1. Financial Analysis, Planning and Forecasting Theory and Application
Chapter 5
Determination and Applications of Nominal and Real Rates-Of-Return in Financial Analysis By
Cheng F. Lee
Rutgers University, USA
John Lee
Center for PBBEF Research, USA

2. Outline
5.1 Introduction
5.2 Theoretical justification of paying interest
5.3 Rate-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

3. 5.3 Rates-of-return measurements and types of averages
Discrete rates-of-return and continuous rates-of-return
Types of averages
Power means

4. 5.3 Rates-of-return measurements and types of averages
(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.

5. 5.3 Rates-of-return measurements and types of averages
(5.2)
(5.3)
(5.4)

6. 5.3 Rates-of-return measurements and types of averages
(5.5)
(5.6)
(5.7)

7. 5.3 Rates-of-return measurements and types of averages
(5.8)
(5.9)
(5.10)

8. 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
0.988
-1.2
2006
64.78
1.455

9. 5.3 Rates-of-return measurements and types of averages
(5.11b)
(5.11c)
(5.12)

10. 5.4 Theories of the term structure and their applications
Table 5.2 Treasury Market Bid Yields at Constant Maturities: Bills, Notes, and Bonds
04/01/15
0.02
0.03
0.12
0.27
0.55
0.86
1.32
1.65
1.87
2.23
2.47
04/02/15
0.10
0.25
0.87
1.35
1.69
1.92
2.29
2.53
04/03/15
0.04
0.21
0.49
0.80
1.26
1.60
1.85
2.24
2.49
04/06/15
0.51
0.83
1.31
1.67
2.31
2.57
04/07/15
0.22
0.52
0.85
1.66
1.89
2.27
2.52
04/08/15
0.54
1.68
2.28
04/09/15
0.56
0.89
1.40
1.73
1.97
2.35
2.61
04/10/15
0.01
0.09
0.24
0.57
0.91
1.41
1.96
2.33
2.58
04/13/15
0.11
0.23
1.38
1.71
1.94
04/14/15
0.53
1.34
1.90
2.54
04/15/15
0.08
1.33
1.91
2.30
2.55
04/16/15
0.50
0.81
1.64
2.56
Sources: U.S. Department of the Treasury.

11. 5.4 Theories of the term structure and their applications
Table 5.2 Treasury Market Bid Yields at Constant Maturities: Bills, Notes, and Bonds
04/17/15
0.03
0.01
0.08
0.23
0.51
0.84
1.31
1.63
1.87
2.26
2.51
04/20/15
0.10
0.24
0.55
0.86
1.33
1.65
1.90
2.31
2.56
04/21/15
0.02
0.09
1.35
1.67
1.92
2.33
2.58
04/22/15
0.57
0.91
1.41
1.75
1.99
2.42
2.66
04/23/15
0.87
1.37
1.70
1.96
2.38
2.63
04/24/15
0.54
1.34
1.68
1.93
2.36
2.62
04/27/15
0.25
1.36
1.69
1.94
2.61
04/28/15
0.00
0.56
0.90
1.39
2.00
2.68
04/29/15
0.07
1.43
1.80
2.06
2.49
2.76
04/30/15
0.06
0.58
1.79
2.05
2.75
05/01/15
0.05
0.60
0.97
1.50
2.12
2.57
2.82
05/04/15
0.96
1.51
2.16
2.88
05/05/15
0.62
1.00
1.54
2.19
2.64
2.90
05/06/15
0.65
1.03
1.58
1.97
2.25
2.72
2.98
Sources: U.S. Department of the Treasury.

12. 5.4 Theories of the term structure and their applications
(5.13)
(5.14)
(5.15)
(5.16)

13. 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= 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’).
eq.(5.16)
F2 = (2)(0.0487) - (1)(0.0491) = eq.(5.16’)
For the further information, please see chapter 15 of the book entitled “Investments” by Bodie, Kane, and Marcus, 9th ed.

14. 5.4 Theories of the term structure and their applications
Et = rt − Ft (5.17) ∆Rt = A + Bet (5.18)

15. 5.5 Interest rate, price-level changes, and components of risk premium
Imperfect-foresight case
Perfect-foresight case

16. 5.5 Interest rate, price-level changes, and components of risk premium
(5.19)
where
=Annual rate of change in the GNP deflator;
Y* = Level of real GNP;
= Rate of change in real GNP;
= 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

17. 5.5 Interest rate, price-level changes, and components of risk premium
(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.

18. 5.5 Interest rate, price-level changes, and components of risk premium
(5.21)
(5.22)
Equation (5.22) can be rewritten as (5.22’)
This equation is referred to Fisher effect.

19. 5.5 Interest rate, price-level changes, and components of risk premium
(5.23)
(5.24) B0 B1 B2
(5.23)
0.0007
(0.003)
-0.98
(0.10) -
(5.24)
(0.0003)
-0.87
(0.12)
0.11
(0.07)

20. The debtor-creditor hypothesis
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.

21. 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 12 and 13), leasing (Chapter 14), and cost of capital determinations (Chapter 11). The mathematical concepts of arithmetic, geometric, and mixed means will also be important for estimating growth of dividends (Chapter 16) and financial planning and forecasting (Chapters 22 and 23). 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.

22. a) Compound Future Sum (Terminal Value) b) Present Value
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

23. Appendix 5A. Compounding and discounting processes and their applications
End of
Year 1
Year 2
Year 3
Year N
Amount
P(0)(1 + i)
P(0)(1 + i)(1 + i)
P(0)(1 + i)(1 + i)(1 + i)
Received
P(0)(1 + i)2
P(0)(1 + i)3 
P(0)(1 + i)N

24. Appendix 5A. Compounding and discounting processes and their applications
(5.A.3a)

25. Appendix 5A. Compounding and discounting processes and their applications

26. Appendix 5A. Compounding and discounting processes and their applications

27. Appendix 5B. Taylor-series expansion and its applications to rates-of-return determination
(5.B.2a)
(5.B.2b)
(5.B.2c)
Here we show the continuously compounded HPY is less than the discrete case of the HPY by using Taylor-series expansion.

28. Appendix 5B. Taylor-series expansion and its applications to rates-of-return determination
(5.B.2d)
Or (5.B.2n)

29. Appendix 5B. Taylor-series expansion and its applications to rates-of-return determination

30. Appendix 5B. Taylor-series expansion and its applications to rates-of-return determination

31. Appendix 5B. Taylor-series expansion and its applications to rates-of-return determination

32. 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, where x is HPR.

33. Appendix 5B. Taylor-series expansion and its applications to rates-of-return determination
Where y is HPR with lognormal distribution, and x is HPYc with normal distribution.
If the time horizon is very short, equation (5.B.16) can be reduced to y=1+x (eq. 5.B.17).

34. Appendix 5B. Taylor-series expansion and its applications to rates-of-return determination
By using Taylor-series expansion, the relationship between HPR (y) with lognormal distribution and HPYc (x) with lognormal distribution can be shown as follows
(5.B.22)

35. Appendix 5B. Taylor-series expansion and its applications to rates-of-return determination
By using Taylor-series expansion, in the short time horizon,
the discrete holding-period yield will equal to continuous
holding-period yield.