Bond valuation is a technique for determining the theoretical fair value of a particular bond. Bond valuation comprises calculating the present value of future interest payments and face value to determine a bond’s theoretical fair value. The bond’s future interest payments also known as its cash flow, and the bond’s value upon maturity also known as its face value or par value. Because bonds’ par value interest payments are fixed, an investor uses bond valuation to determine what rate of return is required for a bond investment to be worthwhile.

Present value approach:

To understand the concept of the present value formula concerning bonds, let’s look at a bond valuation formula with the following example.

The formula used to calculate the Present Value of an Annuity (PVA) to interest payments is;

PVA=I[1-(1+K)^n]/k

Where PVA is the present value of an annuity, ‘I’ is annual interest, k is the required rate of return per period (annual/semiannual, etc.), and n is the number of times periodic interest is paid.

The variables in the formula require you to use the interest payment amount, the discount rate (or required rate of return), and the number of years remaining until maturity.

Assume that a bond has a face value of Rs.1000 and a coupon rate of 6%. It means the annual interest is Rs. 60 for an investment of Rs1000.

Divide the annual interest amount by the number of times interest is paid per year. This calculation is I, the periodic interest paid. For example, if the bond pays interest at 6 percent for Face Value (FV) of Rs.1000 semiannually, I = Rs.30 per period. Each period is 6 months (semiannual).

Determine discount rate. For example, if you require a 5% annual rate of return for a bond paying interest semiannually, k = (5% / 2) = 2.5%. Divide the discount rate required by the number of periods per year to arrive at the required rate of return per period, k.

Calculate the number of period’s interest paid over the life of the bond, or variable n. Multiply the number of years until maturity by the number of times per year interest is paid. For example, assume that the bond matures in 10 years and pays interest semi-annually. In this case, n = (10 X 2) = 20 interest-paying periods.

The present value annuity PVA=I[1-(1+K)^n]/k to arrive at the present value of interest payments. In this example, the present value of interest payments is Rs.30[1-(1+0.025)^20]/0.025 = Rs.467.67.

Computation of the present value of the principal payment on maturity:

The principal is a single repayment to the investor at maturity as interest paid semiannually throughout 10 years. You will receive a single payment of Rs.1000 after 10 years from now. You use a discount rate to discount (reduce) that single payment into a value today.

The formula uses some of the same values you used in the annuity formula. Use the annuity formula first then apply those same variables to the principal payment formula.

Plug into the present value (PV) formula. Use the formula

PV=FV/(1+k)^{n} to arrive at the present value of the principal at maturity. For this example;

Present Value PV = Rs.1000/ (1+0.025)^20 = Rs.781.20.

Add the present value of interest to the present value of the principal to arrive at the present bond value. For our example, the bond value = (Rs.467.67 + Rs.781.20), or Rs.1248.87 at a discount rate of 5%.

Investors use the present value to decide whether or not they want to invest in a particular bond.

Relative price approach:

Relative Price Approach: This technique involves comparing the bond with similar ones in the market, adjusting for varying risk, interest rates, and creditworthiness of the issuer. It’s commonly used for valuing corporate bonds.

Relative value can be defined as the expected price convergence of contracts or portfolios with similar risk profiles. For fixed-income this means similar exposure to duration, convexity, and credit risk. The causes of relative value are limited arbitrage capital and aversion to the risk of persistent divergence. Relative value in the fixed-income space has been pervasive and persistent. Relative value trades can be based on parametric estimation of yield curves or comparisons of individual contracts with portfolios that replicate their essential features. The latter appears to have been more profitable in the past.

Arbitrage-free pricing approach:

In the above example of the Present value approach, we have used a flat discount rate of 5% from the beginning till the maturity of the bond. Arbitrage-free valuation of an asset is based solely on the value of the underlying asset without taking into consideration derivative or alternative market pricing. It can be calculated for various types of assets using financial formulas that account for all of the cash flows generated by an asset.

Let’s take an example. Suppose we want to calculate the value of a Rs1000 par bond, 5% coupon, with a 5-year maturity bond. We also have the following spot rates for the next 5 years:

Fixed-rate bonds are discounted by the market discount rate but the same rate is used for each cash flow . Suppose that

the 1-year spot rate is 4.00%;

the 2-year spot rate is 4.30%;

the 3-year spot rate is 4.51%;

the 4-year spot rate is 4.7%; and

the 5-year spot rate is 4.80%;

Assuming this is an annual pay bond, the bond will have the following cash flows (coupon).

Year 1: Rs50, Year 2:Rs50, Year 3: Rs50, Year 4: Rs50 Year 5: Rs1000 + Rs50

The value of the bond can be calculated by discounting these cash flows by their respective spot rate.

The value of the bond will be calculated as follows:

PV=PMT/(1+S1)^1+PMT/(1+S2)^2+……………..PMT+FV/(1+Sn)^N

Where, PV=Present value, PMT is the periodic coupon, FV is the face value, S1, S2, and Sn are the spot rates for periods 1 to N

You can use the above formula to value any bond with any maturity. All you need is the spot rate for the respective maturity.

We can compare this bond value with the market price of the bond to identify an arbitrage opportunity

Bond Value = 50/(1.04)^1+50/(1.0430)^2+50/(1.0451)^3+50/(1.047)^4+1050/(1.048)^5

Bond Value = Rs1010.033

If the market price of this bond is Rs1020, then there is an arbitrage opportunity. Whenever an asset is traded in multiple markets, its possible prices will temporarily fall out of sync. It’s when this price difference exists that pure arbitrage becomes possible. The arbitrageur will earn a risk-free profit of Rs9.967 (Rs1020 – Rs1010.033) in the above case.

