Lecture 2: The Basic Theory of Interest

1. Lecture 2: The Basic Theory of Interest • Interest is frequently called the time value of money • Compound interest – determine future sums of money resulting from an investment • Discounting (the calculation of present value) – inversely related to compounding, is used to evaluate the cash flow associated with the valuation of financial assets

2. Future Values – Compounding • A dollar in hand today is worth more than a dollar to be received tomorrow • Compounding interest – interest earns interest Fn = P(1+i)n = P.FVIFi,n Fn = Future value = the amount of money at the end of year n P = principal i = annual interest rate n = number of years FVIF = future value interest factor for $1 (Table 2.1)

3. Future Values – CompoundingExample 1 • Nita placed $1000 in a saving account earning 8% interest compounded annually. How much money will she have in the account at the end of 4 years? • Fn = P(1+i)n F4 = 1000(1+0.08)4 = 1000 . FVIF8,4 From Table 2.1, the FVIF for 4 years at 8% is 1.36, therefore, F4 = 1000(1.36) = $1360

4. Table 2.1 Compounded Future Value of $1

5. Present Values – Discounting • Present worth of future sums of money • Opposite of finding the compounded future value = Fn.PVIFi,n where PVIFi,n represents the present value interest factor for $1 (Table 2.2)

6. Present Values – Discounting Example 2 • Daniel has been given an opportunity to receive $20,000 6 years from now. If she can earn 10% on investments, what is the most she should pay for this opportunity? P = 20000 [1/(1+0.1)6] = 20000 (PVIF10,6) = 20000 (0.564) = $11280 Daniel could invest $11280 today at 10% and have $20000 in 6 years

7. Table 2.2 Present Value of $1 (PVIF)

8. Bond and Stock Valuation • The process of determining security valuation involves finding the present value of an assets expected future cash flows using the investor’s required rate of return • Basic model: where V = intrinsic value or present value of an asset Ci = expected future cash flows in period t=1,…n r = investor’s required rate of return

9. Bond Valuation • Three basic elements 1. The amount of the cash flows to be received by the investor, which is equal to the periodic interest to be received and the par value to be paid at maturity 2. The maturity date of the loan, and 3. The investor’s required rate of return (the periodic interest can be received annually or semi annually)

10. Bond Valuation……. • The value of a bond is simply the present value of the cash flows • If the interest payments are annually: • I = interest payment each year = coupon interest rate x par value • M = par value or maturity value, typically $1000 • r = investor’s required rate of return • n = number of years to maturity • PVIFA = present value interest factor of an annuity of $1 (Table 2.3); PVIF - Table 2.2

11. Table 2.3 Present Value of an Annuity of $1 (PVIFA)

12. Bond Valuation…….Example 3 • Consider a bond, maturity in 10 years and having a coupon rate of 8%. The par value is $1000. Investors consider 10% to be an appropriate required rate of return in view of the risk level associated with this bond. The annual interest payment is $80 (8% x $1000). The present value of this bond is

13. Bond Valuation…….Example 3…… Continue.. = = $80(6.145) + $1000(0.386) = $877.60

14. Common Stock Valuation • The value of a common stock is the present value of all future cash inflows expected • The cash inflows expected to be received are dividends and the future price at the time of the sale of the stock • For an investor holding a common stock for only 1 year, the value of the stock would be the present value of both the expected cash dividend to be received in 1 year (D1) and the expected market price per share of the stock at year-end (P1).

15. Common Stock Valuation…… • If r represents an investor’s required rate of return, the value of the common stock (P0) • Since common stock has no maturity date and is held for many years, a more general, multiperiod model is needed. The general common stock valuation model is:

16. Common Stock Valuation…… • There are 3 cases of growth in dividends: 1. zero growth 2. constant growth 3. supernormal growth • In the case of zero growth, if D0=D1=….D, then the valuation model becomes:

17. Common Stock Valuation……Constant Growth • In the case of constant growth, assuming dividends grow at a constant rate of g every year {I.e. Dt=D0(1+g)t }, then the valuation model is is know as the Gordon growth model

18. Common Stock Valuation……Example 4 • A common stock that paid a $3 dividend per share at the end of last year and is expected to pay a cash dividend every year at a growth rate of 10%. Assume that the investor’s required rate of return is 12%. The value of stock would be: D1=D0(1+g) = $3(1+0.10) = $3.30

