**Bond valuation is made on the basis of principle** that works behind valuation of direct claim securities. The principle is that bond’s value is derived from the associated cash flows it generates e.g. coupon receipt and par recovery at maturity. The value of bond is directly associated with the value of securing real assets of the company. The operations of the company generate cash inflows from sale of its products. The income of bond is actually linked with the income of the business that is generated from working of real assets. The company pays coupon income against **Bonds** or to the bond holders on the basis of its cash flows generated from the real assets. The bond’s value is calculated with the help of formula of net present value or present value. The value calculated is termed as the intrinsic value or fair value of the bond. This intrinsic value can be compared with the market value of the bond in order to ascertain any difference.

**How to Find Present Bond Valuation**

The present value of bond is calculated in order to ascertain its intrinsic value. The present value formula of the bond is given below.

PV = ∑CF_{t} / (1+r_{D})^{t} = CF_{1} / (1+r_{D}) + CF_{2} / (1+r_{D})^{2} + . . . + CF_{n} / (1+r_{D})^{n} + PAR / (1+r_{D})^{n}

In the above formula, PV = Fair price of the bond or intrinsic value that is paid to invest in the bond. The fair price is a only theoretical concept and it is different from the market price which a practical concept.

r_{D} = Investor’s (Bond holder) required rate of return for investing in bond. This rate is selected to be minimum interest rate and is derived from market interest rate. Intrinsic rate is different from coupon rate.

CF = Cash flows which include two types of cash flows. One is fixed coupon receipts which are received throughout the entire life of the bond and second is the principal amount or par value of the bond which is received at maturity date of the bond along with the last interest coupon receipt.

n = maturity of life of bond (in years)

The intrinsic value of the bond is the expected value of the bond. This intrinsic value is compared with the market value or price of the bond. The market value of the bond changes on the basis of demand & supply of bond in market & interest rate of the bond. The comparison of these two values provides the clear picture to either invest in the bond or not.

The price of the bond is affected by the market interest rate prevailing in the market. The reason behind this effect is that the market rate of return influence the required rate of return (r_{D}) expected by the bond investor. When Market Interest Rate (Required rate of return of investor) increases, the Value or Price of the bond decreases. This is called Interest Rate Risk. The relationship is simple because (r_{D}) is in the denominator of the equation and it will fall & rise with the general interest rate.

The interest coupon rate is fixed & bond issuer have to pay according to tat fixed rate but the market interest rate changes on daily and even hourly basis. So when Market Interest rate < Coupon Interest Rate then price of Bond or Market Value > Face Value. The simple reason behind that relationship is that when market is offering lower rate of return then the coupon rate of bond, the bond becomes value able. This is called Premium Bond. In case the Required Rate of Return = Coupon Rate then the Market Value = Face value (Par value)

The life of the bond is limited and when bond reaches its maturity date then the market value of the bond reaches the face value (par value) of the bond.

**Long Bond –Risk Theory:**

The long term bond (i.e. 10 years) has more Interest Rate Risk than short term bond (i.e. 1 year) under the condition that the both bonds have similar coupon rate. When long term bond is purchased by an investor then he is locked up in single investment for longer duration of time and there is much possibility of changes in the interest rate & fluctuation rate. The change in the interest rate greatly affects the long term bond. The prices of long term bond fluctuate more because their coupon rates are fixed for longer duration of time while the market interest rate are changing on daily basis. Therefore the price of the long term bonds must keep adjusting constantly. The price of long term bond changes more as compared to short term bond the coupon rate of long term bond is fixed but the market interest rate is changing many times within single year.

**Bond Portfolio Theory:**

Fluctuation in the market interest rate/market seriously affects the portfolio of bondholder in 2 ways.

**Interest Rate Risk:**

With this effect the value portfolio of bond declines if interest rates rise.

**Reinvestment Risk:**

With this effect the overall rate of return or yield on the portfolio of bond rises when interest rate rise. The bond holder’s opportunity cost has changed. Reinvestment risk is higher for short period bonds.

**Interest Rate Trade Off:**

The two effects cancel out each other. When there is rise in the market interest rates, the prices of bonds decline (Interest rate risk increases) but overall returns on future investments in bonds increases (reinvestment risk decreases).

