?
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Using Time Series of Return
tt rn
r 1
2)(1
1)(
ttt
rrn
rVar
)(trVarSTD
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Risk Premium and Risk Aversion
Risk-free rate: Risk premium:
Speculation vs. Gamble
Excess return:
Risk Aversion: Risk averse investors reject investment portfolios that are
fair games or worse
These investors are willing to consider only risk-free or
speculative prospects with positive risk premiums Intuitively one would rank those portfolios as more attractive
with higher expected returns
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Utility Function
Where
U= utility
E ( r ) = expected return on the asset or portfolio
A= coefficient of risk aversion
2= variance of returns
Types of investors:
1. Risk Averse2. Risk Neutral
3. Risk lover (risk seeking)
21
( ) 2U E r A
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Utility Values of Possible Portfolios for an Investorwith Risk Aversion, A = 4
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The Trade-off Between Risk and Returns of aPotential Investment Portfolio, P
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The Indifference Curve
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Indifference Curves
Given A, there is one indifference curve for each level of utility.
Curve 1
Curve 2
Increasing
Utility
E(r)
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Mean-Variance Criterion
A dominates B if E(rA) E(rB) and AB, withat least one strict inequality.
Goal of Risk-Averse Investors: Either
(i) Maximize expected return for a given level ofrisk (), or(ii) Minimize risk () for a given expected return.
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Mean-Variance Dominance
Expected Return
Variance or Standard Deviation
B
A
RiskyC
D
RF
A dominates B
A dominates D
C dominates D
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Asset Allocation Across Risky and Risk-freePortfolios
Asset Allocation: Portfolio choice among broadinvestment classes.
John Bogle of the Vanguard Group of InvestmentCompanies:
The most fundamental decision of investing is theallocation of your assets: how much should you hold instock? how much in bonds? how much in cash reserves.
That decision can account for an astonishing 94% of thedifferences in total returns achieved by institutionallymanaged pension funds.
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Asset Allocation Across Risky and Risk-freePortfolios
Example: Assume that the total market value of aninvestors portfolio is $300,000. Of that $90,000 isinvested in shares of Ready Asset money market fund(a risk-free asset). The remaining $210,000 is in risky
securities, say, $113,400 in shares of Vanguards S&P500 Index Fund and $96,000 in shares of FidelitysInvestment Grade Bond Fund.
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Portfolio Expected Return and Risk
Example: Suppose that, in the previous example,E(rp) = 15%, p= 22%, and rf= 7%. Draw the CapitalAllocation Line and calculate the Sharpe ratio.