3. Dollar return
Suppose that you bought a bond for $1050 a year
ago. You have received two semiannual coupons of
$50 each. The market price of the bond today is
$1100. What is your total dollar return?
Income = 50 + 50 = $100.
Capital gain = 1100 – 1050 = $50.
Total dollar return = 100 + 50 = $150.
Total dollar return = income + capital gain (loss).
4. Percentage return
It is more intuitive to think of returns in terms of
percentages.
Return = (ending value – beginning value) /
beginning value.
Return = ((1100 + 100) – 1050) / 1050 = 14.29%.
Return = income yield (return) + capital gain yield
(return) = 100 / 1050 + 50 / 1050 = 9.52% + 4.76% =
14.29%.
5. Holding period return
Holding period return: the cumulative
return that an investor would obtain
when holding an investment over a
period of N years.
6. Holding period return example
Suppose that an investment yielded 4%, 7%,
8%, 0%, and 10% for the past 5 years. What
is the holding period return for the 5 years?
Holding period return = (1 + 4%) × (1 + 7%) ×
(1 + 8%) × (1 + 0%) × (1 + 10%) – 1 =
32.2%.
7. Why capital market returns?
Examining capital market returns helps us
determine the appropriate returns on non-
financial assets (e.g., firm projects).
Lessons from capital market return history:
– There is a reward for bearing risk.
– Return is positively related to risk.
– This is called the “risk-return tradeoff.”
8. Average return, I
The simple average of a series of returns is
called arithmetic average return.
Geometric average return: average
compound return per period over multiple
periods.
9. Average return example
Year Ret Arithmetic R 1 + Ret Power of 1/N Geometric R
1 0.08 0.076 1.08 1.074811381 0.07481138
2 -0.01 0.99
3 0.12 1.12
4 0.13 1.13
5 0.06 1.06
1.4344
Multiplication term
GR < AR
10. Average return, II
The geometric average will be less than the
arithmetic average unless all the returns are equal.
The arithmetic average is overly optimistic for
multiple horizons.
If the investment horizon is only 1 period, the
arithmetic average is a better measure of likely
return.
Arithmetic average return based on historical returns
are often used as an estimate of expected return in
real life.
11. Portfolio risk statistics
An investor cares about the return variability
of her/his portfolio.
This variability at the portfolio level is usually
measured by variance and/or standard
deviation (std.).
12. Portfolio risk statistics, example
Year Ret Arithmetic R Diff Diff^2 Var Std
1 0.08 0.076 0.004 0.000016 0.00313 0.055946
2 -0 -0.086 0.007396
3 0.12 0.044 0.001936
4 0.13 0.054 0.002916
5 0.06 -0.016 0.000256
Sum 0.01252
13. Historical performance, 1926-2008
Average Return Std
Large stocks 11.7% 20.6%
Small Stocks 16.4% 33.0%
L-T Corporate Bonds 6.2% 8.4%
L-T Government Bonds 6.1% 9.4%
U.S. Treasury Bills 3.8% 3.1%
Inflation 3.1% 4.2%
14. If normally distributed?
If we are willing to assume that asset returns
are normally distributed, can you draw a
return distribution based on the previous
slide?
Hint: about 95% within +/– 2 std.
