Financial Analysis, Planning and
Forecasting
Theory and Application
Chapter 5
Determination and Applications of Nominal
and Real Rates-Of-Return in Financial Analysis
By
Alice C. Lee
San Francisco State University
John C. Lee
J.P. Morgan Chase
Cheng F. Lee
Rutgers University
1
Outline









5.1
5.2
5.3
Introduction
Theoretical justification of paying interest
Rates-of-return measurements and types of averages
Discrete rates-of-return and continuous rates-of-return
Types of averages
Power means
5.4 Theories of the term structure and their applications
5.5 Interest rate, price-level changes, and components of risk premium
Imperfect-foresight case
Perfect-foresight case
5.6 Three hypotheses about inflation and the value of the firm: a review
The debtor-creditor hypothesis
The tax-effects hypothesis
Operating-income hypothesis
The relationship among the three hypotheses
5.7 Summary and concluding remarks
Appendix 5A. Compounding and discounting processes and their applications
Appendix 5B. Taylor-series expansion and its applications to rates-of-return
determination
2
5.3 Rates-of-return measurements and types of averages



Discrete rates-of-return and
continuous rates-of-return
Types of averages
Power means
3
5.3 Rates-of-return measurements and types of averages
Pt  Dt
Pt  Pt 1 Dt
HPY 
1 

Pt 1
Pt 1
Pt 1
(5.1)
where
HPYD = Discrete holding-period yield,
Pt = Price per share in period t,
Pt-1 = Price per share in period t - 1,
Dt = Individual dividends per-share
in period t.
4
5.3 Rates-of-return measurements and types of averages
Pt  Dt
HPYc  ln(
)
Pt 1
(5.2)
Pt  Dt
HPR  HPYD  1 
Pt 1
X
H
X X
G
(5.3)
A
(5.4)
5
5.3 Rates-of-return measurements and types of averages
Xi
X 
1 N
N
A
(arithmetic)
1/ N
X G   XJ 
 J 1 
N
N
X  N
1

1 XJ
H
(5.5)
(geometric)
(5.6)
(harmonic)
(5.7)
6
5.3 Rates-of-return measurements and types of averages
x 
1N
2
(X J  X ) 


N  J 1

Pt
HPRt 
Pt 1
(5.8)
(5.9)
T  N A N 1 G
X 
X 
X
(5.10)
T 1
T 1
where T =number of observations used to estimate
M
the means and N = number of periods which investors
decide to hold their assets.
7
5.3 Rates-of-return measurements and types of averages
TABLE 5.1 Johnson & Johnson stock price and dividend data
Year
Closing Price
Annual Dividend
Annual HPR
Annual HPY
1997
$27.94
$0.43
-
-
1998
36.04
0.49
1.307
30.7%
1999
40.54
0.545
1.140
14.0
2000
46.31
0.31
1.150
15.0
2001
52.8
0.7
1.155
15.5
2002
48.64
0.795
0.936
-6.4
2003
47.63
0.925
0.998
-0.2
2004
59.62
1.095
1.275
27.5
2005
57.61
1.275
0.988
-1.2
2006
64.78
1.455
1.150
15.0
HPR2002
$48.64  $0.795 $49.44


 0.936
$52.8
$52.8
8
5.3 Rates-of-return measurements and types of averages
10.10
X 
 1.1221
9
(5.11a)
X G  9 2.6781  1.1157
(5.11b)
A
X
M
95
5 1

(1.1221) 
(1.1157)
9 1
9 1
 0.5611  0.5578  1.1189
1
r 
X  X j 
N 1

N
r
(5.11c)
1/ r
(5.12)
9
5.4 Theories of the term structure and their applications
Table 5.2 Treasury Market Bid Yields at Constant Maturities: Bills, Notes, and Bonds
Period
1-mo.
3-mo.
6-mo.
1-yr.
2-yr.
3-yr.
5-yr.
7-yr.
10-yr.
20-yr.
30-yr.
July
5.02
5.1
5.18
5.11
4.97
4.93
4.91
4.93
4.99
5.17
5.07
Aug
5.12
5.05
5.11
5.01
4.79
4.71
4.7
4.7
4.74
4.95
4.88
Sept
4.6
4.89
5.02
4.91
4.71
4.62
4.59
4.6
4.64
4.84
4.77
Oct
5.18
5.08
5.13
4.99
4.71
4.62
4.57
4.57
4.61
4.81
4.72
Nov
5.22
5.03
5.1
4.94
4.62
4.52
4.45
4.45
4.46
4.66
4.56
Dec
4.75
5.02
5.09
5
4.82
4.74
4.7
4.7
4.71
4.91
4.81
Jan
5
5.12
5.16
5.09
4.94
4.85
4.82
4.82
4.83
5.02
4.93
Feb
5.24
5.16
5.12
4.96
4.65
4.55
4.52
4.53
4.56
4.78
4.68
Mar
5.07
5.04
5.06
4.9
4.58
4.54
4.54
4.58
4.65
4.92
4.84
Apr
4.8
4.91
5.03
4.89
4.6
4.54
4.51
4.55
4.63
4.88
4.81
May
4.78
4.73
4.96
4.95
4.92
4.88
4.86
4.87
4.9
5.1
5.01
June
4.28
4.82
4.93
4.91
4.87
4.89
4.92
4.96
5.03
5.21
5.12
End of Month
2006
2007
Sources: Office of Debt Management, Office of the Under Secretary for Domestic Finance
10
5.4 Theories of the term structure and their applications
1/ N


