Valuing Bonds

The value of a bond is obtained by discounting the bond's expected cash flows to the present using an appropriate discount rate. Yield to maturity is the discount rate at which the sum of all future cash flows from the bond is equal to the price of the bond. "Time to maturity" refers to the length of time before the par value of a bond must be returned to the bondholder. This section will show you how to calculate a bond's yield to maturity and calculate the price of a bond.

Impact of Payment Frequency on Bond Prices

Payment frequency can be annual, semi annual, quarterly, or monthly; the more frequently a bond makes coupon payments, the higher the bond price.


LEARNING OBJECTIVE

  • Calculate the price of a bond

KEY POINTS

    • Payment frequency can be annual, semi annual, quarterly, monthly, weekly, daily, or continuous.
    • Bond price is the sum of the present value of face value paid back at maturity and the present value of an annuity of coupon payments. The present value of face value received at maturity is the same. However, the present values of annuities of coupon payments vary among payment frequencies.
    • The more frequent a bond makes coupon payments, the higher the bond price, given equal coupon, par, and face.

TERM

  • annuity

    A specified income payable at stated intervals for a fixed or a contingent period, often for the recipient's life, in consideration of a stipulated premium paid either in prior installment payments or in a single payment. For example, a retirement annuity paid to a public officer following his or her retirement.

The payment schedule of financial instruments defines the dates at which payments are made by one party to another on, for example, a bond or a derivative. It can be either customised or parameterized. Payment frequency can be annual, semi annual, quarterly, monthly, weekly, daily, or continuous.

Bond prices is the present value of all coupon payments and the face value paid at maturity. The formula to calculate bond prices:

P=\left(\frac{C}{1+i}+\frac{C}{(1+i)^{2}}+\ldots+\frac{C}{(1+i)^{N}}\right)+\frac{M}{(1+i)^{N}}

=\left(\sum_{n=1}^{N} \frac{C}{(1+i)^{n}}\right)+\frac{M}{(1+i)^{N}}

=C\left(\frac{1-(1+i)^{-N}}{i}\right)+M(1+i)^{-N}

Bond price formula. Bond price is the present value of all coupon payments and the face value paid at maturity.

F = face value, i_F = contractual interest rate, C = F * i_F = coupon payment (periodic interest payment), N = number of payments, i = market interest rate, or required yield, or observed / appropriate yield to maturity, M = value at maturity, usually equals face value, P = market price of bond.

In other words, bond price is the sum of the present value of face value paid back at maturity and the present value of an annuity of coupon payments. For bonds of different payment frequencies, the present value of face value received at maturity is the same. However, the present values of annuities of coupon payments vary among payment frequencies.

The present value of an annuity is the value of a stream of payments, discounted by the interest rate to account for the payments are being made at various moments in the future. The formula is:

a_{\bar{n} \mid i}=\frac{1-(1+i)^{-n}}{i},

Annuity formula: The formula to calculate PV of annuities.

Where n is the number of terms or number of payments n =1 (annually), n = 2 (semi-annually), n = 4 (quarterly)... and i is the per period interest rate.

According to the formula, the greater n, the greater the present value of the annuity (coupon payments). To put it differently, the more frequent a bond makes coupon payments, the higher the bond price.