## The general idea

As mentioned in a previous article market value is the present value of the expected cash-flows. Normally a bond's cash-flow consists of periodically received interest and at maturity a one time payment of the face value. The present value can be written as:

This can shortly be written as:

P0: This is used instead of PV to show that this is the market value
Ct: The cash-flow of period t
r: The yield expected by investors
n: Number of periods left to maturity
Pn: The face value of the bond

As a reminder, please pay attention to the terms market value and current price as they can be confusing. Market value is not the price you will buy on the market. It is a theoretical amount, which tells you what the bond is worth. It can be used to make decisions about the bond. If theoretical price is higher than the current price (which is the price which you can actually buy the bond on the market) the bond is worth it, if it is lower, it is not worth it.

## Bonds with a fixed coupon rate, face value paid in full at maturity

Essentially the periodically received interest is an annuity. So what we need to do is calculate the present value of the annuity and the present value of the face value. Thus, we do the following:

P0=

The left side of the addition would be the present value of the annuity, while the right is the present value of the face value. Let's dive straight into an example!

Example:
A company issued a bond in 2000. The bond matures in 2010, the face value is \$1000, the coupon rate is 10%. Interest is paid yearly, the face value will be paid back in one sum at maturity. What would be a realistic price to buy this bond in 2004 if the expected yield is 8%?
According to the example:

(always the years left, not the original life-span)

Form that we use the formula:

=

This means that the theoretical price is 1092,3. This is what the bond is actually worth, so if the current price is somewhere around this it is a realistic price to buy at.

Example:
Again lets modify our example. What happens if we take the original example and raise the expected yield to 12%.

= =918,1

Example:
What happens if we take our last example and say that interest is paid every half year?
In six years we would get interest twelve times, so we have twelve periods left. We would get the half of the yearly rate, because of the half-year payments. We would also need half the expected yearly yield. So we would get the following:

==1094,25

You can see that more periods cause prices to rise. This is due to the fact that the risk of the bond is less if it pays more often. It is a more predictable income.

## Explanations

You may ask, why do I have to pay \$92,3 more for the bond that the face value in the first example. The reason is really very simple. The market yield has decreased and is currently 8%. That means that for \$10000 worth of investment we would get an 8% yield in the market today. If we buy this bond however, we are promised (coupon rate) a 10% interest rate. This \$92,3 positive difference is essentially "the cost" of a better investment opportunity,
Using the same logic for example 2 we can say that the price is lower than the face value because the current yield is 12%. If we buy the bond we will only receive a 10% interest.  The \$81,9 negative difference is our price difference gain
In the last example the price is higher because that bond is more secure than ones with 1 year periods because the risk is lower.

## Bonds with a varying coupon rate, face value paid in full at maturity

The good news with these bonds are that you already know how to calculate market value, the bad news is that you have to use the long way.
Remember the equation in the first paragraph? For a fixed coupon rate we were able to use annuities as a short cut, but here we have to use the first equation because coupon rates will differ.

Example:
A company has issued a bond two years ago with a maturity of five years and at a face value of \$1000. because they expect the reduction of inflation they will pay decreasing coupon rates each year as follows: 13%, 12%, 11%, 10%, 9%. The face value will be repaid in one amount at maturity. What is the market value if the yield for similar bonds is 12%?
You will have to calculate the last three interest's present values separately and then add the PV of the face value:

## Perpetual bonds

These bonds will not pay back a face value, but will pay interest continuously. The formula for this has a long mathematical derivation therefore I will only show the formula and not comment on it, it is very easy to use.

Market value and current price should be pretty clear by now. I will not discuss other types (for example bonds where face value is paid back in equal installments). The reason for this is that you have to use the first equation for almost all of these as there is no other way around it. Once you familiarize yourself with the problems in this article you should be able to cope with other bond types as well. Just remember to calculate present value for every cash flow and you can not go wrong!

Information is for educational and informational purposes only and is not be interpreted as financial or legal advice. This does not represent a recommendation to buy, sell, or hold any security. Please consult your financial advisor.