Sunday, July 21, 2013

Pricing of Bonds

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.

Why Do Bond Prices Change

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 ,to see the prices of bonds issued by a national government change drastically in response to bad economic data,as the risk is high ,it makes the bond price go down and the demand for high yield go up.
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.
Premium and Discount

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

 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.  Let’s say when it’s selling at a 5% premium, its price is listed as 105,the bond listed as 105 and having a par value of $1,000 can be calculated as 105% of $1,000, which comes to $1050.

Calculating Bond Prices

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]

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]


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.


Do you like this post?


Post a Comment

Related Posts with Thumbnails