Risk and Return
François Cocquemas
1/10/2021
Risk and return
· Most economic agents are assumed to be risk averse
· They dislike taking risk, but will, for an appropriate risk premium
· Example: fair bet
2/20
Measuring returns
𝑡 𝑡 + ℎ:
HPR𝑡,𝑡+ℎ = capital gains yield𝑡,𝑡+ℎ + other income yield𝑡,𝑡+ℎ
· Holding period returns from to
·
· Capital gains yield = price appreciation
· Other income = dividends for stocks, coupons for bonds, etc.
· (Bonus: what else for stocks?)
3/20
Holding period returns for stocks
· Holding period returns for stocks:
·
𝑡,𝑡+ℎ 𝑃𝑡+ℎ𝑃−𝑡 𝑃𝑡 𝐷𝑃𝑡+ℎ𝑡 where is the stock price and dividends
· If there are more than one dividend, they need to be summed
HPR =
+
𝑃
𝐷
· NB. “Adjusted prices” already corrects prices for dividends, splits,
etc. so be careful when using those for HPR calculations
4/20
Holding period returns for stocks: example
· Suppose a stock is worth $50 today and $55 two years from now. It
pays a $2.5 dividend in one year
· What is the HPR? The capital gains yield? The dividend yield?
·
·
·
HPR0,2 = 55−50+2.5
50 = 15%
CGY0,2 = 55−50
50 = 10%
DY0,2 = 2.550 = 5%
· Wait… What mistake are we making?
· We are assuming we can’t reinvest the dividend
×
· So if risk-free rate is 2%, then we should add $2.5 1.02 and not
$2.5
· In fact,
DY0,2 = 2.55/50 = 5.1% and HPR0,2 = 15.1%
5/20
Measuring returns over multiple periods: example
AUM: Assets under management
Q1
Q2
Q3
Q4
Starting AUM* ($m)
1
1.2
2
0.8
Holding-period return (%)
10
25
−20
20
Total assets before net inflows ($m)
1.1
1.5
1.6
0.96
Net inflow ($m)
0.1
0.5
−0.8
0.6
Ending AUM ($m)
1.2
2
0.8
1.56
(10% + 25% + (−20%) + 20%)/4 = 8.75%
Geometric average: [1.1 × 1.25 × 0.8 × 1.2]1/4 − 1 = 7.19%
· Arithmetic average:
·
6/20
Measuring returns over multiple periods: example
AUM: Assets under management
Q1
Q2
Q3
Q4
1
1.2
2
0.8
Holding-period return (%)
10
25
−20
20
Total assets before net inflows ($m)
1.1
1.5
1.6
0.96
Net inflow ($m)
0.1
0.5
−0.8
0.6
Ending AUM ($m)
1.2
2
0.8
1.56
Starting AUM* ($m)
· Dollar-weighted average:
−0.1
−0.5
0.8
0.96
0 = −1 + 1 + IRR + (1 + IRR)2 + (1 + IRR)3 + (1 + IRR)4
IRR = 3.38%
· What is the best one?
7/20
Interest rate conventions
· APR = Annual Percentage Rate = Per-period rate
year
× Periods per
· Ignores compounding
· EAR = Effective Annual Rate
· Actual rate of growth: does not ignore compounding
· How do we recover the EAR?
𝑛
𝑛
APR
EAR = (1 + 𝑛 ) − 1
where is the number of compounding periods (e.g.
monthly payments)
𝑛 = 12 for
8/20
Inflation
· Nominal interest rate: the rate of growth for the dollar value of
investment
· Real interest rate: the rate of growth for the purchasing power value
of investment
· How do they relate?
𝑖
1 + 𝑟real = 1 +1 +𝑟nom
𝑖
where is the inflation rate
· Log-approximation:
𝑟real ≈ 𝑟nom − 𝑖
· Example: one-year CD rate is 8%, expected inflation 5%. What is the
real rate?
