**Table of Contents**

Chapter 1: Categories of Bonds

Chapter 2: Pricing of Bonds

Chapter 3: Calculating Yield and Understanding Yield Curve

Chapter 4: Duration of Bonds

Chapter 5: Relationship Between Price, Yield and Duration

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Chapter 2: Pricing of Bonds

– Why Do Bond Prices Change

– Premium and Discount

– Understanding Present Value of Future Payments

– Calculating Bond Prices

– Pricing a Plain Vanilla Bond

– Pricing a Zero Coupon Bond

– Dirty and Clean Bond Prices

– Calculating Accrued Interest

When bonds are issued, they are usually sold at their par value, which is also referred to as their face value. For most corporate bond issues, this par value is $1,000, while some of the government bonds can have a par value of $10,000. This is the principal amount of a bond and it is returned to the investors when the bond matures. However, during the term of a bond, market forces make the value of the bond change. At any time, the bond could be selling at a value higher than its par, lower than its par or at its par value.

The main reason behind this change in bond value is change in interest rates. Interest rates in the economy are dynamic and they are constantly adjusted by the Federal Reserve in response to changing economic situation. When the economy is not doing well, the Fed can lower interest rates to encourage lending and to give a boost to economic activity.

But when there are serious inflationary expectations in the economy, the Fed can lower interest rates to cool things down. Such decisions can have a significant impact on the bond market, and prices of bonds always respond to changes in interest rates.

- Inverse Relationship with Interest Rates: Bond prices have an inverse relationship with interest rates. When interest rates in the economy go up (all other things being equal), bond prices go down, and vice versa. It is easy to understand why this happens.
Let’s say you have invested in a plain vanilla bond at a par value of $1,000 and a coupon rate of 5%. When interest rates in the economy go up, future bond issues will have to pay a higher coupon rate, let’s say 6%. In such a scenario, an investor will be willing to buy your bond from you only if you sell it at a value lower than its par such that the buyer is compensated for the lower interest payments.

The opposite of this happens when interest rates go down. Now future bonds will be issued at a lower interest rate and buyers will be willing to pay you more as your bond offers higher interest earnings. This will increase the price of the bond in the market.

- Impact of Creditworthiness: Another reason that can have a huge effect on bond prices is a change in the creditworthiness of the issuer. For example, if a company is facing financial difficulties that can adversely impact its ability to repay its obligations, credit rating agencies can decide to lower its credit rating.
When that happens, markets will react by lowering the prices of bonds issued by the company as there is now a much greater risk of default associated with those bonds. The same thing can happen to countries and it is not uncommon to see the prices of bonds issued by a national government change drastically in response to bad economic data.

It should be noted that the bond market does not always wait for a credit rating agency to lower the rating of the issuer before lowering the price of its bonds. Large market participants are well aware of the risks that an issuer faces and expectations of default are always factored in bond prices.

When a bond is selling at a value higher than its par, it is said to be selling at a premium. On the other hand, when the price of a bond falls below its par, it is said to be selling at a discount. When listing bond prices, the prices are mentioned in terms of percentage of premium or discount, irrespective of what the par value of the bond is. When a bond is selling at par value, it’s price is listed as 100.

When it is selling at a 10% discount, its price is listed at 90. Similarly, let’s say when it’s selling at a 5% premium, its price is listed as 105. This kind of quoting convention allows bonds to be compared directly and easily. Finding the true price of the bond is easy. The price of a bond listed as 105 and having a par value of $1,000 can be calculated as 105% of $1,000, which comes to $1050.

**Understanding Present Value of Future Payments**

Bonds assure a stream of future payments to the investor. For a plain vanilla bond, you receive regular interest payments from the bond issuer until the bond matures, and at maturity, you receive the par value of the bond. Therefore, the price that you should be willing to pay for a bond is the value that you can attach to these future payments.

However, valuing future payments is not very simple. This is where you have to understand one of the most fundamental concepts of finance – time value of money. Think about two straightforward scenarios, one where you get $1,000 immediately and one where you get $1,000 one year later. Any smart investor will prefer the first scenario. Why? Because if you receive the money today, you can invest it somewhere and earn interest on it. If you could get 5% interest on your investment, one year from now, you would have $1,050 in the first scenario, while in the second scenario, you would just have $1,000.

