COMAP Problem 1 Page 852

This problem asks you to compute the "annual yield" on a certain bond bought on a certain date at a certain price. I was unable to find a precise definition of the term "annual yield" in the text. (The term "annual percentage yield" is defined on page 799 and "effective annual rate" is defined on page 799 and again on pages 838 and 850, and the term "annual percentage rate"is defined on page 850. The former two are synonymous.) The answer in the back of the book is the percentage of the purchase price which would give the same annual payments (a very easy problem). This doesn't compute anything reasonable, since it disregards two things:

the first interest payment is six months after purchase, and
the face value of the bond ($10,000) is paid in 29.5 years.

What follows is a correct interpretation.

On August 13, 2001 you buy, for $9802, a $10,000 bond that pays simple interest of 5.51% in February each year through February 2031 and also pays $10,000 in February 2031. What interest rate will yield the same stream of payments?

In other words, we are to imagine starting 31 seperate bank accounts in August of 2001 with initial sizes

A₁, A₂, ..., A₃₀, B,

and each earning the unknown interest rate i. We assume that the amounts in these accounts sum to $9802, i.e.

A₁+ A₂+ ...+ A₃₀+ B=$9802.

We close the first account in February of 2002 and get $551,
We close the second account in February of 2003 and get $551,
....
We close the 30th account in February of 2031 and get $551, and also
We close the 31st account in February of 2031 and get $10,000.

Now the kth account accumulates interest for k-0.5 years and grows to exactly $551 so

$551 = (1+i)^k-0.5 A_k

and hence

A_k = (1+i)^-k+0.5 $551.

Similarly

$1000 = (1+i)^29.5 B

B = (1+i)^-29.5 $1000

Hence byt the equation A₁+ A₂+ ...+ A₃₀+ B=$9802. from above we get

(1+i)^-0.5 551+ (1+i)^-1.5 551+ ...+ (1+i)^-29.5 551+ (1+i)^-29.5 10000 =9802.

Multiply this equation by (1+i)^29.5. We get

(1+i)²⁹ 551+ (1+i)²⁸ 551+ ...+ (1+i)⁰ 551+ 10000 =(1+i)^29.5 9802.

Using the formula for summing a geometric series we can write the last equation in the form

551((1+i)³⁰-1)/i+10,000= (1+i)^29.59802.

It is easy to write a computer program to find i, but this goes beyond the scope of the course. (The method is to guess high, then guess low and then improve the guesses.) The built in RATE function in Excel does not evaluate the answer.