Chapter 03 - Mortgage Loan Foundations: The Time Value of Money 3-1 Solutions to Questions - Chapter 3 Mortgage Loan Foundations: The Time Value of Money Question 3-1 What is the essential concept in understanding compound interest? The concept of earning interest on interest is the essential idea that must be understood in the compounding process and is the cornerstone of all financial tables and concepts in the mathematics of finance. Question 3-2 How are the interest factors (IFs) Exhibit 3-3 developed? How may financial calculators be used to calculate IFs in Exhibit 3-3? Computed from the general formula for compounding for monthly compounding for various combinations of “i” and years. FV = PV x (1+i)n. Calculators can be used by entering $ 1 for PV, the desired values for n and i and solving for FV. Question 3-3 What general rule can be developed concerning maximum values and compounding intervals within a year? What is an equivalent annual yield? Whenever the nominal annual interest rates offered on two investments are equal, the investment with the more frequent compounding interval within the year will always result in a higher effective annual yield. An equivalent annual yield is a single, annualized discount rate that captures the effects of compounding (and if applicable, interest rate changes). Question 3-4 What does the time value of money (TVM) mean? Time value simply means that if an investor is offered the choice between receiving $1 today or receiving $1 in the future, the proper choice will always be to receive the $1 today, because that $1 can be invested in some opportunity that will earn interest. Present value introduces the problem of knowing the future cash receipts for an investment and trying to determine how much should be paid for the investment at present. When determining how much should be paid today for an investment that is expected to produce income in the future, we must apply an adjustment called discounting to income received in the future to reflect the time value of money. Question 3-5 How does discounting, as used in determining present value, relate to compounding, as used in determining future value? How would present value ever be used? The discounting process is a process that is the opposite of compounding. To find the present value of any investment is simply to compound in a “reverse” sense. This is done by taking the reciprocal of the interest factor for the compound value of $1 at the interest rate, multiplying it by the future value of the investment to find its present value. Present value is used to find how much should be paid for a particular investment with a certain future value at a given interest rate. Question 3-6 What are the interest factors (IFs) in Exhibit 3-9? How are they developed? How may financial calculators be used to calculate IFs in Exhibit 3-9? Compound interest factors for the accumulation of $1 per period, e.g., $1 x [1 + (1+i) + (1+i)2 …] etc. Calculators may be used by entering $ 1 values for PMT, entering the desired values for n and i then solving for FV. Question 3-7 What is an annuity? How is it defined? What is the difference between an ordinary annuity and an annuity due? An annuity is a series of equal deposits or payments. An ordinary annuity assumes payments or receipts occur at the end of a period. An annuity due assumes deposits or payments are made at the beginning of the period.Chapter 03 - Mortgage Loan Foundations: The Time Value of Money 3-2 Question 3-8 How must one discount a series of uneven receipts to find PV? Each periodic cash receipt or payment must be discounted individually then summed to obtain present value. That is: PV= CF1 (1/1 + i)1 + CF2 (1/1+ i)2 ….+ CFn (1/1 + i)n where CF is cash inflow and i equals the discount rate. Question 3-9 What is the sinking-fund factor? How and why is it used? A sinking-fund factor is the reciprocal of interest factors for compounding annuities. These factors are used to determine the amount of each payment in a series needed to accumulate a specified sum at a given time. To this end, the specified sum is multiplied by the sinking-fund factor. Question 3-10 What is an internal rate of return? How is it used? How does it relate to the concept of compound interest? The internal rate of return integrates the concepts of compounding and present value. It represents a way of measuring a return on investment over the entire investment period, expressed as a compound rate of interest. It tells the investor what compound interest rate the return on an investment being considered is equivalent to. Solutions to Problems - Chapter 3 Mortgage Loan Foundations: The Time Value of Money Problem 3-1 a) Future Value = FV(n,i,PV,PMT) = FV (7yrs, 6%, $12,000, 0) = $18,044 (annual compounding) b) Future Value = FV(n,i,PV,PMT) = FV (28 quarters, 9% ÷ 4, $12,000, 0) = $22,375 (quarterly compounding) c) Equivalent annual yield: (consider one year only) Future Value of (a) = FV(n,i,PV,PMT) = FV (1yr, 6%, $12,000, 0) = $12,720 ($12,720 - $12,000) / $12,000 = 6.00% effective annual yield Future Value of (b) = FV(n,i,PV,PMT) = FV (1yr, 9%, $12,000, 0) = $13,117 ($13,117 - $12,000) / $12,000 = 9.31% effective annual yield Alternative (b) is better because of its higher effective annual yield. Problem 3-2 Investment A: 6% compounded monthly Future Value of A = FV(n,i,PV,PMT) = FV (12 mos., 6%  12, $25,000, 0) = $26,542 (monthly compounding) Investment B: 7% compounded annually Future Value of B = FV(n,i,PV,PMT) = FV (1yr, 7%, $25,000, 0) = $26,750 (annual compounding)Chapter 03 - Mortgage Loan Foundations: The Time Value of Money 3-3 Investment B should be chosen over A. Investment B pays 7% compounded annually and is the better choice because it provides the greater future value. Problem 3-3 Find the future value of 24 deposits of $5,000 made at the end of each 6 months. Deposits will earn an annual rate of 8.0%, compounded semi-annually. Future Value = FV(n,i,PV,PMT) = FV (24 periods, 8%  2, 0, $5,000) = $195,413 Note: Total cash deposits are $5,000 x 24 = $120,000. Total interest equals $75,413 or ($195,413 - $120,000). The $120,000 represents the return of capital (initial principal) while the $75,413 represents the