In the previous article, I discussed Bond basics such as its definition, features and types. Today I will discuss about pricing of bond – how to value a bond and how does the price varies with the market conditions.

The market or fair value of a bond, like any other security or asset, is the present value (PV) of its future cash flows. PV is calculated by “discounting” the expected future “cash flows” to the present using the “discount rate”. These cash flows are nothing but interest payments made by the bond issuer (borrower) to the bond buyer. In bond lingo these interest payments are called Coupon Payments. Let us understand the concepts of discounting and PV before we proceed further.

Say, you have Rs. 100 today and you deposit it in a bank which pays you an interest of 10%. After 1 year this Rs. 100 will become Rs. 110 (100+ 10%of 100). In other words, if you want to have Rs. 110 in your account a year later, you should deposit Rs. 100 “now” in an account which gives 10% return. Thus, the present value of Rs. 110 is Rs. 100. In finance lingo we call Rs. 110 has Future Value (FV), Rs. 100 has Present Value (PV) and 10% as Discount Rate (Rd). Here, the biggest challenge is to determine Rd because most of the times it is unknown and depends on lot of factors.

Discount rate is the required rate of return on an investment i.e. how much market expects from an investment made to a specific asset. This discount rate varies from asset to asset and depends on riskiness of investment. For example – say I go to a bank to borrow money. If I have a stable earning of Rs. 1,00,000 per months and have never defaulted on any previous loan and have no outstanding debt as well, banks will give me a lower interest rate. This is because I am a less “risky” customer for the bank and hence they will require lower interest from me. However, if I have outstanding debt against me and have defaulted on my previous loans, I am a “risky” investor for the bank. Hence, it will require a higher interest rate from me to compensate for the extra risk.

Remember this – **Higher risk requires higher return.**

**Bond Valuation
**As discussed, the fair value of bond price is PV of all future cash flows generated by the bond. Here,

Cash Flow: are coupon payments, C, generated periodically by the bond and the par of face value, F, paid at the maturity of bond after T years (where T>=1)

Discount rate: is the required yield or rate of return (compounded annually).

Bond Price = P =

Where,

C = Coupon Payment

M = No. of payments period

Rd = Discount rates or yield

F = Par value

Here, you can notice a very important relationship between Bond Price and Required rate of return i.e. price of a bond is inversely proportional to its required rate of return or yield i.e. if rate or yield increases, price of bond goes up while if required rate of return goes up, price of bond goes down.

**Example 1: **Calculate the price of a bond with a par value of Rs.1,00,,000 to be paid in ten years, a coupon rate of 10%, and a required yield of 12%. We'll assume that coupon payments are made semi-annually to bond holders and that the next coupon payment is expected in six months. Here are the steps we have to take to calculate the price:

**1. Determine the Number of Coupon Payments: **Because two coupon payments will be made each year for ten years, we will have a total of 20 coupon payments i.e. M=20

**2. Determine the Value of Each Coupon Payment: **Because the coupon payments are semi-annual, divide the coupon rate in half. The coupon rate is the percentage off the bond's par value. Hence, each semi-annual coupon payment will be Rs. 5,000 (Rs. 1,00,000 X 5%). Hence, C = 5,000

**3. Determine the Semi-Annual Yield:** Like the coupon rate, the required yield of 12% must be divided by two because the number of periods used in the calculation has doubled. If we left the required yield at 12%, our bond price would be very low and inaccurate. Therefore, the required semi-annual yield is 6% (12%/2).

Putting these values in the above equation, we get

Price of Bond = P = Rs. 88,530

From the results, we can figure out that P (Price of Bond) <F(Par value). This means bond is “selling at discount”. This happens when the required rate or return, r, i.e. yield, is more than the coupon rate. What does this mean? This means the bond is paying interest payments at a lower rate (10%) than the expected market return (12%). So why will investors buy this when they are getting less interest than expected? They will invest because the bond is available at a discount i.e. they are paying only Rs. 11,111 for the bond which has par value of Rs. 1,00,000.

When the coupon rate is lower than yield, the price of bond is more than its par value i.e. P>F. This means bond is “selling at premium”. Here, bond is paying a higher return than expected by investors. So investors will bid aggressively for this bond and thus the demand would push up the bond price.

**Coupon Yield**

The coupon yield is simply the coupon payment (C) as a percentage of the face value (F).

**Coupon yield = C / F**

Coupon yield is also called nominal yield.

**Current Yield**

The current yield is simply the coupon payment (C) as a percentage of the (current) bond price (P).

**Current yield = C / P**

**Yield to Maturity
**The yield to maturity (YTM) is the yield promised to the bondholder on the assumption that the bond or other fixed-interest security such as gilts will be held to maturity, that all coupon and principal payments will be made and coupon payments are reinvested at the bond's promised yield at the same rate as the original principal invested. It is a measure of the return of the bond.

Several other ways of explaining YTM are – It is the discount rate which returns the market price of the bond. It is thus the internal rate of return of an investment in the bond made at the observed price. YTM can also be used to price a bond, where it is used as the required return on the bond. This is the rate at which the present value of all future cash flows is equal to the bond’s price.

In other words, it is identical to Rd in the above equation. To achieve a return equal to YTM, the bond owner must:

• Buy the bond at price P,

• Hold the bond until maturity, and

• Redeem the bond at par.

The concept of current yield is closely related to other bond concepts, including yield to maturity, and coupon yield. The relationship between yield to maturity and coupon rate is as follows:

• When a bond sells at a discount, YTM > current yield > coupon yield.

• When a bond sells at a premium, coupon yield > current yield > YTM.

• When a bond sells at par, YTM = current yield = coupon yield amount.

**Bond Price Behavior**

If you read newspapers or watch business news regularly, we will find analysts discussing about changes in bond prices due to macroeconomic events, monetary policies or government policies. Let us discuss few important factors that cause change in prices of bond.

1. The forces of economy influence a bond’s price to a great extent. As we know from the bond valuation equation, price of a bond is inversely proportional to its required rate of return or yield i.e. if rate or yield increases, price of bond goes up. Why this rate of return changes? This may happen because of several reasons such as inflation and changes in risk perception of country or company. Risk of a company is reflected in its credit worthiness i.e. a high risk company (one with high debt or lower profit margin) will have poor credit rating.

We can understand this by this example – Real estate companies in India raised lot of debt 2-3 years back to launch new projects. There was a real estate boom in the country and developers were literally printing cash! Hence, their credit rating was very good and they were able to borrow lot of money at lower interest rate. However, within a year subprime crisis happened and world entered into recession. The real estate market crashed in India as well. This put enormous pressure on real estate developers who were not able to sell properties and generate cash to pay back loan. Thus, their credit worthiness fell and no bank was willing to lend them. This increased the required rate of return on their bonds, causing prices of their bonds to go down.

2. The prices of long-term bonds generally display greater sensitivity to changes in interest rates than do the prices of short-term bonds. This is because chances of changes in economic activities will be more over long-term as compared to short-term. For example inflation may not normally increase by a large amount within 2-3 years; however, it may go up significantly over a period of 10 years.

Thus, if you invest in a bond which has high time to maturity the required rate of return in high i.e. market expect a higher return on these bonds because risk or uncertainty in high over a period of long time period.

In the next article I will discuss about a very important concept called Yield Curve in detail. This curve reflects the short-term and long-term macroeconomic conditions of a country. We will learn how to read and interpret this curve in the next article.

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