Bond Valuation - YouTube

Hi and welcome to this finance lecture for Stockholm Business School.

My name is Anders and I will talk about investment decisions and more specifically bonds

and bond valuation or bond pricing.

First, however, as usual, we have some repetition to do.

We said that financial decisions is about giving up resources today

in the hope of obtaining something good in the future, typically money.

So in order to compare the money we pay today

to the money we get in the future, we introduced time value of money

where we was said that money today is better than money in the future.

Due to inflation, interest rates and some risk or uncertainty.

We provided you with some formulas for calculating future and present values.

I will not go through them now, but you can see previous lectures.

And we also introduced a present value of an annuity.

Where we said at an annuity is cash flow stream with equally big cash flows.

And that has exactly one period between every cash flow

typically years or half years or something.

And the present value of such annuity

is the cash flow divided by the interest rate times the factor here,

which is 1 - 1/1 + the interest rate raised to the power of N,

where N is the amount of cash flows.

And here is the notation as you will see in the formula sheet.

And now we'll talk about bonds.

A bond is a debt instrument in which one party is a corporate corporation,

municipality, or a government,

raise money from investors today in exchange for promised future payments,

which is called coupons.

This sounds pretty much like the investment decisions

as we talked about earlier

we as investors give up resources today,

we buy the bond in order to get future payments or something good in the future

In this case, the cash flows that the bond will give us or, in bond terms, the coupon payments.

The bond contract specifies the coupon rate and the face value,

which in turn determines the coupons or the payments we receive.

Coupons are the fixed principle

that investors receive periodically

until the maturity date of the bond.

When the bond matures

The investor usually also receives the face value.

This contract are offered to the market.

The investors that are willing to pay the most for the contract determines

the price of the bond or the market price.

You can see the market as some kind of auction.

All financial instruments are basically sold via the market auction

where you if you're willing to pay more than someone else,

you will get the instrument in this case, the bond.

So some terminology for bonds, the face value or sometimes called the principle, denoted FV,

is the nominal value usually received when the bond matures.

And it determines together with the coupon rate,

the coupons or interest payments that are received periodically through the life of the bond.

And the coupon rate determines, together with the face value, the coupon, or interest payment.

The coupon payment which I've talked about, is the interest payment received periodically.

So if you enter the bond contract, you will get paid some amount and

typically every half year or so semiannually

in this course, often annually, so once a year

But it can be something else as well.

And we calculate the coupon payment

As we multiply the coupon rate with the face value

And then we divide by the number of coupon payments per year.

So if it's semi-annually it's 2, if it's annually it's 1.

And I think that is the two cases you need to remember in this course,

but of course it's not hard if it's four times here,

it's just to divide by 4, same logic.

The term of the bond is the time remaining until the bond expires.

The matured date is the day or the date when the bond matures or expires,

and usually when the face value is paid to the investors.

The discount rate is the rate at which we discount the future payments.

Just as before, when we calculated net present values,

we use a discount rate to discount money till today and the yield maturity,

the yield maturity is the discount rate that sets the present value of the bond payments

equal to the market price.

So you can say that it's some kind of market discount rate.

It's the discount rate that the market have used to price the bond.

We also have something called Par, which is when the bond is traded at a price equal to the face value.

And this happens when the yield maturity is equal to the coupon rate.

I also have discount which is when the bond is traded at a price below the face value.

And this happens when the yield to maturity is greater than the coupon rate.

And it can be traded to premium,

which then is if the bond is traded at a price above the face value.

And this happens when the yield maturity is below the coupon rate.

We can describe a bond with a timeline

just as we do with the regular NPV calculations.

Where we as investors pay something today, we pay a price today

in order to receive the future payments,

coupons and the face value.

And it could look like this, for example.

So this would be a five year coupon bond

So we pay the price of the bond today

and we received a coupon payment every year for five years.

And the fifth year we also get paid the face value.

When we determine the value of the bond.

we discount the future cash flows we will receive

just as we have done with the NPV calculations.

So we move this back till today.

The simplest kind of bond is a zero coupon bond.

Which basically is a bond without coupon payments.

It only pays out face value.

The price of such a bond is then the present value of that payment.

So for example, if you have a zero coupon bond for three years.

You will be paid the face value in three years.

In order to price the bond,

you just calculate the present value of this cash flow.

