compute the price of a 90-day zero coupon bond with a face value of $100 if the market yield is 6 percent

Question

This is a question from a textbook:

compute the price of a 90-day zero coupon bond with a face value of $100 if the market yield is 6 percent

Unless indicated otherwise, assume that 1 year = 365 days, and that interest is compounded annually

and the textbook answer is = 100/(1+0.06*90/365)

I thought it should be 100/(1+0.06/365)^90.

What is wrong with my thought?

Answer 1

Your answer takes raises the daily interest rate to the 90th power. This compounds the daily interest daily for 90 days.

100/(1+0.06/365)^90
100/(1+0.000164)^90   // The daily interest rate is 0.000164
100/(1.000164)^90
100/1.014903
98.53

The textbook answer is simply 90 days of the daily interest rate with no compounding

100/(1+0.06/365*90)
100/(1+0.000164*90)  // The daily interest rate is 0.000164
100/(1+0.014795)
100/1.014795
98.54

In your answer you pay $98.53 for a $100 bond resulting in $1.47 profit. The textbooks answer you pay $98.54 for a $100 bond resulting in $1.46 profit. The profit on your bond is higher because your calculation compounds the interest each day for 90 days.

Taking the power of something means multiplying it by itself that number of times.

3^4 = 3*3*3*3 = 12

Your answer takes 1.000164 times itself 90 times, which results in 90 periods where the interest is applied, rather than one period where the interest is applied.

1.000164 * 1.000164 * 1.000164 * 1.000164 * 1.000164 * 1.000164 * 1.000164 * 1.000164 * 1.000164 * 1.000164 * 1.000164 * 1.000164 * 1.000164 * 1.000164 * 1.000164 * etc

Illustrated another way like this, considering $100 principle at 50% either applied at the end of the term or compounded each period, and we'll use monthly so the per period interest rate is 0.5/12 = 0.0416

              Simple Interest                   Compound Interest
Period    Principle   Interest Payment    Principle   Interest Payment

  1         $100           4.17             $100.00        4.17
  2         $100           4.17             $104.17        4.33
  3         $100           4.17             $108.50        4.51
  4         $100           4.17             $113.01        4.70
  5         etc.....

As you can see when compounding interest, the prior period's interest payment is included in the principle calculation accelerating the yield at a given rate.