Understanding Risk and Return
Part 1: Risk and Return
Lecturer: Ilaria Piatti
School of Economics and Finance
A Basic Principle of Finance
• A safe pound is worth more than a risky pound!
• Individuals dislike risk. Hence, risky projects’ payoffs should be discounted at higher
rates than safe projects!
PV =
−
−
−
−
C1 is the payoff (cash flow)
r (here) is the risk-free discount rate
α is a measure of risk
What is risk?
1
C1
1+r +α
(1)
Expected Payoff
• It is very rare we can say with certainty what a payoff or return of an investment will
be in the future
• We can identify the possible outcomes that might occur and estimate how likely each
outcome is. . .
• The expected payoff to a project or asset is a probability weighted average of the
payoffs in each state of the world
Expected payoff =
n
X
s=1
− Cs is the payoff (cash flow) in state s
− ps is the probability of state s occurring
− n is the number of states
Cs × ps
Expected Return
• Payoff is different than Return:
• Today, you buy a rare coin for $20 and expect to sell it in twelve months time for $30.
Your expected payoff in one year is $30. Your expected return in one year is
50%=(30-20)/20, that is the gain divided by the initial cost.
• Holding period return (HPR) for a stock:
HPR =
P1 − P0 + DIV1
P0
(2)
Expected Return
• The expected return of an investment is a probability weighted average of the rates of
return in each state of the world:
Expected return =
n
X
s=1
− rs is the return in state s
− ps is the probability of state s occurring
− n is the number of states
rs × ps
(3)
Expected Return Example
Example
Consider a stock with purchase price $100.
State
Boom
Normal
Recession
Probability
0.3
0.5
0.2
P1
129.5
110.0
80.5
D1
4.5
4.0
3.5
Expected Return Example
Example
The holding period return (HPR) for the stock in each state is:
• Boom
HPR =
• Normal
129.5 − 100 + 4.5
= 0.34
100
HPR =
• Recession
HPR =
110 − 100 + 4
= 0.14
100
80.5 − 100 + 3.5
= −0.16
100
The expected return of the stock is:
E (r ) = 0.3 × 0.34 + 0.5 × 0.14 + 0.2 × (−0.16) = 0.14 = 14%
Risk
• The standard deviation of the rate of return, σ, is a measure of risk
• The standard deviation is the square root of the variance σ 2
• The variance is the expected value of the squared deviations from the expected return
n
X
[rs − E (r )]2 × ps
(4)
rs2 × ps − E (r )2 = E r 2 − E (r )2
(5)
σ2 =
s=1
• This is equivalent to:
2
σ =
n
X
s=1
• Standard deviation:
σ=
√
σ2
(6)
Expected Return Example
Example
From the previous example,
• The variance is given by:
σ 2 = 0.3 × 0.342 + 0.5 × 0.142 + 0.2 × (−0.16)2 − 0.142 = 0.03
• The standard deviation is given by:
√
σ = 0.03 = 0.1732 = 17.32%
Risk
• Larger values of σ imply a wider dispersion of returns: it measures the total variability
of the return on the stock in isolation
• We can thus use σ as a measure of risk:
− High σ implies high risk
− Low σ implies low risk
• Works well if the distribution is symmetric around the mean
Risk-Free Rate
• Note that, because of inflation and a general preference for immediate consumption,
investors will always require some return even if there’s no risk
• We call this return: Risk-free rate rf
• The risk free rate: usually short term government loans (Ex: T-bills)
− The default risk is close to zero
− Since the maturity is short, they are ’immune’ to changes in the interest rates
− Inflation may still be an issue!
Risk-Premium
• Most investors are risk averse: they dislike risk, so between two investments giving the
same return they prefer the one with the lower risk
• The only way to persuade a risk averse investor to take on more risk is to give an
extra return above the risk free rate: a Risk premium!
• The risk premium is the expected value of the excess return, i.e. the difference
between the actual rate of return and the risk free rate.
Risk-Premium
Definition
The risk premium is the difference between the expected return from investing in a risky
asset and the return from investing in a risk-free asset:
Risk Premium = E (r ) − rf
(7)
Example
A stock has an expected return E (r ) of 14% and the T-bill rate (risk-free) is rf = 6%:
Risk Premium = 14% − 6% = 8%
(8)
Scenario vs Time Series Analysis
• In a forward-looking Scenario Analysis:
− you determine a set of relevant scenarios and associated rates of return
− you attribute a probability to each scenario
− you compute risk and risk premium of the proposed investment
• In Time Series Analysis:
− we observe time series of actual returns
− we infer from this data the probability distribution from which the returns are drawn
Learning from Historical Returns
• When using historical data we treat each observation as an equally likely scenario!