And it looks like this.

It's just as in the present value.

Just that, this here, yield to maturity was called r before

in NPV calculations

And here the face value was called C.

So if you see it as C divided by 1 plus r raised to the power of N, capital N.

This is exactly what you did before.

But when we price bonds, we call the cash flow here,

that we will receive the face value if the zero coupon bond.

And the discount rate are called the yield to maturity.

Otherwise, it's exactly the same thing.

N is the term of the bond, so if it's four years until maturity,

hence we get the face value in four years time then N is 4

because we will move it four years.

A coupon bond, on the other hand, pays investor coupon payments

of some periodicity.

In this course often once a year, but in reality

often twice a year.

The price of such a bond

is the present value of the payments, again.

But here is the coupon payments and the face value

discounted with the yield to maturity.

And for some reason they call the yield to maturity

as a small y instead of YTM when it's a coupon bond.

I mean, it doesn't matter.

You can call it whatever you like.

The formula if you say so looks like this.

And perhaps this looks very hard.

It's hard to get, it's long and it's a bit complicated perhaps.

But you will see that it's not that bad.

The formula calculates the present value of the bond.

I think I've said that, but we can split it up into two parts.

We can first look at this part of the formula, the first part.

We take the coupon payments, and discount till today

using the present value of an annuity formula as we have done before.

So we take the cash flow in this case CPN or coupon payment.

We multiply with one divided by the discount rate

in this case yield to maturity and the factor 1 - 1/1 + the interest rate, or yield to maturity

raised to the power of N, which is the amount of cash flows or amount of coupon payments

this case it's 1,2,3,4,5, so N is 5.

And the second part, this one here.

Well, it's the present value of the face value.

And it's just as when you calculate the present value of a single cash flow.

The cash flow here is called face value.

And you discount it with the interest rate, or yield to maturity

and you raised this with N, which is the amount of years, so we want to move it.

So in this case you received the face value in year five,

so N is 5.

You move it five years.

And together, its like this.

Take one formula at a time if it seems hard.

But it's just combining two of the formulas we have used before.

So let's take an example.

What is the price of a four year zero coupon bond

with a face value of $1000 if the yield to maturity is 6%?

We can look at this as a cash flow stream, or a time line where we pay something today,

the price of the bond

and received something in the future, the face value.

In order to price it

we discount it till today just as we usually do.

And the formula for this is

as follows, you take the cash flow we will receive here called the face value.

We divide it by 1 plus the interest rate

that we want to discount this cash flow with,

which in this case is called the yield to maturity

and we raise this through power of N

which is the amount of years we want to move the cash flow

since it's in four years time, N is 4.

And if we calculate it, we get the price of 792.094.

So if we should take another example.

A bit more advanced at least.

Then we have the same example,

but we also get a coupon payment every year

and we know that the face value is $1000 and we also now know

that the coupon rate is 5%.

So we should still assume that the yield maturity is 6%.

So in order to get the coupon payment I showed you before

it's the coupon rate times the face value divided by the number of coupons payments per year.

The coupon rate was 5%.

The face value was 1000 and it's once a year, so it's 50.

so we get 50 every year and we get some face value in year 4.

So in order to calculate the price of the bond,

we discount the coupon payments.

And since the coupon payments are an annuity,

we can use the present value of annuity formula

where y here is yield to maturity

or where you usually see the discount rate

and N is the amount of cash flows.

This example, it's four cash flows.

At last we take the face value, which is a single cash flow,

and since it's a single cash flow

we just divide it by 1 plus the interest rate, or here the yield to maturity raised to the power of four in this case, since it's four years.

And you will get this price, 965.349.

And we can also now conclude that this bond is traded to a discount

Since the price is smaller than the face value.

And as we said before, it's because the discount rate is greater than the coupon rate.

So to summarise,

a bond is a debt instrument in which one party,

a corporation, municipality, or government raise money from investors today,

in exchange for promised future payments, which we call coupons and the face value.

And in a timeline it could be represented like this.

You get some coupon payments and you get a face value,

and for this you're willing to pay the present value of these coupons or cash flows.

And you also get a formula for this, basically you used two formulas.

You use the formula for a present value of an annuity

and use the formula for present value of a single cash flow, combined.

And that was all for today.

Thank you very much.