Math 53 – Math of Finance

Wednesday, January 24, 2007

Homework 2 --- Due Wednesday, January 31, 2007

 

These problems all assume constant, or almost constant, interest rates.  It might help to do them all in an Excel workbook and turn in a printed version.

 

 

Just the formulas

 

1.  If you deposit  $18  at  18%  interest, how much do you have after  18  years if…

            (a)  it is simple interest?

            (b)  it is compounded yearly?

            (c)  it is compounded monthly? 

            (d)  it is compounded continuously?

 

2.  An account starting with  $100  is worth  $300  after exactly  20  years.  What is the

            interest rate,  r,  assuming that it is continuously compounded?

 

3.  What “effective interest rate” (or “annual percentage rate”, or APR) is equivalent to

            a continuously compounded rate of  9% ?

 

 

Present value

 

4.  The remaining payments on an old government bond are

            --- four payments of  $1000,  at six month intervals, starting six months from now; and

            --- one payment of  $25,000,  exactly two years from now (at the same time as the

                        last $1000 payment).

            (a)  If the market risk-free interest rate is    (continuously compounded, and

                        constant for the next two years), compute the present value of the bond.

            (b)  From the information given, determine the

                                    original face value, and

                                    original coupon rate

                        of the bond.

 

5.  You have won the lottery, and you have a choice of two payment schedules:

 

                                    t = 1                 t = 2                 t = 3                 t = 4                 t = 5

            Plan A             $20000            $20000            $20000            $10000            zero

            Plan B             $15000            $15000            $15000            $15000            $15000

           

            For this problem, interest rates are compounded annually, so .

 

            (a)  If  R = 0.01,  which plan has the higher present value?

            (b)  For what value(s) (if any) of  R  do the two plans have equal present values?

            (c)  For which value(s) (if any) of  R  does plan A have the higher expected value?

 

6.  In the previous problem, suppose all of the payments are advanced by one year, so that the

            payment schedules become these:

 

                                    t = 0                 t = 1                 t = 2                 t = 3                 t = 4

            Plan A             $20000            $20000            $20000            $10000            zero

            Plan B             $15000            $15000            $15000            $15000            $15000

           

            Do the answers to questions (a), (b), and (c) change?  Find a general principle if you can.

 

 

The Rule of 72.

 

7.  People like to say that if you invest money at constant r% interest, then it doubles in N years

            where  rN = 72.  (Or equivalently,  N = 72/r.)  For example, if you invest at 6%, your money doubles in  72 / 6 = 12 years.  This is obviously an approximation, but it’s a pretty good one.

 

            (a)  Do this part in your head in 2 minutes or less:  If you invest $100 at a constant 6%

                        interest rate, approximately how much do you have after 24 years?

 

            (b)  Now calculate it more accurately, assuming that the interest is compounded

                        continuously.  Are you happy with your estimate in part (a) ?

 

 

OK,  let’s try one variable-rate problem.

 

8.  Suppose that everyone knows (with certainty) that the interest rate is going to remain at 4% from now till time  t = 2,  and then jump suddenly to 6% and stay there forever.  (Rates

            in this problem are continuously compounded.)

            (a)  If you lend $100 at time  t = 0,  how much will you be owed at time  t = 3 ?

            (b)  What would you pay at time  t = 0  for the right to receive $1 at time  t = 3 ?

                        (That is, what is P(3) ? )

            (c)  Write a formula for  P(t)  that is valid when  t ≥ 2.

 

 

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