Mcgill房地产答案chapter3

Solutions to Questions - Chapter 3 The Interest Factor in Financing 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. Question 3-8 Why can’t interest factors for annuities be used when evaluating the present value of an uneven series of receipts?  What factors must be used to discount a series of uneven receipts? With an annuity, interest factors can be summed because the payments are equal in amount and are received at     equal intervals.  If the series of annuities being evaluated is uneven, the interest factors cannot be summed and the     interest factors for annuities are of value mathematically.  In the case of an uneven series of receipts, the     calculation requires the use of individual ordinary present value factors. 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 The Interest Factor in Financing Problem 3-1 a)    Future Value    =    $12,000 (FVIF, 9%, 7 years) =    $12,000 (1.82804) =    $21,936 (annual compounding) b)    Future Value    =    $12,000 (QFVIF, 9% , 7 years) =    $12,000 (1.86454) =    $22,375 (quarterly compounding) c)  Equivalent annual yield:  (consider one year only) Future Value of (a)    =    $12,000 (FVIF, 9%, 1 year) =    $12,000 (1.09) =    $13,080 ($13,080 - $12,000) / $12,000 =     9.00% effective annual yield Future Value of (b)    =    $12,000 (QFVIF, 9%, 1 year) =    $12,000 (1.09308) =    $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:  7% compounded monthly Future Value of A    =    $25,000 (MFVIF, 7%, 1 year) =    $25,000 (1.07229) =    $26,807 (monthly compounding) Investment B:  8% compounded annually Future Value of B    =    $25,000 (MFVIF, 8%, 1 year) =    $25,000 (1.08) =    $27,000 (annual compounding) Investment B should be chosen over A.  Investment B that pays 8% compounded annually is the better choice because it provides the greater future value and therefore the greater effective annual yield. Problem 3-3 Find the future value of 24 end-of-period payments of $5,000 at an annual rate of 8.5%, compounded semi-annually based on an ordinary annuity. Future Value    =    $5,000 (SAFVIFA, 8.5%, 12 years) =    $5,000 (40.36113) =      $201,806 Note:  Total cash deposits are $5,000 x 24 = $120,000.  Total interest equals $81,806 ($201,806 - $120,000).  These semi-annual deposits constitute an annuity.  The $120,000 represents the return of capital of initial principal while the $81,806 represent 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 8.5%, compounded semi-annually based on an annuity due. Future Value    =    $5,000 (SAFVIFA, 8.5%, 12 years) =    $5,000 (42.07648) =      $210,382 Problem 3-4 Find the future value of 4 years of quarterly payments at $1,250 each earning an interest rate of 15 percent annually, compounded quarterly. Future Value    =    $1,250 (QFVIFA, 15%, 4 years) =    $1,250 (21.39274) =      $26,741 Problem 3-5 Year Amount Deposited FVIF   Future Value 1 $2,500 x (FVIF, 9%, 4 yrs.) or 1.41158 $3,529 2 $0 x (FVIF, 9%, 3 yrs.) or 1.29503 $0 3 $750 x (FVIF, 9%, 2 yrs.) or 1.18810 $891 4 $1,300 x (FVIF, 9%, 1 yr.) or 1.09000 $1,417 5 $0     $0           Total Future Value =     $5,837 The investor will have $5,837 on deposit at the end of the 5th year. *Each deposit is made at the end of the year. Problem 3-6 a)  Find the present value of 8 years of monthly payments, or 96 payments, of $750 (end-of-month) discounted at an interest rate of 17 percent compounded monthly. Present Value    =    $750 (MPVIFA, 17%, 8 years) - ordinary annuity =    $750 (52.29728) =        $39,223 should be paid today b)  The total sum of cash received over the next 8 years will be: 8 years x 12 payments per year x $750 per month =     $72,000 c)    Total cash received by the investor    $72,000 Initial price paid by the investor    $39,223 Difference:  Interest Earned    $32,777 The difference represents the total interest earned by the investor on the initial investment of $39,223 if each $750 payment received earns 17 percent compounded monthly. Problem 3-7 Find the present value of 10 end-of-year payments of $2,150 discounted at an annual interest rate of 18 percent. Present Value    =    $2,150 (PVIFA, 18%, 10 years) - ordinary annuity =    $2,150 (4.49409) =    $9,662 should be paid today Find the present value of 10 beginning-of-year payments of $2,150 discounted at an annual interest rate of 18 percent. Present Value    =    $2,150 (PVIFA, 18%, 10 years) - annuity due =    $2,150 (5.30302) =                      $11,401 should be paid today Problem 3-8 Find the present value of $45,000 discounted at an 18% annual rate compounded quarterly for a six year period. Present Value    =    $45,000 (QPVIF, 18%, 6 years) =    $45,000 (.34770) =    $15,647 should be paid today Note that a quarterly interest factor is used in this problem because the investor indicates that an annual rate of 18% is desired. Problem 3-9 Year Amount Deposited MPVIF   Present Value 1 $12,500 x (MPVIF, .75%, 12 months) or 0.91424 $11,428 2 $10,000 x (MPVIF, .75%, 24 months) or 0.83583 $8,358 3 $7,500 x (MPVIF, .75%, 36 months) or 0.76415 $5,731 4 $5,000 x (MPVIF, .75%, 48 months) or 0.69861 $3,493 5 $2,500 x (MPVIF, .75%, 60 months) or 0.63870 $1,597 6 $0 x (MPVIF, .75%, 72 months) or 0.58392 $0 7 $12,500 x (MPVIF, .75%, 84 months) or 0.53385 $6,673