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

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?

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.

• Just to confirm, were you talking about simple (textbook) vs compound (mine) interest rate ?? If so, the textbook does have a statement at the beginning of the question section "Unless indicated otherwise, assume that 1 year = 365 days, and that interest is compounded annually". So does this mean textbook answer is incorrect ?? – B Chen Apr 27 '17 at 2:33
• @BChen by your answer, you are saying that the \$100 is compounded DAILY for 90days. Since the textbook has stated that interest is compounded annually, 0.06*90/365 is correct. – MH.Q Apr 27 '17 at 5:42
• @BChen, the textbook answer is correct because it simply multiplies the daily interest rate (0.06/365) by 90 days. Your answer takes the 90th power of the daily interest which would compound daily for 90 days. This is why your answer returns a slightly lower number which generates a greater yield. Your answer generates a bond price of \$98.53, versus the textbook result of \$98.54. One cent less cost means one cent more yield which occurs because your formula compounds daily. – quid Apr 27 '17 at 6:19
• Thanks guys. Last clarification, hopefully. When would we use the following equation: 100/(1+0.06/1)^(90/365) = 98.57 - is it "6% interest rate, annually compounded, for 90 days ????" – B Chen Apr 27 '17 at 12:29
• @BChen, I'm not sure where you got that last formula from... You're raising the full interest rate ( (1+0.06/1) = 1.06) to the power of .24657 which is the ratio of the portion of the year represented by 90 days. I've never seen the formula laid out this way. The idea is you get the daily, monthly, annual, etc. interest rate, either based off of 360 or 365 days in a year, then multiply it by the number of days or raise it to the power of the number of days. – quid Apr 27 '17 at 18:03