# How to Determined Value of a Bond

• How is the value of a bond determined? What is the value of a 10-year, \$1,000 par value bond with a 10% annual coupon if its required rate of return is 10%?

A bond contains a unique pattern of cash flow containing of a stream of constant payment of interest plus the par return at maturity. The payment of annual coupon is the cash flow which is computed as:

Payment = coupon rate x par value = 0.1 x \$1000 = \$100. With annual coupon bond of 10% for a period of 10 years, the value of bond is found as follows (using financial calculator);

VB = \$100/(1+r)’ + \$100/(1+r)2 + …+ \$100/(1+r)10 + \$1000/ (1+r)10

where r is the coupon rate

= \$(90.91 + 82.64 + … + 38.55 + 385.454) = \$1000

This can also be computed using this formula

VB = \$100(PVIF10%, 10) + \$1000(PVIF 10%, 10)

The bond period is 10 year, with annual coupon rate of 10%, and \$100 annuity per year and a lump of \$1000. The sum payment at a period of 10 years is:

VB= \$100((1-1/(1+0.1)10)/0.10 + \$1000(1/1+0.10)10)

The present value annuity is \$614.64 and the present value maturity value is 385.54. The bond value is therefore the sum of the two which is equal to \$1000

Value of the bond= PV annuity + PV maturity value = \$(614.46 + 385.54) = \$1000

• What would be the value of the bond described in part d if, just after it had been issued, the expected inflation rate rose by 3 percentage points, causing investorsto require a 13% return? Would we now have a discount or a premium bond?

With rate of 13% , after the rate rose by 3% from 10%, the bond value in this case by use of financial calculator will be:

VB(10‑YR) =\$100(PVIFA13%,10)+\$1,000(PVIF13%,10)

= \$100 ((1-  1/(1+0.13)10)/0.13) + \$1,000 (1/(1+0.13)10)

=\$542.62+\$294.59=\$837.21.

Based on this result, the change of rate (increase) resulted to a decline in the value of bond. The rate went above the coupon rate resulted to decline of the bond value below the initial bond value. Thus we will have a discount

• What would happen to the bond’s value if inflation fell and rd declined to 7%?Would we now have a premium or a discount bond?

The coupon rate is reduced from 13% to 7%. The nee bond value is computed as follows;

VB(10‑YR) =\$100(PVIFA7%,10)+\$1,000(PVIF7%,10)

= \$100 ((1-  1/(1+0.07)10)/0.07) + \$1,000 (1/(1+0.07)10)

=\$702.36+\$508.35=\$1,210.71.

Based on the computation, the bond value rises above the par value with the reduction of the rate below the initial coupon rate. In this case we will have a premium.