Nearly all the Math you need to know for this course
I. One Period Returns
A. Simple Rate of Return (r)
r
v1  v0  c Ending  Initial   Payments

v0
Initial
Example 1. Suppose you purchase a car for $400 then sell the car the next day for $500.
Calculate the simple rate of return from your investment.
r
500  400 100

 .25  25%
400
400
Example 2. Suppose you purchase 200 shares of Chrysler Corporation stock at a price of
$30 per share. One year later, you sell all 200 shares at a price of $35.50 per share.
During the year, you received a dividend payment of $1.10 per share. What is your rate
of return? {you might consider what factors we might be missing here}
v0  $33  200  $6,600
v1  $35.50  200  $7,100
c  $1.10  200  $220
r
7,100  6,600  220 
6,600
720
 .109  10.9%
6,600
{other factors would be costs associated with trading the stocks. You might also want to
calculate the rate of return on an after tax basis}
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B. Weighted Average Rate of Return (R)
R  WA rA  WB rB  WC rc  
Example 3. suppose you had invested $30,000 in various assets: $10,000 in asset A,
$5,000 in asset B, $15,000 in asset C. During the course of a year, asset A had a rate of
return of 12%, asset B of 10%, and asset C of 8%. Calculate the overall (or, weighted
average) rate of return on your investment of $30,000.
10,000
 .33
30,000
5,000
WB 
 .167
30,000
15,000
WC 
 .5
30,000
WA 
R  (.33  .12)  (.167  .167)  (.5  .08)  .0967  9.67%
II. Multiple Period Returns
A. Arithmetic Mean Return
_
r
r1  r2    rn
n
Example 4. Suppose you had invested $100,000 in a corporation’s stock. You have held
this stock for the past four years. You calculate the simple rate of return on the stock for
each year and found the following:
Year:
Rate of return:
1
25%
2
15%
3
-10%
4
20%
Calculate the simple arithmetic mean rate of return during the holding period.
_
r
.25  .15  .10  .20 .50

 .125  12.5%
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Historical Fact 1. The arithmetic mean rate of return on “large stocks” (S&P 500 stock
index) was 12.49% for the period 1926-2001. The arithmetic mean rate of return on
“small stocks” (Russell 2000 index) was 18.29% for the same period. The arithmetic
mean rate of return on long-term (20 years to maturity or more) U.S. government treasury
bonds was 5.53% during the period.
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B. Geometric Mean Return
1  rG N
 (1  r1 )(1  r2 )  (1  rN ) 
rG  (1  r1 )(1  r2 )  (1  rN 
1
N
1
Example 5. Given the information from example 4, calculate the geometric meana rate of
return for the stock.
rG  (1  .25)(1  .15)(1  .10)(1  .20)
1
4
 1  .116  11.6%
Historical Fact 2. The geometric mean rate of return on “large stocks” (S&P 500 stock
index) was 10.51% for the period 1926-2001. The geometric mean rate of return on
“small stocks” (Russell 2000 index) was 12.19% for the same period. The geometric
mean rate of return on long-term (20 years to maturity or more) U.S. government treasury
bonds was 5.23% during the period.
Example 6. Suppose you have invested $200 in a stock for two years. The first year, the
value of the stock fell by half (decreased by 50%). The second year, the value of the
stock doubled (increased by 100%). Calculate the arithmetic mean and geometric mean
rates of return. Would one of these measures provide more information than the other?
_
r
 50%  100%
 25%
2
rG  (1  .5)(1  1) 2  1  0%
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{the geometric mean might be a better measure of the return here. After the first year,
your investment was down to $100, but the second year that $100 doubled to $200.
Hence, after two years you still have your $200 – it hasn’t grown at all and yet the
arithmetic mean would seem to suggest that it has increased by 25% on average}
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III. Expected Rate of Return
A. Expected Rate of Return on an Asset
E (r )  p1r1  p2 r2    pn rn
{note, p represents the probability of a given return}
Example 7. Suppose you are thinking of making an investment in a real estate venture.
The venture will cost $100,000. Having done some research, you sit down and write out
what you think the possible returns will be along with their likelihood of occurring.
r
p
-20%
.2
10%
.5
30%
.3
What is the expected rate of return from this investment?
E (r )  .2(.20)  .5(.10)  .3(.30)  .04  .05  .09  .10  10%
B. Weighted Expected Rate of Return on a Portfolio
E ( RP )  W1 E (r1 )  W2 E (r2 )    WN E (rN )
Example 8. Suppose you have invested $10,000 in 3 stocks (1, 2, and 3). You have
purchased $3,000 worth of stock 1, $5,000 of stock 2, and $2,000 of stock 3. You have
performed the necessary calculations (like in the previous example) to obtain the
expected return from each individual stock: E(r1) = 6%, E(r2) = 10%, E(r3) = 8%.
3,000
 .3
10,000
5,000
W2 
 .5
10,000
2,000
W3 
 .2
10,000
W1 
E ( RP )  .3(.06)  .5(.10)  .2(.08)  .018  .05  .016  .084  8.4%
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C. Expected Rate of Return on a Portfolio
E ( RP )  p1 E ( R1 )  p2 E ( R2 )    p N E ( RN )
Example 9. You have composed a portfolio of assets (stocks, bonds, real estate, cash,
etc.). Depending on the state of the economy, you have developed the following
probability distribution.
E(R)
p
-5%
.10
0%
.40
20%
.40
10%
.10
Calculate the expected rate of return on the portfolio.
E ( R)  .1(.05)  .4(0)  .4(.20)  .1(.10)  .085  8.5%
IV. Risk and Variability
A. The simple (population) variance and standard deviation for rates of return
N
2 
_
 (ri  r ) 2
i 1
N
  2
Example 10. Suppose you have held the same portfolio for the past four years. You have
calculated the annual rate of return on the portfolio for each year: 10%, 20%, 0%, 10%.
Calculate the variance and standard deviation for the portfolio.
_
.10  .20  0  .10
 .10  10%
First, you must calculate the average rate of return: r 
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Next, you can calculate the variance:
(.10  .10) 2  (.20  .10) 2  (0  .10) 2  (.10  .10) 2 0  .01  .01  0
 

