Transcript Overview of Risk and Return

Overview of Risk and Return
Timothy R. Mayes, Ph.D.
FIN 3600: Chapter 2
Risk and Return are Both Important

It is important to consider both risk and return when
making investment decisions. Over long periods of time
(more than a year or two), risk and return tend be highly
correlated as shown in the table below.
10 Years Ended Dec 31, 2001
Money-Market Funds
Treasuries1
Corporate Bonds2
Dow Jones Industrials
S&P 500
Nasdaq Composite
Average
$10,000 in 1992
Annual Return
Grows To
4.5% $
15,530
7.0%
19,672
7.7%
20,997
17.3%
49,315
14.2%
37,728
17.6%
50,591
Note: Figures are total return except Nasdaq data, which don't included dividends
I have ranked investments, roughly, from lowest risk to highest
1
A basket of Treasury Securities
2
A baskey of investment-grade-rated corporate bonds
Source: Wall Street Journal, 28 January 2002, p. R6
Best Year
Worst Year
2001
Return
Return
Return
5.9%
2.7%
3.7%
18.5%
-3.3%
6.2%
21.6%
-3.3%
9.5%
36.9%
-5.4%
-5.4%
37.6%
-11.9%
-11.9%
85.6%
-39.3%
-21.1%
Sources of Returns

Returns on investment can come from one of two
sources or both:



Capital Gain – This is the increase (or decrease) in
the market value of the security
Income – This is the periodic cash flows that an
investment may pay (e.g., cash dividends on stock, or
interest payments on bonds)
Note that your total return is the sum of your
capital gains and income
Measuring Returns for One Period



Investors look at returns in various ways, but
the most basic (not necessarily the best) is the
single period total return
A period is defined as any appropriate period
of time (year, quarter, month, week, day, etc.)
This measure is known as the Holding Period
Return (HPR):
Ending Price  Cash Flows
HPR 
1
Beginning Price
HPR Example

Suppose that you purchased 100 shares of XYZ
stock for $50 per share five years ago. Recently,
you sold the stock for $100. In addition, the
company paid a dividend each year of $1.00 per
share. What is your HPR?
100  1  1  1  1  1
105
HPR 
 1
 1  110
.  110%
50
50
Annualizing HPRs


If a calculated HPR is for a non-annual holding
period, we generally annualize it to make it
comparable to other returns
The general formula is:
HPRAnnualized  (1  HPRNon  Annual )m 1


Where m is the number of periods per year
Note that m will be > 1 for less than annual
periods and < 1 for greater than annual periods
Annualizing HPRs (cont.)

As an example, suppose that you earned a return
of 5% over a period of three months. There are 4
three-month periods in a year, so your
annualized HPR is:
HPRAnnualized  (1.05)4 1  21.55%

Note that this calculation assumes that you can
repeat this performance every three months for a
year
Annualizing HPRs (cont.)


For another example, suppose that you earned an
HPR of 47% over a period of 5 years. In this
case, your annualized HPR would be 8.01% per
year
Note that in this case, we use an exponent (m) of
1/5 because a year is 1/5th of a five-year period
HPR Annualized  (1.47 )
1
5
 1  8.01%
Multi-period Returns

HPRs provide an interesting bit of data, but they
suffer from some flaws:




The HPR ignores compounding
The HPR is usually not comparable to other returns
because it isn’t an necessarily annualized return
The solution to these problems is to calculate the
IRR of the investment
A security investment’s IRR is usually referred
to as its Holding Period Yield (HPY)
Calculating the HPY



Since the HPY is the same as the IRR, there is no
general formula for finding the HPY
Instead, we must use some iterative procedure
(or a financial calculator or spreadsheet function)
For the XYZ investment, the HPY is 16.421%
per year:
50 
1
1
1
1
1  100




2
3
4
5
116421
.
116421
.
116421
.
116421
.
116421
.

 
 
 

Problems with the HPY

Generally, the HPY is superior to the HPR as a
measure of return, but it also has problems:



The HPY assumes that cash flows are reinvested at
the same rate as the HPY
The HPY assumes that the cash flows are equally
spaced in time (i.e., every year or every month)
The HPY makes no provision for stock splits, stock
dividends, or partial purchases or sales of holdings
The Reinvestment Assumption

To see that the reinvestment assumption is
implicit in the calculation of the HPY, let’s try a
few different reinvestment rates and see what the
compound average annual rate of return is:
Period
Cash Flow
0
-50.00
1
1.00
2
1.00
3
1.00
4
1.00
5
101.00
Total FV
Avg. Ann. Return
FV at 10%
FV at 16.421%
FV at 20%
1.464
1.331
1.210
1.100
101.000
106.105
16.239%
1.837
1.578
1.355
1.164
101.000
106.935
16.421%
2.074
1.728
1.440
1.200
101.000
107.442
16.531%
The Timing Assumption