𝑟real = 1.08/1.05 − 1 = 2.86%
9/20
Inflation (change in CPI)
10/20
Inflation (change in PCE, excluding food and energy)
11/20
Continuously compounded rates
· For financial markets, we often compute returns using natural
logarithms ( )
ln
· These log-returns are nice because:
𝑃
−
𝑃
𝑃
𝑡+1
𝑡
ln(1 + 𝑃𝑡 ) = ln( 𝑃𝑡+1𝑡 ) = ln 𝑃𝑡+1 − ln 𝑃𝑡
· In other words, you can just sum log returns from each period from
1 to to get the overall log return from 1 to
𝑁
· Note that
ln(1 + 𝑥) ≈ 𝑥 for 𝑥 small
𝑁
12/20
Measuring risk
· Risk can be good or bad; excess returns are compensating for
uncertainty
· It is important to measure risk properly depending on the context
· This will be a major point of discussion throughout this class
13/20
Variance and volatility
· Variance and its square root, standard deviation, are common
measures of dispersion for a distribution
· You can apply them in different contexts, e.g., to assess the
accuracy of your forecast ex post
· Commonly, they are used on realized (excess) returns as a measure
of risk for an asset. The variance is:
𝑇𝑡=1 (𝑟𝑡 − 𝑟̄)2
∑
𝑉𝑎𝑟(𝑅) = 𝑇 − 1
where 𝑟𝑡 is the return at time 𝑡 and 𝑟̄ = 𝑇1 ∑ 𝑟𝑡 the average return
· NB. If the returns are already known, this is called Realized Variance
(RV), otherwise it is the Expected Variance
· Similarly for the volatility:
−−−−−−
𝜎𝑅 = √−𝑉𝑎𝑟(𝑅)
· Benefit of using volatility over variance?
14/20
Normal distribution
1.0
μ = 0,
μ = 0,
μ = 0,
μ = −2,
0.6
2
φμ,σ (x)
0.8
σ 2 = 0.2,
σ 2 = 1.0,
σ 2 = 5.0,
σ 2 = 0.5,
0.4
0.2
0.0
−5
−4
−3
−2
−1
0
x
1
2
3
4
5
Source: Wikipedia
15/20
Value-at-Risk
· VaR = Value-at-Risk, not to be confused with Var or var which is
variance (and VAR in finance also commonly means Vector AutoRegression)
· Based on a distribution (e.g. normal), what would be the loss in the
worst % of scenarios?
𝑘
· Typically
𝑘 = 5, 𝑘 = 1, or 𝑘 = 0.1
· However, the normal distribution has thin tails – may
underestimate catastrophic risk
· Also worth noting that during extreme events, correlations between
assets tend to go up
· The simple VaR measure does not factor in changes in the
distribution/in the correlation with other positions
16/20
Deviations from normality
· The returns on the S&P 500 over the last 50 years:
8.8% average (ann.), 15.2% volatility (ann.), skewness -1.04
(normal?), kurtosis 6.45 (normal?)
· Models using a lognormal distribution may be misspecified, maybe
even more so for other asset classes
17/20
Alternative measures of risk
· For assets that have extreme risks, worth measuring not just the
volatility, but also the skewness and the excess kurtosis
· Lower semi-deviation:
−∑−−−−−−−−−−−−−−−−−−−
𝑇𝑡=1 min(𝑅𝑡 − 𝑟𝑓 ,0)2−
𝑇
√
· Many other measures exist, e.g. lower semi-covariance
18/20
Relating risk and reward
· Once we know how to measure risk and returns, we need some
measures to combine them
· This way, we can rank investment opportunities ex-ante (expected)
or ex-post (realized)
· The Sharpe ratio is a reward-to-risk ratio, a simple and common
way of measuring this:
𝑅
−
𝑟
Risk
premium
Sharpe ratio = Volatility = 𝜎𝑅 𝑓
· Same as before: if expected Sharpe ratio, use expected returns;
otherwise, realized Sharpe ratio
· It is usually a good starting point to move on just looking at returns
· For next time: look up the Sharpe ratio of Bitcoin over the last
year
19/20
More measures of performance
· We’ll revisit performance measurement after we discuss the CAPM
framework
· We’ll talk about betas, M-squared, Treynor ratios, Information
ratios, and more
20/20