19. Common Stock Valuation……Supernormal growth • Firms typically go through life cycles (part of which their growth is faster than of the economy and then falls sharply) • The value of stock can be found: Step 1: compute the dividends during the period of supernormal growth and find their PV Step 2: find the price of the stock at the end of the supernormal growth period and compute its PV

20. Common Stock Valuation……Supernormal growth… • Step 3: add these two PV figures to find the value (P0) of the common stock Example 5: A common stock whose dividends are expected to grow at a 25% for 2 years, after which the growth rate is expected to fall to 5%. The dividends paid last period was $2. The investor desires a 12% return. Find the value of this stock.

21. Example 5: Solution • Step 1: Compute the dividends during the supernormal growth period & find their PV. Assuming D0 is $2, g is 25% and r is 12%. D1 = D0(1+g) = $2(1+0.25) = $2.50 D2 = D1(1+g) = $2.50 (1+0.25) = $3.125 PV of dividends = $4.72

22. Example 5: Solution…. • Step 2: Find the price of stock at the end of the supernormal growth period. The dividends for the third year is D3 = D2(1+g’), where g’ = 5% The price of the stock is: PV of stock price = $46.86 (PVIF12%,2) = $46.86 (0.797) = $37.35

23. Example 5: Solution…. • Step 3: Add the two PV figures obtained in steps 1 & 2 to find the value of the stock P0 = $4.72 + $37.35 = $42.07

24. Simple Interest and Compounded Interest Simple Interest • when money of value P on a given date increases in value to S at some later date P – principal S – amount or accumulated value of P, and I = S – P (interest) • when only the principal earns interest for the entire life of the transaction, the interest due at the end of the time is called simple interest.

25. Simple Interest • The simple interest on a principal P for t years at the rate r is: I = Prt and the simple amount is S = P + I = P + Prt = P(1+ rt) The amount grows linearly with time.

26. Simple Interest Example 6 • Find the simple interest on $750 at 4% for 6 months. What is the total amount? P = 750, r = 0.04, and t = 0.5, then I = Prt = 750(0.04)(0.5) = $15 S = P + I = $750 + $15 = $765

27. Compounded Interest • If the interest due is added to the principal at the end of each interest period and thereafter earns interest, the interest is said to be compounded. • The sum of the original principal and total interest is called the compound amount or accumulated value. • The difference between the accumulated value and the original principal is called the compound interest.

28. Compounded Interest….. • The interest period, the time between 2 successive interest computations, is called conversion period. • The rate of interest is usually stated as an annual interest rate (nominal rate of interest). • The phrases ‘interest at 12%’ or ‘money worth 12%’ – 12% compounded annually.

29. Compounded Interest….. The following notation will be used: P = original principal, or the PV of S, or the discounted value of S S = compound amount of P, or the accumulated value of P n = total number of interest (or conversion) periods involved m = number of interest periods per year, or the frequency of compounding’

30. Compounded Interest….. • jm = nominal (yearly) interest rate which is compounded (payable, convertible) m times per year. • i = interest rate per interest period. The interest rate per period, i, equals jm/m. For example, j12 = 12% - a nominal (yearly) rate of 12% is converted (compounded, payable) 12 times per year, i = 1% = 0.01 being the interest rate per month.

31. Compounded Interest….. • Let P represents the principal at the beginning of the first interest period and i the interest rate per conversion period. • It is necessary to calculate the accumulated values at the ends of successive interest periods for n periods. • At the end of the first period: interest due Pi accumulated value P+Pi = P(1+i)

32. Compounded Interest….. • At the end of the second period: interest due [P(1+i)]i accumulated value P(1+i)+ [P(1+i)]i = P(1+i)(1+i) = P(1+i)2 • At the end of the third period: interest due [P(1+i)2]i accumulated value P(1+i)2+ [P(1+i)2]i = P(1+i)2(1+i) = P(1+i)3

33. Compounded Interest….. • Continuing in this manner, the successive accumulated values, P(1+i), P(1+i)2, P(1+i)3, ….. • Form a geometric progression whose nth: S = P(1+i)n where S is the accumulated value of P at the end of the n interest periods.