Pricing Coupon Bonds with Different Coupon and Yield Frequencies

Summary

The document explains how to discount a fixed-coupon bond when coupons are paid more often than once a year. It distinguishes the coupon cash flow per payment from the annual coupon amount, then shows that each payment must be discounted using a rate and time period consistent with the yield’s compounding convention. The example considers a five-year bond with semiannual coupons and an annualized yield, assuming settlement on a coupon date.

If yield is compounded semiannually, the annual yield is divided across two periods and the ten coupon payments are discounted at those period intervals. If yield is compounded annually, payments are instead discounted using fractional-year exponents. The document gives formulas for both conventions but does not calculate a final price. Its method assumes fixed cash flows and coupon-date settlement; other settlement dates or yield conventions may require additional adjustments.

Key ideas

• Use the coupon amount per payment period as each cash flow in the bond valuation.
• Match the discounting intervals to the yield’s compounding convention.
• A semiannual-compounded annual yield is converted to a per-period rate for semiannual payments.
• With annual compounding, discount semiannual cash flows using fractional-year exponents.
• The formulas assume settlement on a coupon date.

Tags

# Compute bond price with more coupon payments in a year


# Compute bond price with more coupon payments in a year












If I have a 5-year bond, which pays every six months a coupon of 2.5% with a yield of 1.5%, should I split up the yield to compute the bond price?

Or is below the way to compute it?

$\displaystyle PV = \frac{2.5}{1.015} + \frac{2.5}{1.015^2} + ... + \frac{100 + 2.5}{1.015^{10}}$

## Answer by Helin (score 1, accepted)

https://quant.stackexchange.com/a/17387

The general bond pricing formula for fixed-coupon bonds, assuming settlement on a coupon date, is as follows:

$$ P = \sum_{i=1}^N \frac{c/f}{(1 + y/n)^{nt}}, $$ where $c$ is the size of the cash flow, $f$ is the coupon frequency per year, $y$ is the annualized yield, and $n$ is the compounding frequency per year.

In your case, $c$ should be $2.5/2=1.25$. Assuming yield is also semi-annually compounded, then it should be $$ PV = \frac{1.25}{(1 + 0.75\%)} + \frac{1.25}{(1 + 0.75\%)^2} + \cdots $$

If yield is annually compounded, however, this would become $$ PV = \frac{1.25}{(1 + 1.5\%)^{0.5}} + \frac{1.25}{(1 + 1.5\%)^1} + \cdots $$

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