Solutions to Practice Problems – Chapter 3

 

1.         The payments from the trust fund will be:

                                                  30   30   30             30

          |       |       |       |     ….      |      |      |      ….     |

          0      1      2      3              15   16   17             24

 

PV at time 14 of payments = 30,000/1.08 + 30,000/1.082 + … + 30,000/1.0810

                                         = 201,302.44

PV at time 0 of payments = 201,302.44/(1.08)14 = 68,535.64

 

2.         The savings run from time 1 through 40.  The expenses start at time 40 and end at time 61.  Cashflows are as follows:

 

                                        60    60  99+30  30 .... 30    30

           S     S           S     S                

    |      |      |   .....    |        |       |        |        |  .....   |       |

    0     1     2          39     40    41      42    43       60    61

 

Solve for S by setting the PV of the expenses at time 40 = the FV of the savings at time 40.

 

PV40(Expenses) = 60,000 + 60,000/1.08 + 99,000/1.082 +

                             [30,000/1.08 + … + 30,000/1.0820]/1.08

                          = 60,000 + 55,555.56 + 84,876.54 + 294,544.42/1.08 = 473,158.42

[Note: you can compute the PV at time 40 of the 20 year annuity by first computing its PV at time 41 – one year before the first payment – and then discounting that back one year.]

                                                                       

FV40(Savings) = S + S*1.08 + S*1.082 + … + S*1.0839 = 473,158.42

 

            Solving for S will give $1,826.47

 

 

3.         When cashflows are spread out over many years, rate of return means IRR.  We have to find what discount rate makes the PV of the inflows equal to the PV of the outflows (at some common point in time).  Let’s use time 10:

 

We need to set FV10(annuity) = PV10(Perpetuity)

 

=>        100 [1 – 1/(1+R)10]*(1+R)10   = 100  

             R                                                 R   

=>        [1 – 1/(1+R)10]*(1+R)10   = 1

 

=>        (1+R)10 = 2   =>  R = 7.1774%

 

(initially sounded like a great investment, but the return’s not really all that high.)  

 

4.  At time 5, her salary = 50,000*1.054, and her savings are 8% of this salary, or 4,000*1.054.

 

If she saved 10% of her salary at time 5, her time 5 savings would be 5,000*1.054, and her savings would represent a 15-year growing annuity as follows:

 

                      5000        5000*1.05     5000*1.052    5000*1.053        . . .     5000*1.0514

      |----------------|----------------|----------------|----------------|--------------- . . . -------|

      0                  1                  2                   3                  4                                 15

 

Her actual savings stream is this 15 year growing annuity minus a time 5 cashflow of 1,000*1.054.

 

=>  FV15(Savings) =  PV0(Growing annuity) * (1+r)t  - 1000*1.054*(1+r)10   

                              =   C1  [1- (1+g)t/(1+r)t] *(1+r)t - 1000*1.054*(1+r)10 

                                 (r-g)

=   5000  (1.115 - 1.0515) - 3,152.71= 100,000 (4.1772 - 2.0789) - 3,152.71 =  206,679.29

  .10 - .05

 

 

5 a) If the annual repayment is x, then the PV of these repayments at 2.3% needs to equal 408,000,000.  In other words, x is the solution to the following equation:

            x/1.023 + x/1.0232 + … + x/1.0235 = 408,000,000

From a financial calculator, x = 87,315,735.82

b)  The true PV of these repayments Beej will actually make is:

87,315,735.82/1.046 + 87,315,735.82/1.0462 + … 87,315,735.82/1.0465 = 382,248,236.00

c) Beej gets $408,000,000 today.  In return, he pays back cashflows whose value is 382,248,236.00.  The value of the parting gift is therefore the difference, i.e. $25,751,764.

d) If Beej were charged the market interest rate of 4.6%, his repayments would be given by:

            x/1.046 + x/1.0462 + … + x/1.0465 = 408,000,000

From a financial calculator, x = 93,198,128.49

In effect, the gift he gets is the difference between these repayments and the actual repayments he’s asked to make => the gift is equal to a five year annuity of 93,198,128.49 - 87,315,735.82 = 5,882,392.67.  The vale of the gift is the PV of this annuity:

5,882,392.67/1.046 + 5,882,392.67/1.0462 + … 5,882,392.67/1.0465 = 25,751,763.99

 

 

6.         Since payments are quarterly, we need to discount using a 3-month effective rate.

Stated rate = 12% compounded quarterly => actual rate = 3% over 3 months.  So no conversion is required.