• With n observations, the expected return is the arithmetic average of the historical
rates of return:
E (r ) =
n
X
s=1
n
rs × ps =
• we replace equal probabilities 1/n for each ps !
1X
rs = r¯
n
s=1
(9)
Learning from Historical Returns
• If the time series of returns fairly represents the true underlying distribution, the
arithmetic average return provides a reasonable forecast of the expected future return
• The compound rate of return is referred to as the geometric average return
• If you have a portfolio over a period of 5 years from 20 quarterly rates of return, the
geometric average is the constant quarterly return over the 20 quarters that yields the
same total or cumulative return:
(1 + rG )20 = (1 + r1 ) × (1 + r2 ) × . . . × (1 + r20 )
• (1 + rG )20 is the compounded value of a $1 investment earning rG each quarter!
Historical Variance
• Using historical data, with n observations, we can estimate the variance as the sum of
squared deviations from the mean r¯, i.e. our estimate of the expected return:
n
1X
(rs − r¯)2
σ̄ =
n
2
s=1
(10)
Learning from Historical Returns: Example
Look at the returns of 3 portfolios of US securities with different degree of risk:
1. Treasury Bills: maturing in less than one year with effectively no risk of default and
price relatively stable because of the short maturity
2. Treasury Bonds: long term bonds, more risky as their price fluctuates with the interest
rate (higher duration!)
3. Common Stocks: the riskiest as their return is affected by the ups and downs of the
corporation
Learning from historical returns
Learning from Historical Returns: Example
I How an investment of $1 at the end of 1899 would have
This is how an investment
of $1 in the three classes at the end of 1899 would have grown
grown by the end of 2017.
by the end of 2017!
(Figure 1)
[source: E.Dimson, P.R. Marsh, M. Staunton (2002) ”Triumph of the
[Source: Dimson, Marsh and Staunton (2002), “Triumph of the optimists. 101 years of
optimists. 101 years of global investment returns”, Princenton University Press]
global investment returns”, Princeton University Press]
Learning from Historical Returns: Example
• Average annual rates of returns on US Treasury Bills, Government Bonds and
Common Stocks, over the period 1900-2017:
T-Bills
Gov Bonds
Stocks
r¯
3.8
5.3
11.5
σ̄
2.9
9.0
19.7
Risk Premium
0
1.5
7.7
• Higher returns are associated with higher risk and risk premium!
Normality Assumption
• Investment management is far more tractable when returns can be well approximated
by a normal distribution! Why?
− Standard deviation is a good measure of risk when returns are symmetric, i.e. positive
and negative deviations from the mean are equally likely
− If assets have normally distributed returns, also the return of a portfolio of those assets
will be normally distributed
− Mean and standard deviation are sufficient to describe the entire distribution of returns
(which makes scenario analysis easier)
− The statistical relation between returns of different normally distributed assets can be
simply summarized by the regression coefficient
Skewness
and Kurtosis
Learning from
historical returns
• If returns are not normally distributed, standard deviation is no longer a complete
I If returns are not normally distributed =)Standard deviation
measure of risk!
is no longer a complete measure of risk!
• Need to consider Skewness and Kurtosis:
I Need to consider skewness and kurtosis
(Skew and Kurtosis)
Skewness Formula
• Skewness is the standard measure of asymmetry in a probability distribution
n
1 X (ri − r¯)3
Skew =
n
σ3
i=1
• If skewness is positive, standard deviation overestimates risk
• If skewness is negative, standard deviation underestimates risk
(11)
Kurtosis Formula
• Kurtosis measures the peakedness of the distribution and the degree of fat tails
• It is a measure of Tail risk, i.e. risk of extreme outcomes!
n
1 X (ri − r¯)4
Kurt =
n
σ4
(12)
i=1
• A normal distribution has Kurt = 3, so we measure Excess Kurtosis as Kurt − 3.
• If excess kurtosis is positive, the standard deviation underestimates the likelihood of
extreme events
Discuss the Following Example
Problem
The next three pictures show log returns (rt = log (Pt ) − log (Pt−1 ) of US, Japanese and
Chinese stock markets observed every 30 minutes. What do you think about these three
markets?
Discuss the Following Example
The next 3 pictures: Log Returns on the US, Japanese and
Chinese
stocks
[ rt = pt pt 1 ] observed every 30 minutes. What
US Stock
Market
Returns:
do you think about these 3 markets?
(US Log Returns 30min)
Problem
Discuss the Following Example
Japanese Stock Market Returns:
(Japan Log Returns 30min)
Problem
Discuss the Following Example
Chinese Stock Market Returns:
(China Log Returns 30min)