In practice, investments often do not pay all cash
flows at convenient equally spaced time periods
This will cause most calculator and spreadsheet
functions to not work properly unless
adjustments are made
The adjustment is to change to a common
definition of a period, and to include cash flows
of $0 for periods without a cash flow
An Example of a Timing Problem

In this example, we simply
change the timing of the
dividends (Note that the
period 3 dividend was
omitted).
Period Cash Flow
0.0
-50.00
0.5
1.00
1.0
1.00
2.0
1.00
4.0
1.00
5.0
101.00
HPY
16.421%
This is wrong!!!!
Period
Cash Flow
0.0
-50.00
0.5
1.00
1.0
1.00
1.5
0.00
2.0
1.00
2.5
0.00
3.0
0.00
3.5
0.00
4.0
1.00
4.5
0.00
5.0
101.00
HPY per period
7.964%
Annualized HPY
16.563%
This is correct!!!
Handling Stock Splits, etc.



Stock splits and stock dividends complicate the finding
of the true HPY.
For example, suppose that XYZ split 2 for 1 immediately
after period 3. In this case, your dividends would be only
$0.50 per share in periods 4 and 5, and you would be
selling the stock for $50 in period 5 (but you will have
the same wealth).
Your true HPY is the same, but if you don’t adjust for the
split you will get an incorrect HPY of 1.61% per year
(and, your HPR would be 8.00%).
Arithmetic vs. Geometric Returns



When they need to calculate a rate of return over a
number of periods, people often use the arithmetic
average. However, that is incorrect because it ignores
compounding, and therefore tends to overstate the return.
Suppose that you purchased shares in CDE two-years
ago. During the first year, the stock doubled, but it fell
by 50% in the second year. What is your average annual
rate of return (it should be obvious)?
Arithmetic: R  100 %  (50%)  25%
2

Geometric:
R
111 (0.50) 1  0.00%
Returns on Foreign Investments


Calculating the return on a foreign investment is very
similar to domestic investments, except that we must take
the change in the currency into account. So, we actually
have two sources of return.
For example, suppose that you purchased shares of
Pohang Iron & Steel (POSCO) on the Korean Stock
Exchange (KSE) on Jan 3, 1997 and sold them on Dec
27, 1997. Here are the details:
Exch. Rate (Won
per Dollar)
Date
Price (Won)
1/3/97
37,300
842.60
12/27/97
45,900
1,500.00
Returns on Foreign Investments (cont)


Now, if you were a Korean investor your return
for the year would have been 23.06%
However, as a U.S. investor your return was a
negative 30.88%! Quite a difference, and it was
entirely due to the loss in value of the won
relative to the dollar during the “Asian
Contagion” currency crisis that began in
Thailand in June 1997
Returns on Foreign Investments (cont)

To calculate this return, we first need to calculate your
investment in dollar terms:
Dollar Cost P0 

1
FC0
Where P0 is the cost in foreign currency, and FC0 is the
exchange rate (foreign currency unit/dollar). Your
proceeds from the sale are calculated the same way:
DollarPr oceeds  P1 

1
FC1
Combining the equations into a rate of return, and
rearranging we get the return in local currency (RLC):
R LC 
P1  FC0
1
P0  FC1
Returns on Foreign Investments (cont)

Now, we can see that your return in dollar
terms was -30.88%
45900  842 .60
 1  0.3088  30.88%
37300 1500

So, you made money on the stock, lost on the
currency, and overall you lost a lot of money
on this investment
Returns on Foreign Investments (cont)


Here’s another example. On Jan 27, 1999 Diageo PLC
(LSE: DGE) was selling for 630p. One year earlier it
was selling for 542p, so a British investor would have
earned a return of 16.24%. However, an American
investor would have made 17.78%
The American made more because the British pound (£)
appreciated against the dollar over that year. Note that
the American originally paid $8.87, but received $10.45
and the return is 17.78%.
6.30  .6108
 1  17.78%
5.42  .6028
Negative Returns



All of the examples we’ve seen so far assume that your investment
appreciates in value. However, its very likely that you will lose
money occasionally.
The formulas that we’ve seen work just as well for negative returns
as for positive returns.
For example, assume that you purchased a stock for $50 three
months ago, and it is now worth $40. What is your HPR and
annualized HPR? Assume no dividends were paid.
40  0
 1  0.20  20.00%
50
HPRAnnualized  (1  (0.20))4  1  0.5904 59.04%
HPR 
Negative Returns (cont.)