In other words, any future payments are not worth the same amount that they would be if the payment was made today. To determine the ‘present value’ of a future payment, you would have to discount it. The farther you go in future, the more this discounting would be. The present value of a $1,000 payment one year from now should be the amount that if you had today and invested in the market would yield $1,000 one year later. This amount would always be lower than the future payment.

In the previous example, it is clear that the rate of discounting a future payment should be the interest that you can realistically earn on your investment. This interest is known as the required rate of interest or the required yield. If you had $1,000 today, it would be worth $1,050 one year from now if the required rate of interest is 5%. This is equivalent to saying that the present value of a payment of $1,050 one year from now is $1,000. The present value of any payment can be calculated using this formula:

PV = F/ (1+r)^n

Here:

PV is present value

F is future payment

r is required rate of return

n is the number of periods after which the payment would be made

Applying this formula in the previous example, we can easily calculate the present value of a $1,000 payment one year from:

PV = $1,000 / (1+.05) ^ 1 = $953.38

The price of a bond is equal to the present value of all its future interest payments and the repayment of par value at maturity. We can use the formula for present value of future payments to determine the value of a bond. But keep in mind that as coupon payments come at different points in time, the discounting factor for each of them will be different, with payments coming later having a heavier discount. The price of a bond can be represented as the following formula:

Price = [I / (1+r)] + [I / (1+r)^2] + … + [I / (1+r)^n] + [Par Value / (1+r)^n]

Here:

I is the interest or coupon payment paid at the end of every period

r is the required rate of return

n is the number of periods after which the bond will mature

This series of periodic payments in a plain vanilla bond is referred to as an ordinary annuity. This formula assumes that the first coupon payment will be made one period from the present time and the end of every subsequent period, the next coupon payments will be made.

Note that period here could be anything, but typically bonds pay coupon semi annually or annually, so one period will be 6 months or 12 months long. Also note that the last coupon payment and the par value of the bond are paid together. It is clear from the formula that the payments that come farther in the future have a lower present value.

Another thing evident from the formula is the inverse relationship between bond prices and interest rates. As interest rates go up in the economy, the required rate of return (r) also goes up. This increases the discounting factors in the formula and the price of the bond will be lower.

The bond pricing formula given above can be simplified as:

Price = I x [1- [1 / (1+r)^n ] ] / r + [Par Value / (1+r)^n]

When you use this formula, you wouldn’t have to calculate the present value of each coupon payment separately, and the price can be determined simply by plugging in the value of these variables. The same formula can be used irrespective of the nature of the bond. Here is an illustration of how this formula can be applied when determining the price for different kinds of bonds.

Let’s consider a plain vanilla bond with a par value of $5,000, maturity period of 5 years, and a coupon rate of 5%, paid semi-annually. Let’s assume that the required rate of return is 10%. Here are the values of different variables that we’ll need in the formula.

n = 10 (Coupon payments are made with a periodicity of 6 months. There are 10 such periods in 5 years)

I = $5,000 * 2.5% = $125 (Although coupon rate is 5%, this is the annual interest rate. For semi annual payments, coupon rate will be half of the annual rate)

r = 5% (For a 10% annual required rate of return, the semi-annual required rate will be 5%)

Par Value = $5,000

Plugging these values in the bond price formula:

Price = $125 x [1- [1 / (1+.05)^10 ] ] / .05 + [$5,000 / (1+.05)^10] = $4,034. 7

You can see this value in light of our previous discussion on bonds selling for a premium or a discount. In this case, the required rate of return is significantly higher than the coupon paid by the bond. That is why the bond is selling at a heavy discount, as otherwise investors will have no reason to purchase this bond.

Now, let’s see what happens when the coupon rate of the bond is 15%. The coupon payment in this case (I) will be $5,000 * 7.5% = $375. All the other variable for the formula remain the same. This will result in the bond being priced as:

Price = $375 x [1- [1 / (1+.05)^10 ] ] / .05 + [$5,000 / (1+.05)^10] = $5,965.2.

The bond is offering a higher coupon rate than the interest rate investors can earn in the market, which is why the bond is now selling at a premium.

A zero coupon bond does not make any interest payments throughout the life of the bond. There is only a single cash flow, at the time of maturity of the bond, when the par value of the bond is returned to the investors. Pricing such a bond is much simpler.

