# Compound Interest Chapter 8

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2 8-2 Compound Interest Chapter 8

3 8-3 Learning Objectives After completing this chapter, you will be able to: > Calculate maturity value, future value, and present value in compound interest applications, by both the algebraic method and the pre-programmed financial calculator method > Calculate the maturity value of compound interest Guaranteed Investment Certificates (GICs) > Calculate the price of "strip" bonds > Calculate the redemption value of a compound interest Canada Savings Bond > Adapt the concepts and equations of compound interest to cases of compound growth > Calculate the payment on any date that is equivalent to one or more payments on other dates > Calculate the economic value of a payment stream

4 8-4 Compound Interest (Future Value) Compounding - involves the calculation of interest over the life of the loan or investment Compound interest - the interest on the principal plus the interest of prior periods Future value (compound amount) - is the final amount of the loan or Present Value - the value of a loan investment at the end of the last period or investment today

5 8-5 Interest earned in the first year Interest earned in the second year \$100 \$110 A m o u n t \$1000 \$1000 Beginning principal for the second year (\$1100) \$1100 Beginning principal for the third year (\$1210) Compounding period Time (years)

6 8-6 Table 8.1 Compounding Frequencies and Periods Compounding Number of Compounding frequency compoundings per year period Annual 1 1 year Semiannual 2 6 months Quarterly 4 3 months Monthly 12 1 month Daily day

7 8-7 Determining values for i and n Number of periods (n) # of years * # of times the interest is compounded per year Rate for each period (i) Annual interest rate / # times interest is compounded per year If you compounded \$100 for 3 years at 6% annually, semiannually, or quarterly What are values for n and i? Periods (n) Annually: 3 * 1 = 3 Semiannually: 3 * 2 = 6 Quarterly: 3 * 4 = 12 Rate (i) Annually: 6% / 1 = 6% Semiannually: 6% / 2 = 3% Quarterly: 6% / 4 = 1.5%

8 8-8 Formula for Future Value FV = PV(1+i) n Calculator Steps: i + 1 = y x n = * PV =

9 8-9 Steve Smith deposited \$1,000 in a savings account for 4 years at an semiannual compounded rate of 8%. What is Steve s interest and compounded amount? PV = \$1000 n = 4 x 2 = 8 i =.08/2 =.04 FV= 1000(1 +.04) 8 = \$1,000 * = \$1, Interest: I = \$1, \$1,000 = \$ / = y x * = Compounded amount = Future Value Investment

10 The Components of the Future Value of \$ S=P(1+i) n Interest on Interest 200 Future Value. S 150 S=P(1+rt) Interest on Original Principal Original Principal Time (years)

11 8-11 Simple Versus Compound Interest Simple Al Jones deposited \$1,000 in a savings account for 5 years at an annual simple interest rate of 10%. What is Al s simple interest and maturity value? Compounded Al Jones deposited \$1,000 in a savings account for 5 years at an annual compounded rate of 10%. What is Al s interest and compounded amount? I = Prt I = \$1,000 *.10 * 5 = \$500 MV = P + I MV = \$1,000 + \$500 = \$1,500 FV = PV(1 +i) n n = 5 * 1 = 5 i =.10/5 =.02 FV = 1000(1.02) 5 = \$1,000 * = \$1, I = FV - PV I = \$1, \$1,000 = \$610.50

12 8-12 Future Values of \$100 at Various Rates of Interest Compounded Annually % Future Value, S or FV % 8% 6% Years to Maturity, n

13 8-13 Nominal Rates of Interest Compared Beginning Nominal rate Compounding End balance of interest period balance \$1, % Annual Semiannual Quarterly Daily \$1, \$1, \$1, \$1,061.80

14 8-14 Future Values of \$100 at the Same Nominal Rate but Different Compounding Frequencies 2500 Future Value, S or FV % Compounded monthly 12% Compounded annually Time (years)

15 8-15 Compounding Daily Interest Calculate the future value of \$2,000 compounded daily for 7 years at 6% n = 7 * 365 = 2555 i =.06 / 365 = FV = 2000(1+.06/365) 2555 =\$2,000 * = \$3, Calculator steps:.06 / = y x = * 2000 =

16 8-16 Using Financial Calculators n i PV FV PMT comp represents the number of compounding periods represents the periodic interest rate, j/m represents the principal or present value represents the maturity value or future value represents the periodic annuity payment (not used until chapter 10) used to tell calculator to compute (CPT)

17 8-17 Using your financial calculator to solve the previous example: Calculate the future value of \$2,000 compounded daily for 7 years at 6% Using your Financial calculator: 7 *365 = n FV = \$2,000 * = \$3, /365 = comp i PV PMT FV

18 8-18 Heather invested \$6000 at 7% compounded quarterly. After 2 years, the rate changed to 8.2% compounded monthly. What amount will Heather have 3 1/2 years after the initial investment? 0 2 years 3.5 years \$6000 i=.07/4 n=8 S1 = P2 i=.082/12 n=1.5*12=18 S 2 S 1 = 6000(1+.07/4) 8 = 6000(1.1489) = S 2 = (1+.082/12) 18 = (1.1304) =