Stochastic calculus approach:

Stochastic calculus is the area of mathematics that deals with processes containing a stochastic component and thus allows the modeling of random systems. Many stochastic processes are based on functions that are continuous, but nowhere differentiable. Short of that, if you are simply trading an asset to gain a specific kind of exposure, stochastic calculus is not used very much.

**Bond valuation theorems**:

There are five bond valuation theorems:

Theorem1: **Price and interest rates move inversely**

When all else being equal, if new bonds are issued with a higher interest rate than those currently on the market, the price of existing bonds will decline as demand for those bonds falls. Equally, if new bonds are issued with a lower interest rate than bonds currently on the market, the price of existing bonds will increase in line with demand. For example, you are holding a bond that pays only 5% interest, and if newly issued bonds pay 5.5% interest, no one would want to buy your bonds in a secondary market with a lower interest rate because its return would be lesser. If you want to sell your bond in the secondary market, you would have to offer to sell your bond at a discount—meaning less than face value—for someone to take it off your hands. The reduction of the bond price offsets the higher interest rates available on newly issued bonds. This is the major factor in understanding how and why bond prices and interest rates move in opposite directions.

Example 1:

Let us assume, that Rs.1000 face value bond offers Rs.40 coupon per annum. Hence, yield = (40/1000) x 100% = 4%. Supposing the price of the bond increases from Rs1000 to Rs1100 due to strong investor demand, it means, the bond now trades for 10% above the issue price. However, the coupon amount remains the same at Rs40.

But, the yield changes from 4% to (40/1100) x 100% = 3.63%. So, when the bond price has gone up, the yield on the bond decreases.

Supposing, the price of the same bond decreases from Rs1000 to Rs900 due to low investor demand, it means, the bond now trades for 10% below the issue price. However, the coupon amount remains the same at Rs40.

But the yield changes from 4% to (40/900) x 100% = 4.44%. So, when the bond price falls, the yield on the bond increases.

Conclusion: The yield and bond price have an important but inverse relationship. When the bond price is lower than the face value, the bond yield is higher than the coupon. When the bond price is lower than the face value, the bond yield is higher than the coupon rate.

Theorems 2: **A decrease in rates raises prices more than a corresponding increase lowers them.**

When interest rates rise, prices of existing bonds tend to fall, even though the coupon rates remain constant, and yields go up. Conversely, when interest rates fall, prices of existing bonds tend to rise, their coupon remains constant – and yields go down.

Example 2:

Let us assume, that Rs.1000 face value bond offers Rs.40 coupon per annum. Hence, yield = (40/1000) x 100% = 4%. Supposing the price of the bond increases from Rs1000 to Rs1100 due to strong investor demand, it means, the bond now trades for 10% above the issue price. However, the coupon amount remains the same at Rs40.

But, the yield changes from 4% to (40/1100) x 100% = 3.63%. So, the price change is 4%-3.63= -0.37% It means when the bond price has gone up, the yield on the bond is reduced.

Supposing, the price of the same bond decreases from Rs1000 to Rs900 due to low investor demand, it means, the bond now trades for 10% below the issue price. However, the coupon amount remains the same at Rs40.

But the yield changes from 4% to (40/900) x 100% = 4.44%. So, change in price 4.44%-4%= +0.44 % when the bond price falls, the yield on the bond increases.

Conclusion: When interest rates rise, prices of existing bonds tend to fall, even though the coupon rates remain constant, and yields go up. Conversely, when interest rates fall, prices of existing bonds tend to rise, their coupon remains constant – and yields go down.

Theorem 3: **Price volatility is inversely related to coupon**

Let us assume, that Rs.1000 face value bond offers Rs.40 coupon per annum. Hence, yield = (40/1000) x 100% = 4%. Supposing the price of the bond increases from Rs1000 to Rs1100 due to strong investor demand, it means, the bond now trades for 10% above the issue price. However, the coupon amount remains the same at Rs40.

But, the yield changes from 4% to (40/1100) x 100% = 3.63%. So, the price change is 4%-3.63= -0.37% It means when the bond price has gone up, the yield on the bond is reduced.

Supposing, the price of the same bond decreases from Rs1000 to Rs900 due to low investor demand, it means, the bond now trades for 10% below the issue price. However, the coupon amount remains the same at Rs40.

But the yield changes from 4% to (40/900) x 100% = 4.44%. So, change in price 4.44%-4%= +0.44 % when the bond price falls, the yield on the bond increases.

Conclusion: Price volatility is inversely related to coupons i.e. when the Bond price rises from 1000 to 1100, the yield is reduced from 4% to 3.63% and when the bond price falls from 1000 to 900, the yield increases from 4% to 4.44%. It means When the price of a bond goes up, its yield goes down, and vice versa.

Theorem 4: **Price volatility increases with maturity**

A bond’s volatility depends on two factors: its coupon rate and when it will be retired (at maturity or call date). Other things being equal, the general rule is that: The longer the time until retirement, the greater the price volatility, and the lower the coupon rate, the greater the price volatility.

Bond owners receive their principal back at maturity, so bond prices converge toward par value as the bond approaches maturity. For example, a discount bond will increase in price toward par value as it nears maturity, all else equal. Meanwhile, a premium bond will decrease in price toward par value as maturity nears. Then at maturity, the owner receives the bond’s par value and any remaining interest payment.

Theorem 5: **Price volatility increases at a diminishing rate as maturity increases:**

There is a direct relation between bond volatility and maturity. If maturity rises, there are higher chances of change (volatility) in the interest rate and vice-versa. When the maturity decreases, the bond’s price volatility decreases at a decreasing rate.

*Percentage price changes are more when the discount rate goes down than when it goes up by the same amount.*

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