Solutions to Problem Set #1

 

 

 1.         The savings run from time 1 through 30.  He will retire at time 30.  The first 3 payments of $120,000 will fall at times 30 through 32 (beginning of the first three years of retirement).  The $200,000 to buy the house is paid at time 33; the 37 payments of $65,000 also start at time 33 (and therefore run through time 69).  The timeline is as follows:

 

                                       120   120    120  200+65  65 ....   65    65

           S     S           S       S                

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

    0     1     2          29     30      31      32      33       34        68    69

 

Solve for S by setting the PV of the expenses at time 30 = the FV of the savings at time 30.

 

PV30(Expenses) = 120,000 + 120,000/1.06 + 120,000/1.062 + 200,000/1.063 +

                             [65,000/1.06 + … + 65,000/1.0637]/1.062

                          = 120,000 + 113,207.55 + 106,799.57 + 167,923.86 + 957,890.72/1.062 = 1,360,450.31

[Note: the PV at the 65,000 annuity is being computed in two steps; the formula gives PV one year before the first payment, i.e. time 32; that is then discounted back two years to time 30.]

                                                                       

FV30(Savings) = S + S*1.06 + S*1.062 + … + S*1.0629 = 1,360,450.31

 

            Solving for S will give $17,208.22  

 

 

2.         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 12:

 

We need to set FV12(annuity) = PV12(Perpetuity)

 

=>        500 [1 – 1/(1+R)12]*(1+R)12   = 500  

             R                                                 R   

=>        [1 – 1/(1+R)12]*(1+R)12   = 1

 

=>        (1+R)12 = 2   =>  R = 5.9463%

 

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

 

 

3a) Stated Rate = 6.16% p.a. compounded annually.

      Actual rate = 6.16 over 12 months

      Effective annual rate = 6.16%

 

b)   Stated Rate = 6.08 p.a. compounded semi-annually.

      Actual rate = 6.08/2 = 3.04% over 6 months

      Effective annual rate = 1.03042 - 1 = 6.1724%

     

c)   Stated Rate = 6.02% p.a. compounded quarterly.

      Actual rate =  6.02/4 = 1.505% over 3 months

      Effective annual rate = 1.015054 - 1 = 6.1573%

     

d)   Stated Rate = 6.01% p.a. compounded monthly.

      Actual rate = 6.01/12 = 0.5008% over 1 month

      Effective annual rate = 1.00500812 - 1 = 6.1783%

     

e)   Compounded every 5 days means 365/5 = 73 times a year

      Stated Rate = 6% p.a. compounded 73 times a year.

      Actual rate = 6/73 = 0.0822% over 5 days

      Effective annual rate = 1.00082273 - 1 = 6.1810%

     

f)    Stated Rate = 5.99% p.a. compounded daily.

      Actual rate = 5.99/365 = 0.0164% over 1 day

      Effective annual rate = 1.000164365 - 1 = 6.1725%

 

The lowest rate is offered by plan c: 6.02% p.a. compounded quarterly.

 

 

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

Stated rate = 5% compounded monthly =>   Actual rate = 5/12 = 0.4167% over 1 month

12-month effective rate = (1.004167)12 - 1 = 5.1162%

PV0 = 4000/1.051162 + 4000/1.0511622 + . . . + 4000/1.0511624 = 14,145.60

 

5)        First, reverse the process followed in question 3, to compute the stated rate from the EAR:

Effective annual rate with semi-annual compounding = 9.25%.

EAR = (1+ Actual rate)2 - 1 => Actual rate = (1 + .0925)0.5 - 1 = 4.5227%

Stated Rate (APR) = .045227*2 = 9.0454%

 

Actual rate with quarterly compounding = 9.0454/4 = 2.2614%

EAR with quarterly compounding = 1.002261412 - 1 = 9.3569%

 

 

6 a) Since the nominal cashflows grow at the inflation rate, the real cashflows will be constant.

For the time 1 cashflow we have real CF1 = (Nominal CF1)/(1+i) = 12,000/1.02 = $11,764.71

Thus, all the real cashflows must be $11,764.71.

We can confirm this for time 2: real CF2 = (Nominal CF2)/(1+i)2 = (12,000*1.02)/(1.02)2  = 12,000/1.02 = $11,764.71

 

b) The real discount rate = 1.075/1.025 – 1 = 4.8780%

Discounting real cashflows at the real rate, we get 5,000/1.04878 + 10,000/1.048782 + 15,000/1.048783 - 20,000 = $6,861.66

 

The nominal cashflows will be:

Time 1:   5,000 * 1.02   =  5,125

Time 2: 10,000 * 1.022 = 10,506.25

Time 3: 15,000 * 1.023 = 16,153.36

Discounting nominal cashflows at the nominal rate, we get 5,125/1.075 + 10,506.25/1.0752 + 16,153.36/1.0753 - 20,000 = $6,861.66

 

   

7 a) P0 = D1/(r – g)   =>   32.80 = 0.45/(.10 - g)

       g =  .10 - 0.45/32.80 = .10 - .01372 = 8.6280%

b) plowback ratio * ROE = g = .08628

    plowback ratio = 1 - payout ratio = .6

    => ROE  = .08628/.6 = 14.3801%

 

8 a)   D1 = 0

         D2 = 3

         D3 = 4

         D4 = 5

         D5 = 5*1.07 = 5.35

         D6 = 5*1.072

         etc.

 

Treat D5 onwards as a growing perpetuity     =>    P4 =  D5/(r - g)  = 5.35/(.11 - .07) = 133.75

=>   P0  =      D2    +    D3    +     D4 + P4

                   1.112      1.113          1.114 

            = 3/1.112 + 4/1.113 +(5+133.75)/1.114 = 96.76

 b)  To come up with the expected stock price at time 1, we can compute the PV at time 1 of D2 onwards:

=>   P1  =      D2    +    D3    +     D4 + P4

                  1.11        1.112          1.113 

            = 3/1.11 + 4/1.112 +(5+133.75)/1.113 = 107.40

        The smarter way to do this is to recognize that E(R) = div yield + price appreciation

        For the first year, there is no dividend, so the dividend yield is 0.  This means price appreciation over the first year = E(R) = 11%.

        => P1  =  96.76 * 1.11 = 107.40

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