Compound Interest: The future value (FV) of an investment of present value (PV) dollars earning interest at an annual rate of r compounded m times per year for a period of t years is: Show
FV = PV(1 + r/m)mtor FV = PV(1 + i)n where i = r/m is the interest per compounding period and n = mt is the number of compounding periods. One may solve for the present value PV to obtain: PV = FV/(1 + r/m)mt Numerical Example: For 4-year investment of $20,000 earning 8.5% per year, with interest re-invested each month, the future value is FV = PV(1 + r/m)mt = 20,000(1 + 0.085/12)(12)(4) = $28,065.30 Notice that the interest earned is $28,065.30 - $20,000 = $8,065.30 -- considerably more than the corresponding simple interest. Effective Interest Rate: If money is invested at an annual rate r, compounded m times per year, the effective interest rate is: reff = (1 + r/m)m - 1. This is the interest rate that would give the same yield if compounded only once per year. In this context r is also called the nominal rate, and is often denoted as rnom. Numerical Example: A CD paying 9.8% compounded monthly has a nominal rate of rnom = 0.098, and an effective rate of: r eff =(1 + rnom /m)m = (1 + 0.098/12)12 - 1 = 0.1025. Thus, we get an effective interest rate of 10.25%, since the compounding makes the CD paying 9.8% compounded monthly really pay 10.25% interest over the course of the year. Mortgage Payments Components: Let where P = principal, r = interest rate per period, n = number of periods, k = number of payments, R = monthly payment, and D = debt balance after K payments, then R = P � r / [1 - (1 + r)-n] andD = P � (1 + r)k - R � [(1 + r)k - 1)/r] Accelerating Mortgage Payments Components: Suppose one decides to pay more than the monthly payment, the question is how many months will it take until the mortgage is paid off? The answer is, the rounded-up, where: n = log[x / (x � P � r)] / log (1 + r) where Log is the logarithm in any base, say 10, or e.Future Value (FV) of an Annuity Components: Ler where R = payment, r = rate of interest, and n = number of payments, then FV = [ R(1 + r)n - 1 ] / r Future Value for an Increasing Annuity: It is an increasing annuity is an investment that is earning interest, and into which regular payments of a fixed amount are made. Suppose one makes a payment of R at the end of each compounding period into an investment with a present value of PV, paying interest at an annual rate of r compounded m times per year, then the future value after t years will be FV = PV(1 + i)n + [ R ( (1 + i)n - 1 ) ] / i where i = r/m is the interest paid each period and n = m � t is the total number of periods. Numerical Example: You deposit $100 per month into an account that now contains $5,000 and earns 5% interest per year compounded monthly. After 10 years, the amount of money in the account is: FV = PV(1 + i)n + [ R(1 + i)n - 1 ] / i =
Value of a Bond: V is the sum of the value of the dividends and the final payment. You may like to perform some sensitivity analysis for the "what-if" scenarios by entering different numerical value(s), to make your "good" strategic decision. Replace the existing numerical example, with your own case-information, and then click one the Calculate. Chapter 3: The Time Value of MoneyJust click on the button next to each answer and you'll get immediate feedback.Note: Your browser must support JavaScript in order to use this quiz. 1. You want to buy an ordinary annuity that will pay you $4,000 a year for the next 20 years. You expect annual interest rates will be 8 percent over that time period. The maximum price you would be willing to pay for the annuity is closest to$32,000. $39,272. $40,000. $80,000. 2. With continuous compounding at 10 percent for 30 years, the future value of an initial investment of $2,000 is closest to$34,898. $40,171. $164,500. $328,282. 3. In 3 years you are to receive $5,000. If the interest rate were to suddenly increase, the present value of that future amount to you wouldfall. rise. remain unchanged. cannot be determined without more information. 4. Assume that the interest rate is greater than zero. Which of the following cash-inflow streams should you prefer? Year1 Year2 Year3 Year4 $400 $300 $200 $100 $100 $200 $300 $400 $250 $250 $250 $250 Any of the above, since they each sum to $1,000. 5. You are considering investing in a zero-coupon bond that sells for $250. At maturity in 16 years it will be redeemed for $1,000. What approximate annual rate of growth does this represent?8 percent. 9 percent. 12 percent. 25 percent. 6. To increase a given present value, the discount rate should be adjustedupward. downward. True. Fred. 7. For $1,000 you can purchase a 5-year ordinary annuity that will pay you a yearly payment of $263.80 for 5 years. The compound annual interest rate implied by this arrangement is closest to8 percent. 9 percent. 10 percent. 11 percent. 8. You are considering borrowing $10,000 for 3 years at an annual interest rate of 6%. The loan agreement calls for 3 equal payments, to be paid at the end of each of the next 3 years. (Payments include both principal and interest.) The annual payment that will fully pay off (amortize) the loan is closest to$2,674. $2,890. $3,741. $4,020. 9. When n = 1, this interest factor equals one for any positive rate of interest.PVIF FVIF PVIFA FVIFA None of the above (you can't fool me!) 10. (1 + i)nPVIF FVIF PVIFA FVIFA 11.You can use to roughly estimate how many years a given sum of money must earn at a given compound annual interest rate in order to double that initial amount .Rule 415 the Rule of 72 the Rule of 78 Rule 144 12.In a typical loan amortization schedule, the dollar amount of interest paid each period . increases with each payment decreases with each payment remains constant with each payment 13.In a typical loan amortization schedule, the total dollar amount of money paid each period .increases with each payment decreases with each payment remains constant with each payment  Multiple-Choice Quiz questions are Copyright © by Pearson Education Limited. Used by permission. All rights reserved.
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