Time Value of Money and Discounted Cash Flows

1. Time Value of Money and Discounted Cash Flows

2. Compounding: Finding a future value for a current cash flow

3. Discounting: finding a present value for a future cash flow

4. Annuity: • Annuity: a sequence of equal cash flows, which pays at the end of each equal period. • Annuity Due: a sequence of equal cash flows, which pays at the beginning of each equal period.

5. Finding future value for an annuity

6. Finding present value for an annuity

7. Present and future values for annuity due

8. Perpetuity: an annuity with infinity periods, so the present value for a perpetuity becomes

9. How to use Excel to compute FV • FV(rate,nper,pmt,pv,type) • Rate   discount rate, if it is an annual rate, then put in 10%. If it is monthly rate, then state as 10%/12, or 0.83%. • Nper   number of period. • Pmt   amount of annuity payment. • pv   amount of present value. • Type   0 or 1. 1 for annuity due, zero (or omission for annuity)

10. How to use Excel to compute PV • PV(rate,nper,pmt,fv,type) • Rate   discount rate, if it is an annual rate , then put in 10%. If it is monthly rate, then state as 10%/12, or 0.83%. • Nper   number of period. • Pmt   amount of annuity payment. • Fv   amount of future value. • Type   0 or 1. 1 for annuity due, zero (or omission for annuity)

11. How to use Excel to compute rate and time period • RATE(nper,pmt,pv,fv,type) • NPER(rate,pmt,pv,fv,type) • Note: When calculating rate and number of period, you need to specify the correct directions of the cash inflows (+) and cash outflows (-). At least one of the cash flows need to have a negative value.

12. PV of annuity and amortization of a mortgage loan • If someone borrow $300,000 from a bank for five years. The term is to pay an equal money at the end of each year, and interest rate is 10%.

13. PV of annuity and amortization of a mortgage loan

14. Effective Interest Rate

15. Effective Annual Rate (EAR) - Formula

16. Bond Valuation

17. One firm issues a bond with 10-year, $10,000 par, paying 8% annual coupon, if the required rate for the bond is 10%, what will be the price you are willing to pay？ • PV(rate,nper,pmt,fv,type) = • PV(10%,10, 800, 1000,0) = $8771.1

18. Yield to Maturity: the rate of return will be obtained if bond holders decide to hold the bond until maturity • One firm issues a bond with 10-year, $10,000 par, paying 10% annual coupon, if the current bond price is $9,500, then the yield to maturity is? • RATE(nper,pmt,pv,fv,type) = • RATE(10,1000, -9500, 10000, 0) = 10.84%

19. Holding period return: the average rate of return that an bond investor has from purchasing the bond to selling the bond. • Four years ago, you purchase a 10-year, $10,000 par, 8% annual coupon for $9,000. Today after received the fourth coupon, you sell the bond for $11,089. What is your holding period return? • RATE(4,800,-9000,11089,0) =15.71%

20. Preferred Stock Valuation • Preferred stock pays fixed dividend, so the value of stock similar to present value of perpetuity.

21. A preferred stock with par of $100 paying 12%preferred dividend, if the required rate of return for the stock is 15%, then its price is？ • Preferred dividend per year $100*12%=$12

22. Cont. Holding period return for a preferred stock • If an investor purchases the preferred stock from last page. Four years later, after the investor received the fourth dividend, he sell the stock. At the time of selling, the stock’s required rate of return raised to 20% due to the increased competition for the company issued the stock, what will be the investor’s holding period return?

23. Holding period return for preferred stock • The selling price for the stock after four years • So the rate of return can be calculated from RATE(4,12,-80,60,0) =9.58%

24. Common stock valuation • We can not predict all dividends for a stock over its entire life, so we have to make assumptions toward dividend behavior: Assuming • Firm pay fixed dividend. • Dividend increases in a constant growth rate • Dividend first increases in high growth rates, then increases in a constant growth rate.

25. Public utility stocks often pay fixed dividend • ABC Utility usually pays $3 dividend per share each year. If the required rate of return for ABC is 14%, what will be its reasonable price?

26. Large and established firms usually pay constant growth dividends.

27. Dividend growth, payout ratio and return on equity g = retention * ROEAssuming ROE=20%, payout=50%, beginning equity = 100

28. Dividend growth, payout ratio and return on equity • Firm A has EPS of $1, assuming that market required rate of return toward its stock is 15%. Observe the stock prices on different ROEs (10%、15%、20%) and different payout ratios:

30. Dividend growth, payout ratio and return on equity • When ks > ROE, the higher the payout, the higher stock price. • When ks < ROE, the lower the payout, the higher stock price. • When ks = ROE, payout is irrelevant to stock price. • When payout ratio is 100%, thus no retention, the growth rate becomes zero, so the ROE is irrelevant to stock price.

31. Dividend first increases in high growth rates, then increases in a constant growth rate. • An up and rising firm this year issues $2 dividend (D0). The analyst expects the dividend will increase in 30% for two years, and 20% for another year before the dividend grows perpetually in 10%. If the required rate of return for the stock is 15%, how much will you pay for the stock?

32. 30% 10% 30% 20% 10% $D4 =4.06(1+0.1) =4.46 $D2 =2.6(1+0.3) =3.38 $D3 =3.38(1+0.2) =4.06 $D0 =2 $D1 =2(1+0.3) =2.6