Why to 2 methods to calculate bond price with semi annual return give different answers?

Question

I am confused as 2 methods give different answers. The difference lies in the "to the power" numbers for discounting.

Example: 2 year semi annual bond (4 periods), $1m annual Coupon Payment, 5% Yield (forget about repayment of principal for simplicity)

Standard formula for discounting coupon payments:

[0.5 * $1m / (1.025)^1] + [0.5 * $1m / (1.025)^2] + [0.5 * $1m / (1.025)^3] + [0.5 * $1m / (1.025)^4] $

ie 4 payments, all discounted with n going from 1 to 4, and cash flow and yield have been halved to £0.5m and 2.5%.

BUT

See Erik Banks' book: "Finance the basics" bottom of page 70 under formula 3.7. It is in Googlebooks: type "time horizon divided by fractional period" into Google

Everything is the same except the values of n (the power that 1.025 is raised). In his version, instead of being 1 to 4 respectively, n is 1/2, 2/2, 3/2, 4/2

So each n is divided by 2, because there are 2 payments each year.

Answer 1

TL;DR: Looks like the book is crap. Switching from annual to semi-annual compounding should give a rather small change in PV.

A payment of 1 at time T, with an annually compounded rate r, is worth $1/(1+r)^T$. With a semi-annual compounded rate s, it is $1/(1+s/2)^{2T}$, which incidentally is $1/(1+s+s^2/4)^T$, so the same as annually compounded to first order (which is just what we want). (While continuously compounded would give you $1/(1+c+c^2/2 + c^3/6 + c^4/24 + \cdots)^T$, again the same to first order).

Now, you have $s = 0.05$ and 4 payments at times $T = 1/2, 1, 3/2, 2$; thus it is $1/1.025^1 + 1/1.025^2 + 1/1.025^3 + 1/1.025^4$.

In this case, with an annual compounded rate of 5% and an annual coupon of \$1m the PV is \$1,849,510; with a semi-annually compounded rate of 5% and a semi-annual coupon of \$0.5m the PV should be \$1,880,987, and not \$1,939,394 as the book appears to be claiming.

(FWIW, what the book actually computed was a semi-annual coupon of \$0.5m discounted with an annually compounded rate of 2.5%, which is perfectly fine per se, but probably not what the author had in mind.)

Answer 2

The method for computing yields/prices for US Treasury bonds is highly standardized in the marketplace. It uses semi-annual compounding, like your formula does (as well as specific day count conventions, etc). For the book, annual interest rates are apparently being used; resulting in an unacceptable (wrong) answer as far as bond trading is concerned. (However, for one's own analysis of a problem everyone can of course use the method they prefer, or even make up a new one). For trading the standards of the Securities Industry Association should be followed.