Let’s start at the beginning
Returns can be measured two ways:
Dollar return = Amount received – amount invested
= $1,200

–

$1,000

= $200

Rate of return =

Amount received – amount invested amount invested

=

($1200 - $1,000)/$1,000

=

20%
1

Other rate of return concepts

Beginning of year

End of year

%return

$ 80

$120

50%

120

60

-50%
0%

But the investor began the first year with $80 and wound up with $60. How can the rate of return be zero?

2

Arithmetic vs. geometric means
The 0% return is an arithmetic average return.

A geometric return would be computed as:
[(1 + r1)(1 + r2)]1/2 – 1
In this case: [(1.5)(.5)]1/2 – 1 = -13.4%

Arithmetic returns overstate investment returns or multiperiod liability costs.
3

More generally

If an asset, A, grows at a rate r for n years:
An

=

A0(1 + r)

n

So the annualized compounded growth rate can be computed as: r = (An /A0)

1/n

-1

For example, suppose a portfolio is initially worth $100 and after 7 years and 9 months has grown to $249. The average annualized rate of return earned on this portfolio is:
1/7.75

r = (249/100)
= 12.49%

4

-1

Expectations

n Let

X be a random variable. The mean of X, denoted by
E[X] is defined as follows.