An often overlooked problem with losses is that you must earn a
higher percentage return than you lost just to get even.
Using our example, you lost 20%. If the stock now rises by 20%
you are not back to $50.
40(1.20)  48  50

To figure the “gain to recover” use the formula (%L is the loss):

1
%GTR 
1
1 %L
So, you would need to earn a return of 25% to get back to $50:
1
%GTR 
 1  0.25  25%
1  0.20
What is Risk?




A risky situation is one which
has some probability of loss
The higher the probability of
loss, the greater the risk
If there is no possibility of
loss, there is no risk
The riskiness of an investment
can be judged by describing
the probability distribution of
its possible returns

Types of Risk









Default Risk
Credit Risk
Purchasing Power Risk
Interest Rate Risk
Systematic (Market) Risk
Unsystematic Risk
Event Risk
Liquidity Risk
Foreign Exchange (FX)
Risk
Probability Distributions


A probability distribution
is simply a listing of the
probabilities and their
associated outcomes
Probability distributions
are often presented
graphically as in these
examples
Potential Outcomes
Potential Outcomes
The Normal Distribution


For many reasons, we
usually assume that the
underlying distribution
of returns is normal
The normal distribution
is a bell-shaped curve
with finite variance and
mean
The Expected Value



The expected value of a
distribution is the most
likely outcome
For the normal dist., the
expected value is the
same as the arithmetic
mean
All other things being
equal, we assume that
people prefer higher
expected returns
E(R)
E R  
N
 R
t
t 1
t
The Expected Return: An Example
Suppose that a particular
investment has the
following probability
distribution:




25% chance of 10% return
50% chance of 15% return
25% chance of 20% return
This investment has an
expected return of 15%
Probability

60%
40%
20%
0%
10%
15%
20%
Rate of Return
E (R i )  0.25010
.   0.50015
.   0.250.20  015
.
The Variance & Standard Deviation


The variance and
standard deviation
describe the dispersion
(spread) of the potential
outcomes around the
expected value
Greater dispersion
generally (not always!)
means greater
uncertainty and therefore
higher risk
Less Risky
Riskier
  R
N
 
2
R
t
t 1
R 
 2R
t
R

2
Calculating  2 and  : An Example

Using the same example as for the expected
return, we can calculate the variance and
standard deviation:
 2  0.25010
.  015
.   0.50015
.  015
.   0.250.20  015
.   0.00125
2
2
2
  0.25010
.  015
.   0.50015
.  015
.   0.250.20  015
.   0.03536
2
2
2
The Scale Problem


The variance and standard deviation suffer from
a couple of problems
The most tractable of these is the scale problem:

Scale problem - The magnitude of the returns used to
calculate the variance impacts the size of the variance
possibly giving an incorrect impression of the
riskiness of an investment
The Scale Problem: an Example
Prob
10%
15%
50%
15%
10%
E(R)
Variance
Std. Dev.
C.V.
Potential Returns
ABC
XYZ
-12%
-24%
-5%
-10%
2%
4%
9%
18%
16%
32%
2.0%
4.0%
0.00539 0.02156
7.34%
14.68%
3.6708
3.6708
Is XYZ really twice
as risky as ABC?
The Coefficient of Variation



The coefficient of variation (CV) provides a
scale-free measure of the riskiness of a security
It removes the scaling by dividing the standard
deviation by the expected return
In the previous example, the CV for XYZ and
ABC are identical, indicating that they have
exactly the same degree of riskiness
CV 
R
R
Historical vs. Expected Returns & Risk



The equations just presented are for ex-ante
(expected future) data.
Generally, we don’t know the probability
distribution of future returns, so we estimate it
based on ex-post (historical) data.
When using ex-post data, the formulas are the
same, but we assign equal (1/n) probabilities to
each past observation.
Portfolio Risk and Return


The preceding risk and return measures apply to
individual securities. However, when we
combine securities into a portfolio some things
(particularly risk measures) change in, perhaps,
unexpected ways.
In this section, we will look at the methods for
calculating the expected returns and risk of a
portfolio.
Portfolio Expected Return

For a portfolio, the expected return calculation is
straightforward. It is simply a weighted average
of the expected returns of the individual
securities:
N
E RP    wi E Ri 
i 1

Where wi is the proportion (weight) of security i
in the portfolio.
Portfolio Expected Return (cont.)


