How much should you pay for the bond? and Bond Street station

How much should you pay for the bond?

A municipal bond pays 6.9% interest compounded semiannually on its face value of $ 5,300. The interest is paid at the end of every half-year period. Fifteen years from now the face value of $ 5,300 will be returned. The current market interest rate for municipal bonds is 6.3% compounded semiannually. How much should you pay for the bond?


Bond best answer:

Answer by Robert B
One should pay $ 5,605.69 for this bond.

The semi-annual interest payment is $ 182.85.
We are interested in the 6.9% only for purposes of calculating that payment.

Thereafter, we want to know two things:
(1) the present value of the $ 5,300, paid 15 years hence, discounted at 6.3%, compounded semi-annually; and
(2) the PV of a series of payments of $ 182.85, discounted at the same 6.3%.

The formula for (1) above is:
PV = 5300 / 1.0315^30 = 2090.25.

The formula for (2) above is:
PV = Ipymt / [i * ((1 + i)^n) / (((1 + i)^n) – 1)]
= 182.85 / [ .0315 * 2.535577967 / 1.535577967 ] = 3515.43,
where Ipymt = the amount of semi-annual interest payment,
i = the semi-annual rate of interest, at the relevant rate, and
n = the number of semi-annual compounding periods.

The sum of the two is 5605.69,
and that is your answer.

Please note that I am assuming, for these calculations, that exactly fifteen years remain on the bond and that the bond issuer has only just paid the most recent semi-annual interest due. Otherwise, it would be equitable to add to the purchase price some portion of the $ 182.85, to reflect interest accrued but unpaid, for the portion of the half-year since elspsed.

* * *

You might like to see where formula (2) came from. It derives from the familiar formula for the sum of a geometric series:

S = 1 + x + x^2 + x^3 + … + x^(n – 1) = (1 – x^n) / (1 – x).
{if you multiply the terms of S by 1 – x, the intermediate terms will cancel nicely, you will see that you get 1 – x^n.}

If you let x = 1 + i, then, (1 + i )^n is the amount of accumulated principal and interest on a single deposit of one unit (dollar, pound, or euro) left at interest for n compounding periods. 'i' is the rate of interest for that compounding period. 'S', then, is the amount accumulated on n periodic deposits of one unit each, made at the end of each of n compounding periods. The present value of a series of such payments is S / (1 + i )^n.

Since we are calculating the present value of a series of payments, not of one dollar, but of 'Ipymt', the periodic interest payment, we have the formula we use above,
PV = Ipymt / [i * ((1 + i)^n) / (((1 + i)^n) – 1)].

* * *

I hope this helps you out.

* * *

ED NOTE: Other answerer, please check the following facts, which any banker or bond broker can verify for you:
(1)The value of a bond always goes up, when the market interest rate comes down, and vice-versa; and
(2)The present value of anticipated periodic interest payments must go into the reckoning, along with the present value of the principal sum which is paid at maturity; and
(3)The present market rate of interest is what must be used in calculating present values, not the original rate of interest at the time of issue; and
(4)Nominal interest rates are quoted per annum, even though fractional parts of them may be compounded with greater frequency. Thus, 6%, compounded semi-annually, actually means 3% credited every six months, either paid then (as in the case of many bonds, including this one) or, else, accumulated and compounded. On this basis, then, $ 100.00, left to accumulate with interest, nominally 6% but compounded semi-annually, would become $ 103.00, at the end of six months, and $ 106.09, at the end of twelve months, and so on. This means that 6% compounded semi-annually is actually worth the same as 6.09% compounded only annually. That's how they actually do it!

It hoped that you and the asker, both, will find it of help to you, to read the full explanation of the present value of a series of periodic payments, given with some care, above.

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