Transcript +. . .+ (.1)

History of Interest Rates and Risk Premiums
Chapter 5
McGraw-Hill/Irwin
5-1
Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.
Factors Influencing Rates
Supply
Households
Demand
Businesses
Government’s Net Supply and/or
Demand
Federal Reserve Actions
5-2
»Level of Interest Rates
Interest Rates
Supply
r1
r0
Demand
Q0 Q1
Funds
5-3
Real vs. Nominal Rates
Fisher effect: Approximation
nominal rate = real rate + inflation premium
R = r + i or r = R - i
Example r = 3%, i = 6%
R = 9% = 3% + 6% or 3% = 9% - 6%
Fisher effect: Exact
r = (R - i) / (1 + i)
2.83% = (9%-6%) / (1.06)
Empirical Relationship:
Inflation and interest rates move closely
together
5-4
Financial Return
Total return: the total gain or loss experienced
on an investment over a given period of time
Components
of the total
return
Income stream from the investment
Capital gain or loss due to changes
in asset prices
Total return can be expressed either in dollar terms
or in percentage terms.
5-5
Cash Flow Time Line
Pt-1
cash payments
Pt
+-----------------------------------------------------+
Time t-1
Time t
Pt = Price at time t (today)
Capital Gain = Pt – Pt-1
= Price today – Price last period
Dollar Return = Cash Payments + Capital Gain
= Cash Payments + Pt – Pt-1
5-6
Example 5.1
You purchased a stock last year for $25.
It has paid $1 in dividends and is not
worth $21. What is your Dollar Return?
5-7
Example 5.2
You bought an 11% coupon bond one
year ago for $1125. You can sell that
bond today for $1100. What is your
Dollar Return?
5-8
Holding Period Return (hpr) or Percentage Return
This is the most common way to express
the gains or losses over a period
It is the $Return relative to the amount
invested
CashPmt  Pt  Pt 1
hpr  %Return 
Pt 1
1+hpr is often called the wealth relative
5-9
Example 5.1 (Revised)
You purchased a stock last year for $25.
It has paid $1 in dividends and is not
worth $21. What is your Dollar Return?
What is your hpr?
5-10
Example 5.2 Revised
You bought an 11% coupon bond one
year ago for $1125. You can sell that
bond today for $1100. What is your
Dollar Return?
What is your hpr?
5-11
Measuring Wealth over Time
Year
hpr
$1 Investment
$1 Investment each
year
1
15%
1*(1.15) = 1.15
Vt-1(1+it) = Vt
1*(1.15)=1.15
(1+Vt-1)(1+it) = Vt
2
-10%
1.15*(0.90) =1.035 (1+1.15)*(0.90) =
1.935
3
13%
1.035*1.13 = 1.1696 (1+1.935)*(1.13) =
3.3166
5-12
Arithmetic Average Return
Add the individual hpr’s and divide by the
number of years
15%  10%  13%
x
 6.00%
3
5-13
Geometric Rate of Return
Multiply by the wealth relatives, raise to the
1/N power and subtract 1
Is the constant rate of wealth building over
time that results in the observed future value
g  [(1  r1 )(1  r2 )(1  r3 )...(1  rN )]
 [(1.15)(0.90)(1.13)]
1
3
1
N
1
 1  0.0536  5.36%
By Financial Calculator: P/YR=1
I/YR(FV=1.1696, PV=-1, N=3) = 5.36%
5-14
IRR from a Constant Investment
P/YR=1 t
CF
0
-1
1
-1
2
-1
3
3.3166
•Press  IRR = 5.10%
5-15
Value of $1 Invested in Equities,
Treasury Bonds and Bills, 1900 - 2003
100,000
$
15,579
10,000
1,000
Equities
Bonds
Bills
Inflation
$148
100
61
$22
10
1
1900
1920
1940
1960
1980
2000
2003
Year
5-16
Geometric Return Calculation
A $1 investment in Large Stocks (with
dividends reinvested) was worth 15579
after 103 years. The geometric mean
return can be computed as (use P/Yr = 1)
I/YR(FV=15579, PV=-1, N=103)=9.83% Stocks
I/YR(FV=148, PV=-1, N=76)=4.97% Long US Bond
I/YR(FV=61, PV=-1, N=76)=4.07% US TBill
I/YR(FV=22, PV=-1, N=76)=3.05% Inflation
5-17
Geometric Real Rates of Return
To compute the long run real rate of return one
can divide the ending value of the investment by
the ending value of the inflation figure to
determine the purchasing power of the
investment. Then compute the return using
“Real Dollars”
Real value of the Large Stocks at end of period
is 15579/22 = 708.14
I/YR(FV=708.14, PV=-1, N=103)=6.58%
I/YR(FV=6.73, PV=-1, N=103)=1.87% Long US Bond
I/YR(FV=2.73, PV=-1, N=103)=1.00% TBill
5-18
Arithmetic versus Geometric Returns (1900-2003)
Nominal Returns
Arithmetic Avg
Nominal Returns
Geometric
Real Returns
Arithmetic Avg
Real Returns
Geometric
Stocks
Bonds
Bills
11.7
5.2
4.1
9.8
5.0
4.1
8.5
2.3
1.1
6.6
1.9
1.0
5-19
Annual Holding Period Returns (1926-2002)
Geom.
Series Mean%
Sm Stk 11.6
Lg Stk 10.0
LT Gov
5.4
T-Bills
3.8
Inflation 3.1
Arith.
Mean%
17.7
12.0
5.7
3.8
3.1
Stan.
Dev.%
39.3
20.6
8.2
3.2
4.4
5-20
Stocks
Annual Returns for Securties
Bonds
T-Bills
50%
40%
Return
30%
20%
10%
0%
-10%
-20%
-30%
1980
1985
1990
1995
2000
Year
5-21
IRR From a Constant Annual Investment
(Jan 1, each year) through Oct 2002
20%
15%
10%
5%
0%
-5%
-10%