PV0 = 2000/1.03 + 2000/1.032 + . . . + 2000/1.0312 = 19,908.01

 

7.         Since payments are made every 4 months, we need to discount using a 4-month effective rate.

Stated rate = 15% compounded monthly =>   Actual rate = 1.25% over 1 month

4-month effective rate = (1.0125)4 - 1 = 5.0945%

PV0 = 1500/1.050945 + 1500/1.0509452 + . . . + 1500/1.05094512 = 13,224.28

 

8 a)      Since payments are made every month, we need to discount using a 1-month effective rate.

Stated rate = 10% compounded monthly =>   Actual rate = 0.8333% over 1 month => no conversion is required.

PV0 = 2500/1.008333 + 2500/1.0083332 + . . . + 2500/1.00833360 = 117,663.42

 

b)         Since payments are made every year, we need to discount using a 12-month effective rate.

Stated rate = 8% compounded semi-annually =>   Actual rate = 4% over 6 months

12-month effective rate = (1.04)2 - 1 = 8.16%

PV0 = 800/1.0816 + 800/1.08162 + . . . + 800/1.08165 = 3,180.74

 

9.         Since payments are made every month, we need to discount using a 1-month effective rate.

Stated rate = 9% compounded quarterly =>   Actual rate = 9/4 = 2.25% over 3 months

1-month effective rate = (1.0225)1/3 - 1 = 0.7444%

PV0 = 900/1.007444 + 900/1.0074442 + . . . + 900/1.00744460 = 43,423.69

 

10 a)      Since payments are made every quarter, we need to use a 3-month effective discount rate.

Stated rate = 9% compounded monthly =>   Actual rate = 0.75% over 1 month

3-month effective rate = (1.0075)3 - 1 = 2.2669%

FV24 = 250*1.02266923 + 250*1.02266922 + . . . + 250 = 7,858.17

 

b)         Stated rate = 9% compounded quarterly =>   Actual rate = 2.25% over 3 months

=> no conversion is required.

FV24 = 250*1.022523 + 250*1.022522 + . . . + 250 = 7,841.85

 

c)         Stated rate = 9% compounded semi-annually =>   Actual rate = 4.5% over 6 months

3-month effective rate = (1.045)0.5 - 1 = 2.2252%

FV24 = 250*1.02225223 + 250*1.02225222 + . . . + 250 = 7,818.04

 

d)         Stated rate = 9% compounded annually =>   Actual rate = 9% over 12 months

3-month effective rate = (1.09)0.25 - 1 = 2.1778%

FV24 = 250*1.02177823 + 250*1.02177822 + . . . + 250 = 7,772.69

   

11.         With Honest Eddie, you’ll pay $80 interest for 1.5 months.  In other words, your actual rate is 80/400 = 20% over 1.5 months.

With Vigo Delimit, your actual rate is  (481/425) - 1 = 13.18% over one month.

To compare the rates, we can express the second rate as an effective 1.5 month rate, which would be:

            (1.1318)1.5 - 1 = 20.4021%.  Thus, Honest Eddie has the lower rate.

 

(Or you can convert both rates to effective annual rates:

Honest Eddie:       (1 + .2)12/1.5 – 1 = 329.98%

Vigo Delimit:          (1 + .1318)12/1 – 1 = 341.65%)  

 

12.            Convert all three rates into an effective annual rate:

6.5% compounded quarterly => EAR = (1 + .065/4)4 - 1 =6.6602%

6.25% compounded monthly => EAR = (1 + .0625/12)12 - 1 = 6.4322%

actual rate of 0.121% a week => EAR = (1 + .00121)52 - 1 = 6.4901%

=> choose 6.5% compounded quarterly, since it represents the highest rate.  