YTM N   (1  Ft )   1
 t 1

where Ft is the forward rate in period t
(5.13)
(1  YTM N ) N  (1  F1 )(1  F2 )  (1  Ft )
(5.14)
(1  YTM N ) N
 1  FN
(1  F1 )(1  F2 ) (1  Ft 1 )
(5.15)
(1  YTM N ) N
1  FN 
(1  YTM N 1 ) N 1
(5.16)
N
11
5.4 Theories of the term structure and their applications
FN = (N)(YTMN) - (N - 1)(YTMN-1)
(5.16′)
Using figure 5.1 if YTM2=0.0487 and YTM1
=0.0491, then the forward rate for year 2 cab be
calculated in terms of either eq.(5.16) or eq.(5.16’).
(1  YTM 2 )2
(1  0.0487) 2
F2 
1 
 1  0.0483eq.(5.16)
1
(1  YTM1 )
(1  0.0491)
F2 = (2)(0.0487) - (1)(0.0491) = 0.0483
eq.(5.16’)
For the further information, please see chapter 15 of the book entitled
“Investments” by Bodie, Kane, and Marcus, 9th ed.
12
5.5 Interest rate, price-level changes, and components of
risk premium


Imperfect-foresight case
Perfect-foresight case
13
5.5 Interest rate, price-level changes, and components of
risk premium
N
rnt   0   a j 1Pt  j  1Yt*  2 Yt*  3M t*
(5.19)
j 0
where rnt  Nominal interest rate
P =Annual rate of change in the GNP deflator;
Y* = Level of real GNP;
Y * = Rate of change in real GNP;
M * = Average change in the real money stock
(nominal money stock deflated by the
GNP deflator);
Both equations (5.16) and (5.19) can be used to forecast
future interest rate
14
5.5 Interest rate, price-level changes, and components of
risk premium
Bt  Bt 1
Rt 
Bt 1
(5.20)
when
Rt = Nominal rate of return in period t,
Bt = Price of the bill in period t - 1,
and
Bt-1 = the price of the bill period t - 1.
15
5.5 Interest rate, price-level changes, and
components of risk premium
Pt v Bt  Pt v1 Bt 1
rt 
(5.21)
v
Pt 1 Bt 1
ptv  ptv1
(5.22)
(Ct 
)
rt  Rt  Ct
v
pt 1
Where rt is the real return, Rt is the nominal return,
and Ct is the loss in purchasing power.
Equation(5.22) can be rewritten as Rt  rt  inflation rate (5.22’)
This equation is referred to Fisher effect.
16
5.5 Interest rate, price-level changes, and
components of risk premium
Ct  B0  B1Rt
(5.23)
Ct  B0  B1Rt  B2Ct 1
(5.24)
B0
B1
B2
(5.23)
0.0007
(0.003)
-0.98
(0.10)
-
(5.24)
0.00059
(0.0003)
-0.87
(0.12)
0.11
(0.07)
17
5.6 Three hypotheses about inflation and the value of the
firm: a review
 The
debtor-creditor hypothesis
 Economic theory suggests that unanticipated inflation should
redistribute wealth from creditors to debtors because the real
value of fixed monetary claims falls.
 The
tax-effects hypothesis
 Since depreciation and inventory tax shields are based on
historical costs, their real values decline with inflation. This, in
turn, reduces the real value of the firm.
 Operating-income
hypothesis
 According to the traditional view of economics, wealth
transfers caused by general inflation are due primarily to
those effects discussed above. We will discuss this
hypothesis in chapter 6 eq. (6.8a) on page 192.
18
5.7
Summary and concluding remarks
In this chapter we have examined several concepts that
will be of importance later. Determination of appropriate
interest rates and risk premiums is very important in capital
budgeting (Chapters 9 and 10), leasing (Chapter 10), and
cost of capital determinations (Chapter 8). The
mathematical concepts of arithmetic, geometric, and mixed
means will also be important for estimating growth of
dividends (Chapter 13) and financial planning and
forecasting (Chapters 16 and 17). A basic understanding
about the relationships between various types of risks
(inflation, liquidity, and default) will be necessary for
analyzing alternative risk premiums in financial analysis,
planning, and forecasting.
19
Appendix 5A. Compounding and discounting processes
and their applications
5.A.1 SINGLE-VALUE CASE
a) Compound Future Sum (Terminal Value)
b) Present Value
 5.A.2 Annuity Case
a) Compound Future Sum of An Annuity
b) Present Value of An Annuity