-15%
Stocks
LongGovBond
-20%
Tbills
2002
2001
2000
1999
1998
1997
1996
1995
1994
1993
1992
1991
1990
1989
1988
1987
1986
1985
1984
1983
1982
1981
1980
-25%
Year of First Investment
5-22
IRR From a Constant Annual Investment
(Jan 1, each year) through Dec 2003
30%
Stocks
LongGovBond
25%
Tbills
20%
15%
10%
5%
0%
2003
2002
2001
2000
1999
1998
1997
1996
1995
1994
1993
1992
1991
1990
1989
1988
1987
1986
1985
1984
1983
1982
1981
1980
-5%
Year of First Investment
5-23
IRR From a Constant Annual Investment
(Jan 1, each year) through Dec 2004
20%
18%
Stocks
LongGovBond
16%
Tbills
14%
12%
10%
8%
6%
4%
2%
2003
2004
1999
2000
2001
2002
1995
1996
1997
1998
1990
1991
1992
1993
1994
1986
1987
1988
1989
1982
1983
1984
1985
1980
1981
0%
Year of First Investment
5-24
Dollar Gain per Dollar Invested From a Constant Annual
Investment (Jan 1, each year) through Dec 2004
6.00
Stocks
5.00
LongGovBond
Tbills
4.00
3.00
2.00
1.00
2003
2004
2000
2001
2002
1997
1998
1999
1994
1995
1996
1991
1992
1993
1988
1989
1990
1985
1986
1987
1982
1983
1984
1980
1981
0.00
Year of First Investment
5-25
Take home message
If you have a short holding period, stocks
are very risky, but from a longer term
perspective they have provided the best
returns both recently and historically
Investment is not about saving money for
the future, its about earning money from
the money you invested so that most of
your portfolio is from the earning of that
portfolio and not from your deposits into
that fund
5-26
Percentage Returns on Bills, Bonds, and
Stocks, 1900 - 2003
Nominal (%)
Asset Class Average
Best Year
Real (%)
Worst Year Average Best Year Worst Year
Bills
4.1
14.7
0.0
1.1
19.7
-15.1
Bonds
5.2
40.4
-9.2
2.3
35.1
-19.4
Stocks
11.7
57.6
-43.9
8.5
56.8
-38
Difference between average return of stocks and bills = 7.6%
Difference between average return of stocks and bonds = 6.5%
Risk premium: the difference in returns offered by a
risky asset relative to the risk-free return available
5-27
Why are Treasury Bills considered risk free?
If the government default on Treasury Bills, your
last concern will be the money you might have
earned on the TB
When you buy a Treasury Bill, you purchase it at
a discount and redeem it at par, so you know
when you buy it, what your return will be
If you buy a stock, you don’t know what you will
sell it for, or what dividends it will pay; thus, it is
risky
The yield on Treasury Bills, is generally taken to
be the risk free return
5-28
Distribution of Historical Stock Returns, 1900 - 2003
Histogram of Nominal Returns on
Equities 1900-2003
<-30
-30 to -20 to -10 to 0 to
-20
-10
0
10
10 to
20
20 to 30 to
30
40
40 to
50
>50
Percent return in a given year
Probability distribution for future stock returns is
unknown. We can approximate the unknown
distribution by assuming a normal distribution.
5-29
Variability of Stock Returns
Normal distribution can be described by its
mean and its variance. A Normal Distribution is
symmetric around the mean
Variance (2) - the expected value of squared
deviations from the mean
N
Variance   2 
 ( R  R)
t 1
2
t
N 1
Units of variance (%-squared) - hard to interpret, so
calculate standard deviation, a measure of volatility
equal to square root of 2
5-30
Characteristics of Probability Distributions
1) Mean: most likely value
2) Variance or standard deviation
3) Skewness
* If a distribution is approximately
normal, the distribution is fully
characterized by its mean and
standard deviation
5-31
The Normal Distribution
5-32
Volatility of Asset Returns
Asset
Class
Equities
Bonds
Bills
Nominal Returns
Real Returns
Average(%) Std. Dev. (%) Average(%) Std. Dev. (%)
11.7
5.2
4.1
20.1
8.2
2.8
8.5
2.3
1.1
20.4
10
4.7
Asset classes with greater volatility pay higher
average returns.
Average return on stocks is more than double the
average return on bonds, but stocks are 2.5 times
more volatile.
5-33
Risk Premiums and Real Returns (1926-2002)
Risk
Series Premiums%
Sm Stk
13.9
Lg Stk
9.3
LT Gov
1.9
T-Bills
--Inflation
---
Real
Returns%
14.6
8.9
2.6
0.7
---
5-34
Mean Scenario or Subjective Returns
Subjective returns
s
E (r )   p s r s
1
ps = probability of a state
rs = return if a state occurs
1 to s states
5-35
Scenario or Subjective Returns: Example
State
1
2
3
4
5
Prob. of State
.1
.2
.4
.2
.1
r in State
-.05
.05
.15
.25
.35
E(r) = (.1)(-.05) + (.2)(.05) +. . .+ (.1)(.35)
E(r) = .15
5-36
Variance or Dispersion of Returns
Subjective or Scenario
  
   p s  r s  E r 
2
s

2
Standard deviation = [variance]1/2
Using Our Example:
Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2 + . . .+ .1(.35-.15)2]
Var= .01199
S.D.= sqrt[ .01199] = .1095
5-37