**Bond Maturity (Life) Tradeoff:**

Short term bonds (i.e. 1 year) have lower interest rate risk as compared to long term bonds (i.e. 10 year) but the short term bonds have higher reinvestment rate risk.

**Bond Valuation Examples**

Suppose Mr. John wants to start a new business but he does not have sufficient cash and he contact with a bank for loan. The bank agrees to provide a loan of $ 200,000 to Mr. Ali through bond. Mr. Ali issues a bond in the favor of bank which has the following features.

- Face Value or Par Value = 200,000
- Coupon rate = 15% (paid annually)
- Maturity = 2 years
- Security = Deed of property for canteen space

The main point to ascertain is the value of investing in a bond with Mr. Ali by the bank.

Par Value = 200,000

Coupon value = coupon rate x par value

Coupon value = 15% x 200,000 = 30,000 p.a.

This means that besides the recovery of principal amount at maturity date, the bank receives interest payment of $ 30,000 each year from Mr. Ali on bond.

Suppose that the required rate of return (r_{D} ) for bank is 10% p.a.

Now for computation of PV or intrinsic value of bond, following formula is used

PV = CF_{1} / (1+r_{D}) + CF_{2} / (1+r_{D})^{2} + PAR / (1+r_{D})^{2}

PV = 30,000/1.1 + 30,000/(1.1)^{2} + 200,000/(1.1)^{2}

PV = 27273 + 24793 + 165289

PV = + 217,355

The above value is for PV not for NPV and shows that the lending of Rs 200,000 to Mr. Ali for 2 years at n15% mark up by the bank worth positive 217,355 today which is net gain in the value for bank.

**Bond Yield To Maturity (YTM):**

The expected rate of return for which bond is held by bond holder till its maturity is called Yield To Maturity (YTM) of bond. In other words it is also the overall return on bond. Besides the calculation of value of the bond, the computation of rate of return of bond is also important. In order to compare the overall rate of return of different bonds, the comparison of the respective YTMs makes the required comparison an easy task.

The calculation of YTM of bond is made in similar way as calculating IRR from the NPV equation. The present value equation for the bond valuation, set equal to the market price of the bond and then r_{D} is solved by trial & error method. The value of r_{D} at which PV of the bond = market value of the bond is the YTM of that bond.

PV = Market Price of bond = CF_{t} / (1+r_{D})^{t} = CF_{1} / (1+r_{D}) + CF_{2} / (1+r_{D})^{2} + . . . + CF_{n} / (1+r_{D})^{n} + PAR / (1+r_{D})^{n}

In the above equation all the variables are obvious (PAR, CF, n etc) except r_{D}. Keeping PV = Current Market Price of bond, r_{D} is solved where

YTM = r_{D}

**Bond YTM Example:**

Suppose the TFC (Term Finance Certificate) of a Company XYZ is traded on stock exchange of Karachi at for Rs 900. The face value of par value of the TFC is Rs 1,000. There is fixed coupon rate of 15% p.a. and the coupon interest is paid on annual basis. After two years, the bond will mature (it is 5 year bond and 3 years has passed). The main issue is to find out the overall expected rate of return (YTM) of the bond.

It is clear that the Market Price of bond (Rs 900) is less than its Par Value (Rs 1,000) which indicates that the bond is selling at a discount.

In order to calculate the **Expected rate of Return** (YTM) of bond, the IRR formula of trial & error is used by keeping the present value of bond equal to its market price.

PV = Market Price = Rs 900

Par Value = Rs 1,000 (which will be received after 2 years)

Coupon Receipt (annually received) = coupon rate x par value = 15% x 1000 = Rs 150

r_{D} = the expected rate of return of investor of bond = YTM

This variable is unknown in the equation. Now

PV = 900 = 150 / (1+r_{D}) + 150 / (1+r_{D})^{2} + 1000/ (1+r_{D})^{2}

PV = 900 = 150 / (1+r_{D}) + 1,150 / (1+r_{D})^{2}

By using trial & error

r_{D} > 15%

By trying r_{D} = 20%, PV of the bond becomes Rs 924 which is close to actual value (Rs 900)

So r_{D} is set as 21%. This makes PV = 909 which is much closer.

By slightly increasing the r_{D} from 21% to 21.7%, the PV = 900.

Hence YTM = 21.7%