Problem 25 setting. A zero coupon note payable in 8 years is offered by
Brown Brothers Box Company. The offering price of the note was
set by the company to provide a rate of return equal to 8 percent
per year compounded annually.

Problem 25 questions.

(a) Find the offering price of the note as a percentage of redemption value.

  Solution:
  
    The growth factor over the 8 years is the annual growth factor
    raised to the 8th power.

    Let A = redemption value.
    Let P = offer price.

    A = P*(1.08^8).

    So P/A = 1/1.08^8 = .540269 = 54%.

(b) After 1 year investors would be willing to buy notes with
    a 7-year maturity which provided a rate of return equal to 7
    percent per year compounded annually.  If Audrey bought $10,000
    of the Brown Brothers notes at the original price and sold them
    1 year later, find her profit.

  (Corrected) solution:

    The posted solutions mistakenly take $10,000 to be the offer
    price. But a $10,000 note is a note with a redemption value
    of $10,000.  So

      A = 10,000
      P = A/1.08^8 = $5,402.69

    Let Q = amount the investors pay.

      Q = A/1.07^7 = $6227.50

    So the profit is

      Q - P = A*(1/1.07^7 - 1/1.08^8)
            = $10,000*(
            = $6227.50 - $5,402.69
            = $824.81.

(c) Find the rate of return to Audrey.

  Solution:

    The rate of return is the profit divided by the purchase price:

      (Q-P)/P = 824.81/5402.69 = .15266 = 15.266%.

Problem 26. In the situation described in exercise 25, suppose
that after 1 year interest rates have increased, and investors
buying zero coupon notes with a 7-year maturity ask for return
equivalent to 8.5 percent per year compounded annually. If Audrey
bought $10,000 of the Brown Brothers notes at the original price
and sold them 1 year later, does she have a profit or a loss?
How much?

  Solution:

    Everything is the same as in problem 25, except that we should
    replace 1.07 with 1.085.

      Q-P = $10,000*(1/1.085^7 - 1/1.08^8)
          = $246.57;

    Since this number is positive, it represents a profit
    rather than a loss.