Let’s consider a zero coupon bond with a par value of $5,000 and a maturity period of 5 years. Let’s assume that the required rate of return is 10%. Plugging these values in the bond pricing formula:

Price = [$5,000 / (1+.05)^10] = $3069.5

Compare this price with the price of the plain vanilla bond that we calculated in the last example. You can see that as there are no coupon payments made by the bond issuer, investors need a much larger incentive, in the form of a bigger discount, to purchase the bond. Of course, in case of zero coupon bonds, there is no question of the bond selling at a premium, or even at par. No investor would be ready to pay $5,000 (or more) today, just to get $5,000 back a few years from now.

You would have noticed that we assumed a periodicity of 6 months in the formula despite the fact that a zero coupon bond pays no interest. This is done so that these bonds can be easily compared with other bonds that pay coupon on a semi-annual basis.

So far, the bond pricing that we’ve discussed misses out on one crucial point – accrued interest. In the formula that we used to find out bond prices, we did not take into account the fact that the price of a bond will change as coupon payment date comes closer. We simply assumed that the next coupon payment is exactly one period away.

At any given date between two coupon payments, the price of a bond should include the interest that has been accrued so far since the last coupon payment. This is the interest that the bond investor has already earned by holding the bond, but it has not bee paid to him yet.

You can also think of this change in price between two coupon payments in terms of time value of money. Let’s say a bond paid coupon on June 31 and the next payment is scheduled for Dec 31. As the next payment and all future cash flows come closer, their time value should increase, which means that the price of the bond should increase. However, this has not been accommodated in the formula so far.

Bond prices that include accrued interest are known as dirty bond prices while those excluding accrued interest are referred to as clean bond prices or flat prices. Typically, quoted prices of bonds are flat prices. The reason behind this is that a clean price allows investors to evaluate the quality of the bond on the basis of issuer risk, interest rates etc. It also enables easier comparison between two bonds, without complicated assessment of when the last coupon was paid and how much interest has been accrued.

The dirty price of a bond moves in a saw tooth pattern throughout the life of the bond (assuming interest rates and issuer risk do not change in that period). The price of the bond keeps increasing from the date of coupon payment as more and more interest gets accrued. On the subsequent coupon payment, the price falls to its minimum level again and starts rising in the same manner from the next day onwards.

As you would have guessed, to be able to calculate the accrued interest, we need to first determine the exact number of days that have passed since the last coupon payment. Different day counting conventions are used for different bonds.

In an actual / actual day-count convention, you need to count the exact number of days that have passed so far since the last coupon payment and evaluate interest assuming that it accrues on every day. This convention is used for treasury securities. Consider a situation where the last coupon payment on a treasury bond was made on July 1 and the next payment is scheduled for January 1. To price the bond on September 1, we’ll have to count the exact number of days between July 1 and September 1. Accrued interest in this case will be calculated for 62 days, which is the number of days that have passed since the last payment.

In a 30 / 360 convention, it is assumed that each month of the year has 30 days and that there are 360 days in a year. This somewhat simplifies the calculation for the number of days, as you don’t have to think about which month has 30 days, which has 31, if it is a leap year, and so on.

This convention is typically used for corporate bonds and municipal bonds. If in our previous example, the bond was issued by a company, the accrued interest would have been calculated on 60 days instead of 62 days. But in this case, the daily accrual of interest will be calculated assuming that there are only 360 days in the year, i.e. interest will be divided into 360 periods.

Accrued interest can be calculated using the following formula:

Accrued interest = I x [d/D]

Here:

I = Coupon payment

d = Number of days since last payment

D = Total number of days between payments

Let’s consider a corporate bond, where the last coupon payment was made on July 1, the next payment is due on January 1, and we are calculating accrued interest on October 1. The coupon rate is 10%, paid semi-annually, and the par value of the bond is $5,000.

I = $5,000 x (10% / 2) = $250

d = 90 days (as all months are assumed to have 30 days and exactly 3 months have passed since last coupon payment)

D = 180 days (coupon payments are semi-annual, so periodicity is 6 months, with each month assumed to have 30 days)

Accrued Interest = $250 x [90 / 180] = $125

This accrued interest should be added to the clean price of the bond (as calculated from the bond pricing formula) to arrive at the true value of the bond on October 1. This is the price that you’ll have to pay if you want to buy the bond from the secondary market.

Next Chapter: Calculating Yield and Understanding Yield Curve