 

13 a)      The cashflows you receive will be as follows:

            C1 = 100

            C2 = 100*1.04

            C3 = 100*1.042

            C4 = 100*1.043

            C5 = 100*1.044

            C6 = 100*1.044*1.06 = 124.01

            C7 = 100*1.044*1.062

            .

            .

            C10 = 100*1.044*1.065

These are the actual dollar cashflows you will get, i.e. nominal cashflows.  Discounting them at the nominal rate of 8%, treat them as the sum of two growing annuities

A = growing annuity starting at time 1 with a payment of $100 and growing at 4%.

PV0(A) = {C1/(r-g)} * [ 1  - (1+g)t/(1+r)t] = {100/(.08-.04)} * [ 1 – (1.04)5/(1.08)5]

       = 2500 * (1 – 0.8280) = 429.92

B = growing annuity starting at time 1 with a payment of $124.01 and growing at 6%.

PV5(B) = {C6/(r-g)} * [ 1  - (1+g)t/(1+r)t] = {124.01/(.08-.06)} * [ 1 – (1.06)5/(1.08)5]

       = 6200.25 * (1 – 0.9108) = 553.22

PV0(B) = 553.22/(1.08)5 = 376.52

 

PV at time 0 of all the cashflows = 429.92 + 376.52 = 806.43

 

b)         Since the nominal cashflows grow at the inflation rate, the real cashflows will just be constant.  The real cashflow corresponding to C1 = 100/1.04 = 96.15.  The real cashflow corresponding to C2 = (100*1.04)/1.042 = 96.15, etc.

The real discount rate for the first 5 years = 1.08/1.04 – 1 = 3.8462%

The real discount rate for the next 5 years = 1.08/1.06 – 1 = 1.8868%%

 

A = ordinary annuity starting at time 1 with a payment of $96.15

PV0(A) = 96.15/1.038462 + 96.15/1.0384622 + … + 96.15/1.0384625 = 429.92

B = ordinary annuity starting at time 6 with a payment of $96.15

PV5(B) = 96.15/1.018868 + 96.15/1.0188682 + … + 96.15/1.0188685 = 454.71

This is now a time 5 value.  To discount it back from time 5 to time 0, we have to use the real discount rate for the first 5 years, namely 3.8462%

PV0(B) = 454.71/(1.018868)5 = 376.52

 

As before, PV at time 0 of all the cashflows = 429.92 + 376.52 = 806.43  

 

 

14 a)    The cashflows represent a growing annuity.  There are 35 payments, starting with $28,000 at time 1 and growing at 4%.  The PV is given by:

    PV0 = [C1/(r-g)] * [ 1  - (1+g)t/(1+r)t]  = (28,000/.06) * [1 - 1.0435/1.135]  = 466,666.67 * .85958 = 401,138.25

    

b)    The monthly cashflows are not a growing annuity.  The first year, there are 12 payments of 28,000/12 = $2,333.33 each.   The second year there are 12 payments of 2,333.33*1.04 = $2426.67 each, etc.  There isn't a constant growth rate per month.

First use the effective monthly rate to compute the FV of each year's salary at the end of the year.  The resulting stream of future values will then be a growing annuity, and the PV can be computed as before.

The effective monthly rate = 1.11/12 - 1 = 0.7974%.  

The FV of the first year's payments at the end of the year = 2,333.33*1.00797411 + 2,333.33*1.00797410 + . . . 2,333.33 = 29,261.25.

The future value of the second year's payments at the end of the second year will be 4% more.  So we have a growing annuity of 35 payments, starting with $29,261.25 at time 1 and growing at 4%.  The PV will be:

    PV0 = [C1/(r-g)] * [ 1  - (1+g)t/(1+r)t]  = (29,261.25/.06) * [1 - 1.0435/1.135]  = 487,687.53 * .85958 = 419,207.41

(Of course, we can simply go (29,261.25/28,000)*401,138.25 = 419,207.41)

 

 

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