Find Bond YTM - annual vs semiannual coupons - YouTube

In this video I would like to explain how we would solve for the yield to

maturity (YTM) for a bond. So, bonds can be corporate

or government bonds. And when, for example, a company sells a bond, it borrows that

amount of money that is paid for the bond from whoever bought that bond from

the company. So, let's say if a company sells a bond for $900 then whoever

bought it lent $900 to the company. And in return - like with any loan - the company

that borrowed this much will need to be making payments back over the next, let's

say, 10 years, and the payments include interest payments and one large lump sum

payment at the end, which in our case is $1,000. So, "yield to maturity" means what?

What is "yield to maturity"? It is the same thing as... this is the bond terminology,

but... it is the same thing as the interest rate, or what we also often call the

discount rate. And in the financial calculator it's the same thing as "I/Y" per

year. So, the annual interest rate (the "I/Y" in the financial calculator) on a bond, the

financial security that represents debt. Okay. So, we need to solve for the

interest rate. Next. It looks like we are given an interest rate here, 8%, but

that's not what we want. 8% coupon rate... a coupon rate in general

is something else. Its only purpose is to figure out how

much money the company will be paying back to the buyer of its bond every year,

like the interest payment. So, the coupon rate of 8% means that... so

we take 8%, we multiply that by $1,000, and we get $80 per year. So, this is the

annual coupon amount that the company will be paying back to the buyer of the

bond. Another thing that we can realize - if you know enough about bonds - is that

the price of the bond being less than $1,000 (right? it's $900), the price being

less than $1,000 means that this is, first, what we call a "discount bond". And

that's always gonna be the case when the coupon rate is smaller than the

yield to maturity. So, coupon rate is less than the yield to maturity. And so you

can right away say that our yield to maturity should be higher than 8%. Okay.

So how would we find it? Let's calculate two different ways. It depends on what we

know about the bond. If the coupons are paid annually then on our timeline this

means that 1... year 1... year 2... year 10, the maturity date, it means that the

company gets $900 for the bond that it sold, and then at the end of year 1 it

pays an $80 coupon. Then it pays another $80 coupon at the

end of year 2, and it keeps doing that until the maturity. And at that time it

will also pay the $1,000 face value. So it pays both. Now, with the semi-annual

coupon payments it's a little bit different! What we have here is... so once

again we are saying that the company receives $900 for the bond that

it sold, bond that it sold. But the coupons are paid semi-annually which

tells us that the $80 annual coupon payment is split into two halves: one

half is paid at the end of the first half a year, the other half is paid at

the end of the second half a year. And so on. So, 40 and 40. So we were... we are gonna

be paying back to the investor who bought the bond $40 at the end of every

half a year. And then at the end of year 10 we will pay the last $40 with the

$1,000 face value, as the last large payment. Okay? And now to find the yield

to maturity, YTM, we need to enter four things in our calculator: future value

payment, the number of payments, and the present value ("PV"). And we need to compute

"I/Y". So, for the case with the annual coupon payments the "Future Value" is

$1,000 which is always the case, and we should enter it with a negative sign in

the calculator. The "Payment" is 80, that's the coupon amount. Same sign. It's

important that everything that's in the future is entered with the same

sign: both the Face Value and the coupons. How many coupons do we have? What is our

"N"? One per year over the next 10 years, so a total of 10. And "Present Value" is nothing

but the bond price which is given: $900. We don't change the sign

to negative. We keep it as is. And you can think of it as 900 dollars

being the money that the company gets, like a cash inflow, and both $1000 face

value in the $80 coupon payment stream being the amounts, the amounts of

money that the company pays, the cash outflows. So, the signs are different for

what's today and what's in the future. Now, for the semi-annual coupons,

"Future Value" is still 1,000 dollars, again make it negative. The "Payment" is

now $40, also negative. How many payments do we have? 10 years, twice per year, so

that's 20. Right? So, 10 years times 2 per year. Then, the "Present Value" is again

$900. The sign is the opposite from the negative sign for both the "Future Value"

and the "Payment" amount. And we are computing "I/Y" for both cases! Okay. Let's

bring up the financial calculator! Let's clear everything that we had stored.

Let's increase the decimal places: "2nd", "format", let's make it 6, "enter".

First, annual coupon payments: "1000", negative, "FV", "80", negative, "PMT", "10", "N", "900", "PV",

"Compute", "I/Y". 9.599563 percent per year.

This is the yield to maturity. Okay? So, the yield to maturity is, "I/Y" equals,

9.5996. This is the answer for the first part of the

problem. So, this is percent per year. Now, let's similarly calculate the yield

to maturity on the semi-annual coupon bond. Let's clear everything, increase the

decimals. "1000", negative, "Future Value", "40", negative, payment "PMT", 20

payments, so I enter "20", "N". "900" is my "Present Value", or the price at which the

bonds are sold. "PV". And I'm computing "I/Y". 4.788070.

Okay. Let's write it down. So, "I/Y" equals 4.78... let's

round it a little... 81. This is percent. And I'm gonna change the color

to emphasize it. Because everything is done on the semiannual frequency this is

percent per half a year! Not per year! Which is what "yield to maturity" really means.

And so to fix that we need to take the interest rate per half a year multiplied

by two, which will then give us the answer.

And 2 times 4.7881 gives

9.5761. This is the answer! And this is percent per year.