More Time Value of Money

Questions:

1. For a given stated annual rate, you are better off borrowing at a(n) __________ compounded rate and better off lending at a(n) __________ compounded rate:
   a.  Monthly;  Annually
   b.  It doesn't matter - they both work out the same.
   c.  It depends on the interest rate and the term of the loan.
   d.  Annually; Monthly
   e.  There is not enough information given to definitely answer this question.

2. You are analyzing a potential investment. The investment is structured as follows: you pay a lump sum up front, and receive  a series of payments of equal amounts in the future. Which of the following would make the investment more appealing. Assume that your discount rate is greater than zero and assume that in each case (a, b, c, and d), only the factors mentioned change, but that all else remains constant.

a. The discount rate increases but the payment pattern remains unchanged.
b. The total amount of the cash flows you receive remains the same, but the cash flows are paid out over a longer time period.
c. The discount rate decreases, but the cost of the investment (the payment up front at t=0) remains unchanged.
d. Answers a and b above are both correct.
e. None of the above are correct.

Problems:

1. 19. NYT Corporations is considering a project that will pay nothing for the first five years, $40,000 in the sixth year, $80,000 in the seventh year, $120,000 in the eighth year, $160,000 in the ninth year, and $200,000 in the tenth year.  The appropriate discount rate is 8.8% and the project requires an  investment tomorrow of $250,000 if we accept the project.  The NPV of this project is closest to:
     a.   $10,000
     b. $20,000
     c. $30,000
     d. $40,000
     e.  None of the above answers are within $5,000 of the actual NPV.

5.  Suppose that the present value of a 2-year ordinary annuity is $100.  If the discount rate is 10 percent, what must be the annual cash flow?
   a. $53.78   
   b. $57.62   
   c. $65.45   
   d. $82.64   
   e. $79.22

3.  Consider the cash flow series shown in the table below.  As you will note, cash flow 3 is missing.  Find the missing Year 3 cash flow that  will make the future value at the end of Year 10 of this cash flow series equal to $6,500 given a nominal discount rate of 12.4 percent.

                                                     Year          Cash Flow
                                                           0                $384.12
                                                          1                $416.42
                                                          2                $584.16
                                                          3                 ?????
                                                          4                $214.73
                                                          5                $962.22

   a.  $   186.98
   b.  $   542.59
   c.  $2,464.11
   d.  $1,218.08
   e.  None of the above answers is within $2.00 of the correct answer.

4. You have invested $3,000 in a CD paying a nominal interest rate of 12.4%, semiannual compounding. How much will you have in your account after 4 years?
a. $4,788.36
b. $4,682.18
c. $4,854.20
d. $4,600.00
e. $6,280.50

5. You have just won a lawsuit. The settlement will pay either a single lump sum settlement today, or the following cash flows (assume the following cash flows come at the end of the period):

Year 1: $10,000
Years 2-4: $ 5,000
Years 5 onward (forever): $ 3,000

You can presently earn 8% on your investments. What is the least you should accept today as a lump sum settlement:

a. $48,753.88
b. $21,190.26
c. $51,190.26
d. $46,712.13
e. $28,150.00

6. You are planning for your retirement. You know that you want to retire exactly 35 years from now, (i.e., end of year 35) and have calculated that you can retire comfortably if you have accumulated $3,010,000 at that time. You plan on making equal annual deposits, with the first one being made today, and the last deposit being made exactly one year before you retire. You have assumed that you can earn 18% annually  on your investments. What is the amount of your annual deposit?

a. $ 1,656.90
b. $ 1,404.14
c. $ 2,308.33
d. $86,000.00
e. $ 1,955.12
 

Answers:

Questions: 1. D, 2. C, yet another variation on "rate down implies price up," 3.  

Problems:

1. D, This is a generic uneven cash flow problem.  The only key is to make sure cash flows 1-5 are 0 (with a TI BAII this means FO1=5 then press Set/Enter).  Cash Flows are as follows, CF0=-250000, CF1-5=0, CF6=40000,CF7=80000,CF8=120000, CF9=160000,CF10=200000, NPV@8.8%=40503.

 2. B, PV=100,N=2,I=10,FV=0 compute PMT=-57.619048.

3.  A, The key to this problem is to get the correct sign on the present value. This problem has several other solution method with fewer steps but higher complexity.  Note this method only works with multiple choice tests. Step 1, Where Fv=6500,N=10, I=12.4, find the PV=2019.53.  Step 2. Enter the 4 possible cash flows and choose the one where NPV=2019.53.  Alternate Step 2, Enter the other known cash flows, asssuming Cash Flow 3 =0, and compute the NPV@ 12.4%=1887.86. Step 3 Take the difference between 2019.53 and 1887.86, and make this your PV=131.67, N=3, I=12.4,PMT=0 and compute FV=186.97.  To check  your work, enter the 186.97 as Cash Flow 3 and see if this equals your PV of 2019.53.

4. C, PV=3000, N=4*2=8, I=12.4/2=6.2,PMT=0, solve for FV.  Alternate method, convert nominal 12.4 with semiannual compounding to effective annual rate of 12.7844, then solve PV=3000,N=4,I=12.7844,PMT=0, solve for FV =4854.196.

5. A, here are two ways to solve. Numerical approximation method, CF0=0, CF1=10,000,FO1=1,CF2=5000,Nj2=FO2=3,CF3=3000,Nj3=FO3=some  big number, e.g. 99, then solve for NPV @8%=48740.35, which is very close to Answer A.  Exact solution method, step 1) find value of perpetuity at time 4 (note the value is at time 4 since the cash flows begin in time 5) = 3000/8%=$37500, Step2 CF1=10,000,FO1=1,CF2=5000,Nj2=FO2=2, CF3=5000+37500=42500, then solve for NPV @8%=48753.88 which is answer A.

6. B, this is an annuity due since first deposit in being made today, so in Begin mode,  PV=0,I=18, FV=3010000,N=35, and compute PMT=1404.1478