BUSI3502: Investments

Week 5: October 7th, 2021

Risk, Return & the Historical Record

Course Professor: Dr. M. Al Guindy

Today: Time Permitting

Return & Risk

1. Investment
Environment 4. Utility and 11. Derivatives
5. Risky
Probability
Portfolios
Theory

2. Financing
Vehicles Factor Models

12. Portfolio
6. CAPM 7. APT
Management

8. Market Efficiency &

Behvaioural Finance

Bonds
3. Trading 13. FinTech
9. Term 10. Duration/
Structure Convexity

2
Overview – Textbook Chapter 5

◼ Interest rate determinants

◼ Rates of return for different holding periods
◼ Risk and risk premiums
◼ Estimations of return and risk
◼ Normal distribution
❑ Deviation from normality and risk estimation
◼ Historic returns on risky portfolios

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Interest rate determinants
◼ Supply
❑ Savers, primarily households
◼ Demand
❑ Businesses
◼ Government’s Net Supply and/or Demand
❑ Central Bank Actions
◼ The expected rate of inflation.

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Real vs nominal interest rate
rnom = Nominal Interest Rate
rreal = Real Interest Rate
i = Inflation Rate

𝒓𝒏𝒐𝒎 − 𝒊
𝒓𝒓𝒆𝒂𝒍 =
𝟏 + 𝒊
d

𝑵𝒐𝒕𝒆 ∶ 𝒓𝒓𝒆𝒂𝒍 ≈ 𝒓𝒏𝒐𝒎 − 𝒊

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T-Bills & inflation 1957-2016
◼ Moderate inflation offsets most nominal gains on
low-risk investments
◼ $1 in T-bills from 1957–2016 grew to $29.14 but
with a real value of only $3.25

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Taxes and the real interest rate
◼ Tax liabilities are based on nominal income
rnom = Nominal Interest Rate
rreal = Real Interest Rate
i = Inflation Rate
t = Tax Rate
rnom  (1 − t ) − i = (r real + i )  (1 − t ) − i = rreal (1 − t ) − i  t

The after-tax real rate falls as the inflation rises!

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 Measuring Historical Return
◼ What are the 2 components of a stock’s
return?
❑ Dividend Yield
❑ Capital Gains (or Loss)

◼ For a single period, how do we calculate

return?
𝐷1 𝑃1 − 𝑃0
𝑟1 = +
𝑃0 𝑃0
r1 = return at the end of period 1
D1 = dividend at the end of period 1
Dividend Yield P1 = stock price at the end of period 1
P0 = stock price at the end of period 0
Capital Gain/Loss

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 Recall: Holding Period Return (HPR)
Rates of return: Single period:
P1 − P0 + D1
❑ HPR = Holding Period Return
❑

❑
P0 = Beginning price
P1 = Ending price
HPR =
❑ D1 = Dividend during period one P0
Example: Ending Price $110
Beginning Price $100
P1 − P0 + 𝐷1 $110 − $100 + $4
HPR = = Dividend $4
P0 $100
$110 − $100 $4
= +
$100 $100
= 10% Capital Gains yield + 4% Dividend Yield
= 14% Holding Period Return

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Securities’ total returns
◼ Suppose prices of zero-coupon Treasuries with $100
face value and various maturities are as follows. We
find the total return of each security
Total Return
Horizon, Price,
[100/P(T)] − 1 for Given
T P(T)
Horizon
100/97.36 − 1 = rf (0.5) =
Half-year $97.36
0.0271 2.71%
100/95.52 − 1 = rf (1) =
1 year $95.52
0.0469 4.69%
100/23.30 − 1 = rf (25) =
25 years $23.30
3.2918 329.18%

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EAR vs APR
Effective Annual Rate (EAR):
1
1 + EAR = 1 + rf (T )  T

Annualized Percentage Rate (APR):

(1 + EAR )
T
−1
APR =
T

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EAR given a 10% APR
Compounding Period # Of Times Effective Annual Rate
Compounded

Year 1 10.00000%
Quarter 4 10.38129%
Month 12 10.47131%
Week 52 10.50648%
Day 365 10.51558%
Hour 8,760 10.51703%
Minute 525,600 10.51709%
Continuous Infinite 10.52%

