find coupon given price, YTM, face value, term

G

Gerard

New member
Joined
Jul 23, 2015
Messages
6
Hello, I'm doing a free online finance course at coursera.
The latest assignment has a question with the following:
price - 81,110
face value - 100,000
YTM - 11.5%
semi annual coupon for 8 years.
How much is the coupon?

The instructor showed how to calculate it with Excel but I'd like to know what the math involved is.
Thanks for any help,
Gerard
 
Gerard said:
Hello, I'm doing a free online finance course at coursera.
The latest assignment has a question with the following:
price - 81,110
face value - 100,000
YTM - 11.5%
semi annual coupon for 8 years.
How much is the coupon?

The instructor showed how to calculate it with Excel but I'd like to know what the math involved is.
Thanks for any help,
Gerard
Click to expand...

The price is what is called Present Value (PV). The face value and the semi annual coupons you get are what is called Future Values (FV). To turn a PV into a FV you accumulate the interest for the lenght of time wanted, that is
FV = PV (1+i)n
where i is the interest rate per period and n is the number of periods. You can turn this around also
PV = FV / (1+i)n

So, let's see what buying the bond will get you: It will get you interest every 6 months at the coupon rate for 8 years or you will get 16 FV payments p for an amount of 100000*(i/2). To turn each of the payments p into a PV at the current interest rate of 11.5% per annum we divide each p by 1.0575n where n is the time until you get you payment, i.e. n= 1, 2, 3, ... 16. The total sum of thaose payments are
PV1 = p (1/1.0575 + 1/1.05752 +1/1.05753 + ... + 1/1.057516) = p (1 - 1/1.057516)/0.575
which is a standard financial formula. Note also that at the end of 8 years you will get a lump sum payment of $100000 whose PV at the current interest rate of 11.5% per annum is
PV2 = 100000/1.057516
The total present value PV1 + PV2 = $81,100 the present price of the bond. Thus you have
$81,100 = 100000/1.057516 + p (1 - 1/1.057516)/0.0575

Solve for p = 100000*(i/2) and then for i.
 
Beer soaked opinion follow.
Denis said:
Isn't that all you need?
100000 * .115 / 2 = 5750 = coupon amount

The question is "How much is the coupon?"
Click to expand...
I be mistaken but methinks 11.5% is the YTM rate not the "coupon" rate.
Accordingly, we have
\(\displaystyle 81,110 = (100,000*\frac{r}{2})\frac{{1 - \left( {1+i} \right)^{ - 16} }}{{i}} + 100,000(1+i)^{-16}\)

where i = .115/2
 
Last edited:
Denis said:
Isn't that all you need?
100000 * .115 / 2 = 5750 = coupon amount

The question is "How much is the coupon?"
Click to expand...
Two different interest rates. In this case i is the (unknown) coupon rate and the payment p=100000*i/2 is what will actually be paid. However, because the current interest rate [actually the yield to maturity (YTM)] is 5.75% each 6 months, that is the interest rate at which the FV of p has to be brought back to a PV. If the YTM were to be the same as the coupon rate as
100000 * .115 / 2 = 5750 = coupon amount
implies, there would be no discount.
 
Denis said:
... This rate has not been given.
The OP is apparently completely unaware of what he's asking :idea:
Click to expand...

I thought I understood what the question is asking, here's the full text:
Suppose Wolverine Steel Company wishes to issue a $100,000 bond
with a maturity of 8 years to raise $81,110. The market requires a yield to maturity
(YTM) of 11.5% for this company's borrowing/debt. How much coupon will the company
have to pay every six months?
 
Last edited:
Ishuda said:

... The total present value PV1 + PV2 = $81,100 the present price of the bond. Thus you have
$81,100 = 100000/1.057516 + p (1 - 1/1.057516)/0.0575
Click to expand...
It looks to me like p here is the coupon's price, and I can see how all the parts fit together then.
Am I right?
 
I think that Ishuda's and Jonah's posts suggests a different answer. As I understand it, they suggest that the coupons are an annuity, the present value of which is: \(\displaystyle coupon / .0575 * \left[1 - 1/1.0575^{16}\right]\)
The present value of the $100,000 is: \(\displaystyle \$100,000/ 1.0575^{16}\)
Together they sum to the price: \(\displaystyle \$81,100 = coupon / .0575 * \left[1 - 1/1.0575^{16}\right] + \$100,000/ 1.0575^{16}\)
Now we can solve for the coupon: \(\displaystyle coupon = (.0575*(\$81,100 - \$100,000/1.0575^{16}))/\left(1-1/1.0575^{16}\right)\)
That apparently gives two solutions for the coupon: (7.878083491, -3966.302971891). The 3966 seems to make more sense even though it's negative, so maybe the correct answer is 3966.30.
Does that make sense?
 
Last edited:
Denis said:
The question is: how much is the coupon?

That has nothing to do with yields, purchase price et al...

A bond has a "coupon rate"....right?!

This rate has not been given.

The OP is apparently completely unaware of what he's asking :idea:
Click to expand...

You are correct in that the question is 'How much is the coupon?' which would be twice the p = 100000*i/2 above. The definitions I've seen say it is the total sum of the actual amount of money received each year from the bond.

However, in this particular case, you are not given the coupon rate [the i above] but must compute the coupon from the information given.
 
I found the silly mistake I made when I used SpeQ Mathematics V3.4 to solve: \(\displaystyle coupon = (0.0575*(81,100 - 100000/1.0575^{16}))/(1-1/1.0575^{16})\) It came up with: \(\displaystyle coupon = (7.878083491, -3966.302971891)\) because I had the comma in 81,100! I don't think I've got what it takes to get really good at math, c'est la vie, huh? Fortunately that doesn't prevent me from enjoying it :)
 
You must log in or register to reply here.
Top