20
Appendix 5A. Compounding and discounting processes
and their applications
PD ( N )  P(0)(1  i)
N
(5.A.1)
End of
Year 1
End of
Year 2
End of
Year 3
Amount
P(0)(1 + i)
P(0)(1 + i)(1 + i)
P(0)(1 + i)(1 + i)(1 + i)
Received
P(0)(1 + i)
P(0)(1 + i)2
P(0)(1 + i)3 
End of
Year N
P(0)(1 + i)N
21
Appendix 5A. Compounding and discounting processes
and their applications
i m N
PD ( N )  P (0)(1  )
m
1 m
lim(1  )  e  2.7183
m
m
i ( m / i )iN
PC ( N )  lim P(0)(1  )
 P(0)eiN Pc
m
(5.A.2)
(5.A.3)
(5.A.3a)
22
Appendix 5A. Compounding and discounting processes
and their applications
P( N )
PD (0) 
[(1  i ) N ]
(5.A.4)
P( N )
PC (0)  iN
e
(5.A.5)
23
Appendix 5A. Compounding and discounting processes
and their applications
N
CFSAD   C (t )[(1  i ) N t 1 ]
t 1
(5.A.6)
N
CFSAC   C (t )[e iT ]dt
(5.A.7)
t 1
n
C (t )
PVAD  
t
t 1 (1  i )
n
C (t )
PVAC   it
t 1[e ]dt
(5.A.8)
(5.A.9)
24
Appendix 5B. Taylor-series expansion and its applications
to rates-of-return determination
Here we show the continuously compounded HPY is less than
the discrete case of the HPY by using Taylor-series expansion.
F (a)
F ( n ) (a)
2
Fn ( x)  F (a)  F (a)( x  a) 
( x  a)   
( x  a) n   (5.B.1)
2!
n!
F (a) 
d
1
ln( x) |x  a 
dx
a
a0
(5.B.2a)
F (a)  a 2
(5.B.2b)
F (a)  2a 3
(5.B.2c)
25
Appendix 5B. Taylor-series expansion and its
applications to rates-of-return determination
F (a)  6a
Or
4
(n 1)!an
if n is even
(n 1)!an
if n is odd
(5.B.2d)
(5.B.2n)
26
Appendix 5B. Taylor-series expansion and its applications
to rates-of-return determination
F n (a)
( x  a) n
n!
(5.B.3)
(n  1)! 1
 n  ( x  a) n
n!
a
1 x 
1

n a 
(5.B.4)
n
(5.B.5)
2
1 x 
x  1 x 
F ( x)  ln( a)    1    1      1
n a 
 a  2!  a 
n
(5.B.6)
27
Appendix 5B. Taylor-series expansion and its applications
to rates-of-return determination
1
1
2
n
F ( x)  ln( 1)  x  1  x  1    x  1
2!
N!
(5.B.7)
 Pt  Dt 
HPYD  
 1
 Pt 1

(5.B.8)
 Pt  Dt 
HPYC  

 Pt 1 
(5.B.9)
28
Appendix 5B. Taylor-series expansion and its applications
to rates-of-return determination
Pt  Dt
x
Pt 1
ln x  HPYC
1
2
HPYC  ln x  ( x  1)  ( x  1)
2
(5.B.10)
(5.B.11)
(5.B.12)
29
Appendix 5B. Taylor-series expansion and its applications
to rates-of-return determination
1
2
HPYC  HPYD  ( x  1)
2
(5.B.13)
Here we show the continuously compounded HPY is less than
the discrete case of the HPY by using Taylor-series expansion,
where x is HPR.
F (0) 2

Fn ( x)  F (0)  F (0)( x) 
( x) 
2!
x
e  1 
1!
x
F ( n ) (0) n

( x) 
n!
(5.B.14)
n
x
 
n!
(5.B.15)
30
Appendix 5B. Taylor-series expansion and its applications
to rates-of-return determination
1 2
1 2 1
E ( y )  E (e )  E (1  x  x )  1  E ( x)    ( E ( x)) 2
2
2
2
x
Pt 1  Dt
yt 1 
Pt
Where y is HPR with lognormal distribution, and x is
HPYc with normal distribution.
2
x
y  ex  1  x  
2!
xn

n!
(5.B.16)
If the time horizon is very short, equation (5.B.16) can be
reduced to y=1+x (eq. 5.B.17).
31
Appendix 5B. Taylor-series expansion and its applications
to rates-of-return determination
y 1  x
x  ln y
Pt 1  Dt
x  ln(
)
Pt
(5.B.18)
(5.B.20)
(5.B.21)
By using Taylor-series expansion, the relationship between
HPR (y) with lognormal distribution and HPYc (x) with
lognormal distribution can be shown as follows
( Pt 1  Dt )
 1  HPYc
Pt
(5.B.22)
32
Appendix 5B. Taylor-series expansion and its applications
to rates-of-return determination
( Pt 1  Dt )
 1  HPYc
Pt
HPYD  HPYC
(5.B.22)
(5.B.23)
By using Taylor-series expansion, in the short time horizon,
the discrete holding-period yield will equal to continuous
holding-period yield.
33