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 Continuous Compounding
◼ The general formula for the future value of an
investment compounded continuously over many
periods can be written as:
𝐹𝑉 = 𝐶0 𝑒 𝑟𝑇
where:
❑ C0 is cash flow at date 0,

❑ r is the stated annual interest rate,

❑ T is the number of periods over which the cash is
invested, and
❑ e is a mathematical constant approximately equal to
2.718. ex is the exponential function and also a key on
your calculator.
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Expected Return & Standard Deviation
Expected returns
❑
❑
p(s) = Probability of a state
r(s) = Return if a state occurs
E ( r ) =  p( s ) r ( s )
❑ s = State s

 =  p( s )  r ( s ) − E (r ) 
2
Risk → Variance 2

Standard deviation = σ

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Example
State Prob. of State r in State
Excellent .25 0.3100
Good .45 0.1400
Poor .25 -0.0675
Crash .05 -0.5200

E(r) = (.25)×(.31)+(.45)×(.14)+(.25)×(-.0675) +(0.05) ×(-0.52)

E(r) = 0.0976 or 9.76%
𝝈𝟐 = .25 × .31 − 0.0976 2 + .45 × .14 − .0976 2
+ .25 × −0.0675 − 0.0976 2 + .05 × −.52 − .0976 2

= .038

𝝈 = .038
= .1949

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Time series analysis of past rates of return

◼ True means and variances are

unobservable because we don’t actually
know possible scenarios like the one in the
examples

◼ So we must estimate the means and

variances

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Mean & Variance of Historical Returns

Return:
𝑛

𝐸(𝑟) = ෍ 𝑟 (𝑠)
𝑠=1

Variance:
𝑛
𝑖
𝜎 = ෍[ 𝑟(𝑠) − 𝑟]2
2
𝑛
𝑠=1

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 Geometric vs. Arithmetic Average Returns
1957-2014
Average Return

Investment Arithmetic (%) Geometric (%) Standard deviation (%)

Canadian common stocks 10.30 9.01 16.50
U.S. common stocks (Cdn $) 11.73 10.42 16.94
Long bonds 8.57 8.16 9.78
Small stocks 12.81 9.83 25.96
TSX Venture stocks 7.16 −2.69 44.35
Inflation 3.82 3.77 3.10

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Sharpe Ratio
"A ratio developed by Nobel laureate William F. Sharpe to measure
risk-adjusted performance. The Sharpe ratio is calculated by
subtracting the risk-free rate - such as that of the 10-year U.S. Treasury
bond - from the rate of return for a portfolio and dividing the result by
the standard deviation of the portfolio returns."

Sharpe Ratio for a single Asset: For Portfolios:

𝑟𝑥 − 𝑟𝑓 𝑟𝑝 − 𝑟𝑓
𝜎𝑥 𝜎𝑝
• 𝑟𝑥 is the asset return • 𝑟𝑝 is the portfolio return
• 𝑟𝑓 is the risk-free rate • 𝑟𝑓 is the risk-free return (3-Month T-
• 𝜎𝑥 is the standard deviation of asset x Bills)
• 𝜎𝑝 is the standard deviation of your
portfolio returns

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The Reward-to-Volatility (Sharpe) Ratio

❑ Excess Return
❑ Risk Premium
❑ Sharpe Ratio

𝑹𝒊𝒔𝒌 𝒑𝒓𝒆𝒎𝒊𝒖𝒎
Sharpe Ratio =
𝑺𝒕𝒅.𝑫𝒆𝒗.𝒐𝒇 𝒆𝒙𝒄𝒆𝒔𝒔 𝒓𝒆𝒕𝒖𝒓𝒏𝒔

20
Using Sharpe Ratio
◼ "The Sharpe ratio is a risk-adjusted measure of return that is often
used to evaluate the performance of a portfolio. The ratio helps to
make the performance of one portfolio comparable to that of another
portfolio by making an adjustment for risk.“

◼ Imagine a portfolio manager "Gambler" with a return of 15% per

annum and a second portfolio manager "Brainy" with a return of
only 8%. One night at a bar Gambler brags about his return being
better than Brainys'. Brainy, correctly, states that we do not know if
his return is better because we don't know about the risk he took.
◼ The next night Brainy comes back to the bar and proudly states that
his "risk" is about 5 in terms of the standard deviation. Gambler
proudly crows that his risk is even larger, "24 baby" he proudly
states! Assume the risk-free rate = 3%

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Sharpe Ratio (cont.)
◼ What were their Sharpe ratios?

