Bond Pricing with a Market Discount Rate
Bond pricing is an application of discounted cash flow analysis that depends on a bond’s cash flow features and the rate (or rates) used for discounting.
The market discount rate is used in the time-value-of-money calculation to obtain the present value.
The market discount rate is the rate of return required by investors given the risk of the bond investment.
It is also called the required yield, or the required rate of return.
The present value (PV) calculation for each bond coupon cash flow (PMT) that occurs in t periods with a market discount rate of rper period should be familiar from an earlier time-value-of-money lesson:
PV(Bond coupon) = PMTt /(1 + r)t .
A general formula for calculating a bond price (PV) given the market discount rate on a coupon date:
PV = PMT1/(1 + r)1 + PMT2/(1 + r)2 + … + (PMTN + FVN)/(1 + r)N,
where FV is equal to the bond’s face value and N is the number of periods to maturity.
Yield-to-Maturity
The yield-to-maturity is the internal rate of return on the cash flows—the uniform interest rate such that when the future cash flows are discounted at that rate, the sum of the present values equals the price of the bond. It is an implied or observed single market discount rate.
A bond investor will earn a rate of return on a bond equal to its YTM if each of the following is true:
- The investor holds the bond to maturity.
- The issuer makes full coupon and principal payments on scheduled dates. The YTM is the promised yield—that is, assuming no issuer default.
- The investor reinvests all coupon payments at the same YTM.
Flat Price, Accrued Interest, and the Full Price
When a bond is priced between coupon payment dates, its price has two components: the flat price (PVFlat) and the accrued interest (AI). The sum of the parts is the full price (PVFull), which also is known as the invoice or “dirty” price.
PVFull = PVFlat + AI.
If the coupon period has T days between payment dates and t days have passed since the last payment, the accrued interest is calculated
where
- t = number of days from the prior coupon payment to the settlement date
- T = number of days in the coupon period
- t/T = fraction of coupon period that has passed since the prior payment
- PMT = coupon payment per period
The full price of a fixed-rate bond between coupon payments given the market discount rate per period (r) is the present value of future cash flows as of the trade settlement date,
simplified by multiplying both the numerator and denominator by the expression (1 + r)t/T.
