Financial Analysis, Planning and Forecasting Theory and Application By Alice C. Lee San Francisco State University John C. Lee J.P. Morgan Chase Cheng F. Lee Rutgers University Chapter 5 Determination and Applications of Nominal and Real Rates-Of-Return in Financial Analysis 1

Outline  5.1Introduction  5.2Theoretical justification of paying interest  5.3Rates-of-return measurements and types of averages Discrete rates-of-return and continuous rates-of-return Types of averages Power means  5.4Theories of the term structure and their applications  5.5Interest rate, price-level changes, and components of risk premium Imperfect-foresight case Perfect-foresight case  5.6Three 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  5.7Summary 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 (5.1) where HPY D = Discrete holding-period yield, P t = Price per share in period t, P t-1 = Price per share in period t - 1, D t = Individual dividends per-share in period t. 4

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

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

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

5.3 Rates-of-return measurements and types of averages TABLE 5.1 Johnson & Johnson stock price and dividend data YearClosing PriceAnnual DividendAnnual HPRAnnual HPY 1997$27.94$ %

5.3 Rates-of-return measurements and types of averages (5.11a) (5.11b) (5.11c) (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. End of Month 2006July Aug Sept Oct Nov Dec Jan Feb Mar Apr May June Sources: Office of Debt Management, Office of the Under Secretary for Domestic Finance 10

5.4 Theories of the term structure and their applications (5.13) (5.14) (5.15) (5.16) 11

5.4 Theories of the term structure and their applications F N = (N)(YTM N ) - (N - 1)(YTM N-1 ) (5.16′) Using figure 5.1 if YTM 2 = and YTM 1 =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) F 2 = (2)(0.0487) - (1)(0.0491) = eq.(5.16’) 12 For the further information, please see chapter 15 of the book entitled “Investments” by Bodie, Kane, and Marcus, 9 th ed.

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 (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); 14 Both equations (5.16) and (5.19) can be used to forecast future interest rate

5.5 Interest rate, price-level changes, and components of risk premium (5.20) when R t = Nominal rate of return in period t, B t = Price of the bill in period t - 1, and B t-1 = the price of the bill period t

5.5 Interest rate, price-level changes, and components of risk premium (5.21) (5.22) 16 Equation(5.22) can be rewritten as (5.22’) This equation is referred to Fisher effect.

5.5 Interest rate, price-level changes, and components of risk premium B0B0 B1B1 B2B2 (5.23) (0.003) (0.10) - (5.24) (0.0003) (0.12) 0.11 (0.07) 17 (5.23) (5.24)

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

5.7Summary 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 (5.A.1) End of Year 1 End of Year 2 End of Year 3 End of Year N AmountP(0)(1 + i)P(0)(1 + i)(1 + i)P(0)(1 + i)(1 + i)(1 + i) ReceivedP(0)(1 + i)P(0)(1 + i) 2 P(0)(1 + i) 3  P(0)(1 + i) N 21

Appendix 5A. Compounding and discounting processes and their applications (5.A.2) (5.A.3) (5.A.3a) 22

Appendix 5A. Compounding and discounting processes and their applications (5.A.4) (5.A.5) 23

Appendix 5A. Compounding and discounting processes and their applications (5.A.6) (5.A.7) (5.A.8) (5.A.9) 24

Appendix 5B. Taylor-series expansion and its applications to rates-of-return determination (5.B.1) (5.B.2a) (5.B.2b) (5.B.2c) 25 Here we show the continuously compounded HPY is less than the discrete case of the HPY by using Taylor-series expansion.

Appendix 5B. Taylor-series expansion and its applications to rates-of-return determination (5.B.2d) Or (5.B.2n) 26

Appendix 5B. Taylor-series expansion and its applications to rates-of-return determination (5.B.3) (5.B.4) (5.B.5) (5.B.6) 27

Appendix 5B. Taylor-series expansion and its applications to rates-of-return determination (5.B.7) (5.B.8) (5.B.9) 28

Appendix 5B. Taylor-series expansion and its applications to rates-of-return determination (5.B.10) (5.B.11) (5.B.12) 29

Appendix 5B. Taylor-series expansion and its applications to rates-of-return determination (5.B.13) (5.B.14) (5.B.15) 30 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.

Appendix 5B. Taylor-series expansion and its applications to rates-of-return determination (5.B.16) 31 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).

Appendix 5B. Taylor-series expansion and its applications to rates-of-return determination (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 (5.B.22) 32

Appendix 5B. Taylor-series expansion and its applications to rates-of-return determination (5.B.22) (5.B.23) 33 By using Taylor-series expansion, in the short time horizon, the discrete holding-period yield will equal to continuous holding-period yield.