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Solving for the Coupon Rate That Prices a Bond at Par

Article Quant Q&A · Author: Roozbe
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Summary

The document explains how to find the fixed coupon rate that makes a coupon bond's value at its starting date equal its face value. Under evenly spaced payment dates, each coupon is the rate multiplied by the accrual interval and face value. The bond price is the present value of principal at maturity plus the present values of all coupon payments, using discount factors for each payment date.

Setting that price equal to face value and rearranging gives the par coupon as the difference between one and the maturity discount factor, divided by the accrual interval times the sum of discount factors for coupon dates. The note also mentions yield to maturity as an alternative route and continuous-compounding yield as a quick approximation. Its derivation assumes equal spacing and a fixed coupon; it does not discuss day-count conventions, irregular schedules, or market-specific discounting details.

Key ideas

  • A bond's value is the discounted principal repayment plus the discounted coupon cash flows.
  • For equally spaced coupons, each payment equals face value times the coupon rate times the accrual interval.
  • Setting the bond price equal to face value yields a par coupon from the discount factors.
  • Yield to maturity provides another route to relate coupon rate and par pricing.
  • The derivation assumes regular payment spacing and omits irregular schedules and day-count details.

Tags

Fixed incomeDerivatives pricing
Full text
# I want to prove Determine the coupon rate $r$, such that the price of the bond, at $T_0$, equals its face value


# I want to prove Determine the coupon rate $r$, such that the price of the bond, at $T_0$, equals its face value












Consider a coupon bond, starting at $T_{0}$ , with face value $K$, coupon payments at $T_1, . . . , T_n$ and a fixed coupon rate $r$. Determine the coupon rate $r$, such that the price of the bond, at $T_0$, equals its face value.

## Answer by user16891 (score 0, accepted)

https://quant.stackexchange.com/a/19236

For simplicity,we let \begin{align} &\delta=\frac{T_n-T_0}{n}\\ &T_i=T_0+i\delta, \end{align} for $i=1,2,...,n$ we have $$c_i=r\,\delta\,K.$$ The price, $p(t)$ at a time $t < T_1$, of the coupon bond is given by $$p(t)=KP(t,T_{n})+\sum_{i=1}^{n}c_i P(t,T_{i}),$$ we know the price of the bond, at $T_0$, equals its face value,thus we have $$p(T_0)=K=KP(T_0,T_{n})+r\,\delta\,K\sum_{i=1}^{n} P(T_0,T_{i}),$$ then $$r=\frac{1-P(T_0,T_{n})}{\delta\sum_{i=1}^{n} P(T_0,T_{i})}.$$ For more details, you can see this link

## Answer by Richi Wa (score 1)

https://quant.stackexchange.com/a/15357

Are you familiar with the concept of yield-to-maturity (YTM)? Here you find all necessary steps. You first calculate using the current price and the cashflows. Then as you can see in the paper provided a bond with coupon rate equal to its YTM is priced at par (100) and thus the price equals its face value.

## Answer by Drew (score 0)

https://quant.stackexchange.com/a/15362

I always thought doing $y(t,T) = \frac{-ln(P(t,T))}{T-t}$ was a quick good approximation, it applies when the bond price is calculated in continuous time

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.