Suppose that we have three securities in the portfolio.
Security 1 has an expected return of 10% and a weight of
25%. Security 2 has an expected return of 15% and a
weight of 40%. Security 3 has an expected return of 7%
and a weight of 35%. (Note that the weights add up to
100%.)
The expected return of this portfolio is:
ERP   0.250.10  0.400.15  0.350.07  0.1095 10.95%
Portfolio Risk




Unlike the expected return, the riskiness (standard deviation) of a
portfolio is more complex.
We can’t just calculate a weighted average of the standard deviations
of the individual securities because that ignores the fact that
securities don’t always move in perfect synch with each other.
For example, in a strong economy we would expect that stocks of
grocery companies would be moderate performers while technology
stocks would be great performers. However, in a weak economy,
grocery stocks will probably do very well compared to technology
stocks. Both are risky, but by owning both we can reduce the overall
riskiness of our portfolio.
By combining securities with less than perfect correlation, we can
smooth out the portfolio’s returns (i.e., reduce portfolio risk).
Portfolio Risk (cont.)
The following chart shows what happens when we combine two
risky securities into a portfolio. The line in the middle is the
combined portfolio. Note how much less volatile it is than either of
the two securities.
30%
25%
20%
15%
Return

10%
5%
0%
-5%
2000
2001
2002
2003
Year
Stock A
Stock B
Portfolio
2004
Portfolio Risk (cont.)




The key to the risk reduction shown on the previous chart
is the correlation between the securities.
Note how Stock A and Stock B always move in the
opposite direction (when A has a good year, B has a not
so good year and vice versa). This is called negative
correlation and is great for diversification.
Securities that are very highly (positively) correlated
would result in little or no risk reduction.
So, when constructing a portfolio, we should try to find
securities which have a low correlation (i.e., spread your
money around different types of securities, different
industries, and even different countries).
Portfolio Risk Quantified



The correlation coefficient (rxy) describes the degree to which two
series tend to move together. It can range from +1.00 (they always
move in perfect sync) to -1.00 (they always move in different
directions). Note that rxy = 0 means that there is no identifiable
(linear) relationship.
Our measure of portfolio risk (standard deviation) must take account
of the riskiness of each security, the correlation between each pair of
securities, and the weight of each security in the portfolio.
For a two-security portfolio, the standard deviation is:
 P  w   w   2r1, 2 1 2w1w2
2
1

2
1
2
2
2
2
The equation gets longer as we add more securities, so we will
concentrate on the two-security equation.
Portfolio Risk Quantified (cont.)


Suppose that we are interested in two securities, but they are both
very risky. Security 1 has a standard deviation of 30% and security 2
has a standard deviation of 40%. Further, the correlation between
the two is quite low at 20% (r1,2 = 0.20).
What is the standard deviation of a portfolio of these two securities if
we weight them equally (i.e., 50% in each)?
P 

0.50 2 0.30 2  0.50 2 0.40 2  20.20 0.30 0.40 0.50 0.50   0.273
Note that the standard deviation of the portfolio is less than the
standard deviation of either security. This is what diversification is
all about.
Determining the Required Return


The required rate of return for a particular investment
depends on several factors, each of which depends on
several other factors (i.e., it is pretty complex!):
The two main factors for any investment are:



The perceived riskiness of the investment
The required returns on alternative investments (which includes
expected inflation)
An alternative way to look at this is that the required
return is the sum of the risk-free rate (RFR) and a risk
premium:
E R i   RFR  Risk Premium
The Risk-free Rate of Return



The risk-free rate is the rate of interest that is earned for
simply delaying consumption and not taking on any risk
It is also referred to as the pure time value of money
The risk-free rate is determined by:

The time preferences of individuals for consumption



Relative ease or tightness in money market (supply & demand)
Expected inflation
The long-run growth rate of the economy



Long-run growth of labor force
Long-run growth of hours worked
Long-run growth of productivity
The Risk Premium



The risk premium is the return required in excess of the
risk-free rate
Theoretically, a risk premium could be assigned to every
risk factor, but in practice this is impossible
Therefore, we can say that the risk premium is a function
of several major sources of risk:




Business risk
Financial leverage
Liquidity risk
Exchange rate risk
The MPT View of Required Returns

Modern portfolio theory assumes that the
required return is a function of the RFR, the
market risk premium, and an index of systematic
risk:

E R i   R f   i E R M   R f


This model is known as the Capital Asset Pricing
Model (CAPM).
Risk and Return Graphically
Rate of Return
The Market Line
RFR
Risk
f(Business, Financial, Liquidity, and Exchange Rate Risk)
Or
 or 