Chapter 5
Risk and Return
Konan Chan
Spring, 2017
Chapter 5 - Risk and Return
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How to calculate return ?
Conventional quotation of returns
How to measure expected return and risk?
– Probability distribution
– Historical data
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Inflation
Asset allocation
Capital market line
Distribution and VaR
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Calculating Return - Single period
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Holding period return (HPR)
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This assumes we only have one investment period.
What about multiple periods?
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Calculating Return - Multi periods
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Arithmetic average
– Arithmetic mean of returns
– Good measure for future performance
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Geometric average
– Geometric mean of returns
– The return measure that gives the same cumulative
performance as actual returns (buy-and-hold)
– Required for mutual fund performance
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Returns - Example
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Arithmetic Average: (0.14 -0.1455 + 0.10) / 3 = 3.15%
Geometric Average: (1 + RG)3 = (1 + 0.14)*(1 - 0.1455)*(1 + 0.10)
RG = [(1 + 0.14)*(1 - 0.1455)*(1 + 0.10)]1/3 – 1=2.33%
RG  RA , RG is a better measure for past performance
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Time-Weighted vs. Dollar-Weighted
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Year
Price/
share
2011
100
2012
2013
90
108
One-yr
Return
Action
Buy one share (cost $100)
-10%
20%
Buy one more share (cost $90)
Selling two shares (wealth $216)
Time-weighted: geometric average
[(1 –10%)*(1+20%)] 0.5 –1 = 8%
Dollar-weighted: solve internal rate of return (IRR)
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Conventions for Quoting Returns
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Annual Percentage Rate (APR)
– APR = periodic rate * number of periods per year
– Frequently quoted rate : mortgages, credit cards, car loans
– Ignores compounding of interest on interest
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Effective Annual Rate (EAR)
– Corrects APR for interest on interest compounding
– EAR= (1 + APR/n)n - 1
– As n approaches infinity (continuously compounding),
EAR = eAPR - 1
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Uncertainty of Investment
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Return and risk tradeoff (every investment has its
uncertainty)
At the time when we measure the expected level of
returns, we need to quantify the uncertainty (risk)
How to estimate the expected return and risk?
– based on probability distribution
– based on historical data
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Expected Return & Risk
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Expected Return (Mean)
– Find out possible future states
– Estimate probability and outcome for each state
– Sum of all possible outcomes by multiplying probabilities
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Risk (Variance or Standard deviation)
– The degree of spreaded outcomes, or deviation from mean
– Standard deviation is the square root of variance
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Expected Return & Risk - Example
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Initial investment : $100
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Expected return = 0.3(0.5)+0.5(0.2)+0.2(-0.4)=17%
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Standard deviation = (0.0981)0.5 = 0.313
Variance = 0.3(0.5-0.17)2+0.5(0.2-0.17) 2
+0.2(-0.4-0.17)2 = 0.0981
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Return & Risk - Historical data
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Treat each historical outcome equally and assign a
probability of 1/n ( n is number of observations)
Return
– Use sample average
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Risk
– Use sample variance
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Return & Risk -Historical data (example)
• Using Excel functions
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Terminology of Return and Risk
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Risk-free rate
– The rate of return that can be earned with certainty
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Risk premium
– Difference between return and risk-free asset return
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Risk aversion
– The degree to which an investor is unwilling to accept risk
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Inflation
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Different rates
– Inflation rate (i) : the rate at which prices are rising
– Nominal interest rate (R) : quoted interest rates.
– Real interest rate (r) : the growth rate of purchasing power
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Relationship
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When inflation is small :
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rR-i
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Risk-Free Asset
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Only the government can issue default-free bonds.
– Risk-free in real terms exists only if price indexed and
maturity equal to investor’s holding period.
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T-bills viewed as “the” risk-free asset
Money market funds also considered risk-free in
practice
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Asset Allocation
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Assume there are only two assets
Risky asset : E(rp) = 15%  p = 22%
Risk-free asset : rf = 7%
What portfolio C can we hold?
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Asset Allocation (continued)
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Capital Allocation Line (CAL)
– varying the weights between a risk-free asset and a risky
portfolio gives us all portfolio combinations, which fall on
a single line
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The slope of the CAL is the Reward-to-Variability
Ratio, or the Sharpe ratio
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Asset Allocation - Capital Allocation Line
• E(rc) = E(rp) * y + rf * (1-y)
•  c = y * p
 p = 22%
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Risk Aversion and Allocation
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Assume investors are risk averse, they invest a risky
security if it provides risk premium.
Greater (lower) levels of risk aversion lead investors
to choose larger (smaller) proportions of the riskfree rate
If the reward-to-variability ratio increase, then
investors might well decide to take on riskier
positions.
– If the slope of CAL increases, for the same level of risk,
investors get higher expected return; they might choose a
higher proportion in the risky portfolio.
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Capital Market Line
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The passive strategy avoids any security analysis
Supply and demand forces may make such a strategy
a reasonable choice for many investors
A natural candidate for a passively held risky asset
would be a well-diversified portfolio of common
stocks such as the S&P 500 (the market).
Capital market line (CML)
– capital allocation line formed from 1-month T-bills and a
broad index of common stocks (e.g. the S&P 500).
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Historical Evidence on CML
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From 1926 to 2013, the passive risky portfolio
offered an average risk premium of 8.34% with a
standard deviation of 20.23%, resulting in a rewardto-volatility ratio of 0.41.
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The Normal Distribution
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The Normal Distribution
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When returns are normally distributed
– Standard deviation is a good measure of risk when returns
are symmetric.
– Future scenarios can be estimated using only the mean and
the standard deviation.
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But if 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 skew and kurtosis
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Skew and Kurtosis
Skew
measures asymmetry of distribution
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Kurtosis
measures fat tail of distribution
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Normal and Skewed Distributions
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Normal and Fat-Tailed Distributions
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Value at Risk (VaR)
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A measure of downside risk
Measures the loss most frequently associated with
extreme negative returns
The 5% VaR, commonly estimated in practice, is the
return at the 5th percentile when returns are sorted
from high to low.
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Value at Risk (VaR)
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From the standard deviation we can find the corresponding
level of the portfolio return: VaR = E[r] + -1.64485
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A $500,000 stock portfolio has an annual expected return of
12% and a standard deviation of 35%. What is the portfolio
VaR at a 5% probability level?
VaR = 0.12 + (-1.64485 * 0.35)
VaR = -45.57%
VaR$ = $500,000 x (-.4557) = -$227,850
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Annual return distribution, 1926-2013
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