Math Forum / Mathematics / Undergraduate Math / September 2008

I think you meant 1000/(1.08^10). Perhaps you used superscript
formatting, which doesn't translate on usenet groups. Anyway, this is
the amount you would need to pay for the bond so that it has a value
of 1000 (at 8%) after 10 years.

This looks at the amount needed to have an additional withdrawal of
£80 each year. (to make the first year's payment you need 80/1.08
extra. For the second payment, you need 80/(1.08^2), and so on)

What are your x_1, ..., x_10 ?

This is the way I read the problem:
Actually, since there are payments each year, it isn't quite that
simple, since the payments will almost certainly come from the
interest earned, rather than an outside source. However, that also
makes the final result easy to find.
Thus, we have:
At the start, the value is X (the amount paid)
At the end of year 1, the value is 1.08 * X
At the START of year 2, the value is 1.08 * X - 80, since 80 was paid
out for the annual coupon. Call this amount X_2

At the end of year 2, the value is 1.08 * X_2.
At the start of year 3, the value is 1.08 * X_2 - 80. Call this X_3

and so on, so at the end of year 10, the value is 1.08 * X_10, where
X_10 is the value at the start of year 10. This amount should be
£1080, so the coupon and the base payment can be made. Working
backwards. we see that X_10 = £1000, and so X_9 = £1000, and so on.
The initial investment is £1000, and you get the principle back at the
end.

Using the formulas you gave above, you need 1000/(1.08^10) = 463.19 to
get the £1000 base value.
You also need  80/(1.08^10) + 80/(1.08^9) + ... + 80/(1.08) = 536.81
to get £80 available to pull out each
year. Adding these indicates you need to pay £1000 initially.

This makes sense, since pulling out £80 is pulling out all the
interest earned, so the principle never grows. If you only had, say, a
£40 coupon each year (so half the interest is reinvested, rather than
paid out), then you would only need to invest an extra £268.4, or a
total of £731.6 initially. Thus, you would get less cash on hand each
coupon, but you also need less investment money.

I don't understand a lot of financial computations either (and as
recent events prove I'm not alone), but:

If the bond pays £80 per year, isn't that the 8% interest.  Shouldn't
the current price be £1000?

If the current price were £463.19 and the bond paid nothing but was
redeemed in 10 years for £1000, wouldn't that represent 8% compounded
annually.

How can you do both and still be at 8%?

Isn't a bond's price "adjusted" to reflect the difference between the
bond's stated interest rate and the current market rate?  If the
current market rate were 5%, shouldn't this bond sell for £1600 so
that the £80/year would be 5%?  I would imagine that somewhere along
the way the remaining life of the bond is also factored into this.

use a finical calculator on line.

this problem is trivial.

In article
<29154225.1222513250826.JavaMail.jakarta@nitrogen.mathforum.org>, ttt
<myimmyim@hotmail.co.uk> wrote:

This is a very odd situation. Ordinarily, at an assumed 8% APR, you
would expect to pay £463.19 to receive £1000 in 10 years OR you would
pay £1000, collect £80 per year for 10 years and then get your £1000
back.

This problem seems to be a combination:

Assuming 8% APR, you pay £463.19 to get your £1000 in ten years.
In addition, evidently you pay 80/1.08 = £74.07 to receive £80 in one
year and 80/(1.08)^2 = £68.58 to receive £80 in two years etc Add all
those up to get £536.80. So, I guess you should pay
536.80 + 463.19 = £1000 just as Barry Schwarz said.