 .005
4
4
2
{note, you cannot restate the variance as a percentage, such as .5% --- NO!!! It is in units
of percent squared at this point --- one reason why the standard deviation is useful}
Now, you can calculate the standard deviation:
  .005  .0707  7.07%
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Historical Fact 3. The standard deviation for large stock (S&P 500) was 20.7% during
the period 1926-2001. The standard deviation for small stocks (Russell 2000) was
39.98% for the same period. The stand deviation for long-term government bonds was
7.94% for the period. {take a look back at the average rates of returns from Facts 1 and 2
now, see a pattern?}
B. Variance and Standard Deviation in terms of Expected Value
2
N
   pi ri  E (r )
2
i 1
  2
Example 11. Calculate the variance and standard deviation of the portfolio in example 9.
{recall, in example 9, we had calculated the expected rate of return on the portfolio –
we’ll use that calculations here}
 2  .10(.05  .085) 2  .40(0  .085) 2  .40(.2  .085) 2  .10(.1  .085) 2 
 .10(.018)  .40(.0029)  .40(.0053)  .10(.0002)  .01
  .01  .1  10%
C. Covariance and Correlation for two assets

_
_

N
_
_
 12  E (r1  r1 )( r2  r2 )   pi (r1,i  r1 )( r2,i  r2 )


i 1

 12
 1 2
Example 13. This will be a longer example for review purposes, then at the end cover the
covariance and correlation. Suppose you purchase one stock (asset 1) and one bond
(asset 2). You have developed two scenarios of what might happen during your
investment horizon.
Scenario:
Probability of Scenario:
Rate of Return on Stock
Rate of Return on Bond
Recession
.40
-.05
.10
Expansion
.60
.20
0
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Calculate the expected rate of return for each asset, covariance between the investments,
and the correlation.
Expected rate of return on stock:
r1  .40(.05)  .60(.2)  .10  10%
Expected rate of return on bond:
r2  .40(.10)  .60(0)  .04  4%
Variance of Stock return:  12  .40(.05  .10) 2  .60(.20  .10) 2  .015
Standard deviation of stock return:
Variance of Bond return:
 1  .015  .12  12%
 22  .40(.10  .04) 2  .60(0  .04) 2  .0024
Standard deviation of bond return:
 2  .0024  .05  5%
Covariance:
 12  .40(.05  .10)(.10  .04)  .60(.20  .10)(0  .04)  .006
Correlation;

 .006
 1
(.12)(.05)
D. Variance and Standard Deviation of a Portfolio
{this we’ll wait for --- it explains the Nearly in the title}
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