A bond is a long-term debt instrument used by governments, corporations, and firms to raise debt financing. Most of the bonds have following three attributes generally.

• Pay interest annually, semiannually, or quarterly
• They have defined maturity period
• They have defined face value usually \$100 or \$1000

Par value or face value is the amount mentioned at the face of the bond representing the sum to be borrowed at which interest is to be paid and also will be repaid at the expiry of maturity period. Coupon rate is the stated rate of interest and interest payable will be calculated by multiplying coupon rate to the par value. Maturity period refers to the number of year after which the principle amount will be repaid or refunded to bondholder.

[adsense1]In financial management the true value of a physical or financial asset is determined by discounting its future expected net benefits to the present values. The value of the bond is the sum of present values of the interest payments (contractually agreed upon at the stated rate of interest) discounted at the required rate of return plus the present value of par value repayable at the end of maturity period. The required rate of return is the commensuration of prevailing interest rate and risk. The key inputs to valuation process are expected returns in terms of cash flows together with their timings and risk in terms of required returns. Bond valuation model can be presented as follows:

B = I x (PVIFA)kd,n + M x (PVIF)kd,n

Where,

B = Value of the bond
I = Annual interest payment
N = life of the bond in years
M = Maturity value of par value
Kd = Required rate of return

### Example

A firm issued a 10% coupon interest bonds for a period of 10 years with a face value of \$1000. The required rate of interest is also 10% and interest is paid annually. Find out the value of the bond.

### Solution

Annual Interest (I) = \$1000 x 10% = \$100
(PVIFA) 10%, 10 = 6.145 (from the annuity table)
(PVIF) 10%, 10 = 0.386

B = (\$100 x 6.145) + 1000 x 0.386 = \$1000
In the above example the bond value is equal to the par vale i.e. \$1000. This is due to the fact that when required return is equal to the coupon rate the bond value equals the par value. But in reality it seldom happens when required rates and coupon rates are equal therefore market value of the bond generally remains lower or greater than the par value.

### Significance of Required Returns to the Bond Value

As mentioned above that market value of the bond will differ from it par value if required return is different from the coupon rate. These two rates differ due to the following reasons:

• Changes in the basic cost of long-term funds
• Changes in the basic risk of the firm

When the required return is more than coupon rate of interest the bond market value is less than par value or bond will sell at discount. On the contrary if required return is less than coupon rate the market value of the bond would be more than its par value or bond will sell at premium. This phenomenon can be more easily understood by the use of examples.

### Example

A firm issued a 10% coupon interest bonds for a period of 10 years with a face value of \$1000. The required rate of interest is 12% and interest is paid annually. Find out the value of the bond.

### Solution

Annual Interest (I) = \$1000 x 10% = \$100
(PVIFA) 12%, 10 = 5.650 (from the annuity table)
(PVIF) 12%, 10 = 0.322

B = (\$100 x 5.650) + 1000 x 0.322 = \$887

Because the required return is greater than coupon rate in the above example the bond value is less than its par value.

### Example

A firm issued a 10% coupon interest bonds for a period of 10 years with a face value of \$1000. The required rate of interest is 8% and interest is paid annually. Find out the value of the bond.

### Solution

Annual Interest (I) = \$1000 x 10% = \$100
(PVIFA) 8%, 10 = 6.710 (from the annuity table)
(PVIF) 8%, 10 = 0.463

B = (\$100 x 6.710) + 1000 x 0.463 = \$1134

Because the required return is less than coupon rate in the above example the bond value is greater than its par value.

The above two examples clearly show that required rate of return is the major determinant of market value of the firm’s bonds.

If the required rate of return remain constant over the life of the bond the bonds market price approaches its par value. On the other hand under the changing circumstances of required returns the shorter the time to maturity, the smaller the impact on bond value caused by given changes in the required return.

### Semiannual Interest and Bond Value

The procedure towards the calculation of bond’s value paying interest semiannually is similar. As interest is two times in a year therefore half yearly interest should be calculated to find the present value. The required return over that the interest payments are discounted is also needed to be divided by two. Number of years in the maturity period is converted to discounting period by multiplying by two. Symbolically:

B = I / 2 x (PVIFA)kd/2 , 2n + M x (PVIF) kd/2 , 2n

### Example

A firm issued a 10% coupon interest bonds for a period of 10 years with a face value of \$1000. The required rate of interest is 14% and interest is paid semiannually. Find out the value of the bond.

### Solution

Semiannual Interest (I / 2) = \$1000 x 10% x 6 / 12 = \$50
(PVIFA) 14% / 2, 10 x 2 = 10.594 (from the annuity table)
(PVIF) 14% / 2, 10 x 2 = 0.258

B = (\$50 x 10.594) + 1000 x 0.258 = \$787.7