# UCLA ECON 103 - Chap003 (1) (8 pages)

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## Chap003 (1)

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- Pages:
- 8
- School:
- University of California, Los Angeles
- Course:
- Econ 103 - Introduction to Econometrics

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Chapter 03 Mortgage Loan Foundations The Time Value of Money 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 3 1 Chapter 03 Mortgage Loan Foundations The Time Value of Money 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 b Future Value FV n i PV PMT FV 7yrs 6 12 000 0 18 044 annual compounding 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 12 720 12 000 12 000 Future Value of b 13 117 12 000 12 000 FV n i PV PMT FV 1yr 6 12 000 0 12 720 6 00 effective annual yield FV n i PV PMT FV 1yr 9 12 000 0 13 117 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 3 2 Chapter 03 Mortgage Loan Foundations The Time Value of Money 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 interest earned on the capital contributions Find the future value of 24 beginning of period payments of 5 000 at an annual rate of 8 0 compounded semi annually based on an annuity due Future Value FV n i PV PMT FV 25 periods 8 2 0 5 000 208 230 Note n is changed to 25 because the deposits are made at the beginning of each period Therefore the first deposit will be compounded 25 times whereas if the 1st deposit was made at the end …

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