19 Shannon borrowed \$6000 at 9% compounded quarterly. On the first and second anniversaries of the loan, she made payments of \$2500. What is the balance outstanding immediately following the second payment? 0 1 year 2 years 8-19 \$6000 i=.09/4 n=4 S i=.09/4 n=4 S 1 = 6000(1+.09/4) 4 = 6000(1.0931) = P 2 = S 2 S 2 = (1+.09/4) 4 = (1.0931) = Outstanding balance: \$

20 8-20 Present Value of \$1 at 8% for Four Periods \$1.20 \$1.10 \$1.00 \$0.90 \$0.80 \$0.70 \$0.60 \$0.50 \$0.40 \$0.30 \$0.20 \$0.10 \$0.00 Present value goes from the future value to the present value Present value \$.7350 \$.7938 \$.8573 \$ \$ Future Value Number of periods

21 8-21 Formula for Present Value PV = FV(1+i) - n Calculator Steps: i + 1= y x n +/- = * FV =

22 8-22 Calculating Present Value Steve Smith needs \$1, in 4 years. His bank offers 8% interest compounded semiannually. How much money must Steve put in the bank today (PV) to reach his goal in 4 years? n = 4 * 2 = 8 i =.08 =.04 PV = (1+.04) -8 = \$1, *.7307 = \$1, Calculator steps: 1.04 y x 8 +/- = * = Invest Today Investment

23 8-23 \$1,200 Figure 8.5 The Present Value Of \$1000 (Discounted at 10% Compounded Annually) \$1,000 \$800 Present value \$600 \$400 \$200 \$0 25 years earlier 20 years earlier 15 years earlier 10 years earlier 5 years earlier Payment date

24 8-24 Two payments of \$4000 each must be made 1 and 4 years from now. If money can earn 8% compounded monthly, what single payment 3 years from now would be equivalent to the two scheduled payments? 0 1 year 2 years 3 years 4 years \$4000 \$4000 i=.08/12 n=2*12=24 FV 1 = 4000(1+.08/12) 24 = 4000(1.1729) = PV 2 = 4000(1+.08/12) -12 = 4000(0.9234) = i=.08/12 n=1*12=12 Final ans: = \$

25 8-25 What amount must you invest now at 6% compounded daily to accumulate to \$5000 after 1 year? Using your Financial calculator: 365 n j= 6% m = 365 S = FV = \$5000 i=.06/365 n = 1 * 365 = 365 6/365= comp i FV PMT PV Ans.: \$

26 8-26 What regular payment will an investor receive from a \$10 000, 5 year, monthly payment GIC earning a nominal rate of 7% compounded monthly? Interest rate per payment interval is: i= j/m =.07/12 = the monthly payment will be: PV * i= \$ * = \$58.33

27 8-27 Guaranteed Investment Certificates (GICs) > purchased from banks, credit unions, trust companies, and caisses populaires (in Quebec). > you are in effect lending money to the financial institution. > the financial institution uses the funds raised from selling GICs to make loans most commonly, mortgage loans. > mortgage rates are typically 1.5% to 2% higher than the interest rate paid to GIC investors. > Guaranteed refers to the unconditional guarantee of principal and interest by the parent financial institution. > maturities in the range of 1 to 5 years. > most GICs are not redeemable before maturity.

28 8-28 Structure of Interest Rates Fixed rate: The interest rate does not change over the term of the GIC. Step-up rate: The interest rate is increased every 6 months or every year according to a pre-determined schedule. Variable rate: The interest rate is adjusted every year or every 6 months to reflect prevailing market rates. There may be a minimum floor below which rates cannot drop

29 8-29 Payment of interest Compound interest version: Interest is periodically converted to principal and paid at maturity. Regular interest version: Interest is paid to the investor every year or every 6 months.

30 8-30 Canada Savings Bonds (CSBs( CSBs)? you purchase from financial institutions, but \$ goes to the federal government to help finance its debt.? usual term is 10 or 12 years? variable interest rates? interest rates is changed on each anniversary, with minimum rates for subsequent 2 years? may be redeemed at any time? both regular and compound interest versions

31 8-31 Valuation Principle: > The fair market value of an investment is the sum of the present values of the expected cash flows. > The discount rate used should be the prevailing market determined rate of return required on this type of investment.

32 8-32 Strip bonds? its owner will receive a single payment (called the face value of the bond) on the bond s maturity date.? The maturity date could be as much as 30 years in the future. No interest will be received in the interim. Suppose a \$1000 face value strip bond matures 18 years from now. The owner of this bond will receive a payment of \$1000 in 18 years. What is the appropriate price to pay for the bond today if the prevailing rate of return is 8.75% compounded semi-annually? FV = \$1000 i=.0875/2 n = 18 * 2 = 36 PV = 1000( /2) -36 = 1000(0.2141) = \$214.06

33 8-33 THE END

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