𝟏𝟓%−𝟑% 𝟖%−𝟑%
𝑮𝒂𝒎𝒃𝒍𝒆𝒓 𝑺𝑹 = 𝑩𝒓𝒂𝒊𝒏𝒚 𝑺𝑹 =
𝟐𝟒 𝟓
= 0.5 = 1.0
❑ For Sharpe ratios the higher the better.
❑ 1 is a decent Sharpe Ratio, 2 is better, and 3 is almost
unattainable in the long-run in the real world!

𝑹𝒊𝒔𝒌 𝒑𝒓𝒆𝒎𝒊𝒖𝒎
Sharpe Ratio =
𝑺𝒕𝒅.𝑫𝒆𝒗.𝒐𝒇 𝒆𝒙𝒄𝒆𝒔𝒔 𝒓𝒆𝒕𝒖𝒓𝒏𝒔

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The Normal Distribution

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The Normal Distribution
◼ Investment management is easier when
returns are normally distributed:
❑ Standard deviation is a good measure of risk
when returns are symmetric
❑ If security returns are symmetric, portfolio returns
will be as well
❑ Only mean and standard deviation needed to
estimate future scenarios

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 Normality & Risk Measures
◼ What if excess 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
R − Rሜ 3
◼ Skewness: Skew = Average
σ3
ෝ
R − Rሜ 4
◼ Kurtosis: Kurtosis = Average 4 − 3
σ
ෝ
◼ Lower Partial Standard Deviation (LPSD)

◼ Sortino Ratio

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Normal & Skewed Distributions

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Normal & Fat-Tailed Distributions

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❑ The second half of the
20th century offered the
highest average returns

❑ Firm capitalization is
highly skewed to the
right: Many small but a
few gigantic firms

❑ Average realized returns

have generally been
higher for small stocks vs.
large stocks

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Next week – continue Risk & Return
Time Permitting

Return & Risk

1. Investment
Environment 4. Utility and 11. Derivatives
5. Risky
Probability
Portfolios
Theory

2. Financing
Vehicles Factor Models

12. Portfolio
6. CAPM 7. APT
Management

8. Market Efficiency &

Behvaioural Finance

Bonds
3. Trading 13. FinTech
9. Term 10. Duration/
Structure Convexity

29
 APPENDIX material

- Arithmetic vs geometric returns

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 Arithmetic Average Return Vs. Geometric
Average Returns
◼ The Arithmetic Average Return (“mean”) answers the question:
“What was your return in an average year over a particular period?”
❑ This is how average is “normally” calculated

◼ The Geometric Average Return (“GAR”) answers the question “What

was your average compound return per year over a particular
period?”
❑ GAR = [(1 + R1) x (1 + R2) x …x (1 + RT)]1/T – 1

◼ Geometric returns will always be smaller than arithmetic returns(1)

Note (1): As long as the returns are not all identical in which case the “averages” will be
the same.

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Arithmetic vs. Geometric Average Example

◼ You invested $100 in a stock five years ago. Over

the last five years, annual returns have been 15%,
-8%, 12%, 18% and -11%. What is your average
annual rate of return? What is your investment
worth today?

Arithmetic Average Return = 𝑹𝑨

𝟏𝟓 + (−𝟖) + 𝟏𝟐 + 𝟏𝟖 + (−𝟏𝟏)
=
𝟓
𝐑 𝐀 = 5.2%

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What is the investment worth today?

FV=$100(1+.15)(1-.08)(1+.12)(1+.18)(1-.11)
FV=$124.44

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Calculating Geometric Average Continued

◼ What equivalent rate of return would you have to earn

every year on average to achieve this same future
wealth?
$124.44
= $100 × (1 + 𝑅𝐺 )5
1ൗ
𝑅𝐺 = 1.2444 5 − 1
𝑅𝐺 = 4.47%

◼ Your average return was 4.47% each year. Notice that

this is lower than the arithmetic average. This is
because it includes the effects of compounding.

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Geometric Average

◼ The general formula for calculating the

geometric average return is the following:

Geometric Average Return

1ൗ
= 1 + R1 × 1 + R 2 ×. . .× 1 + R T 𝑇 −1

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