Derivation of Treasury Bond Price

By FRANTZ

The Future Value, "FV", of an investment is given as FV = PV * g as per Interest Rate Equations

Therefore the Present Value, "PV" is PV = FV / g

Treasury bonds as investments have the following relatively unique properties:

1. They're always zero-coupon

2. They will always be paid

That means

1. Interest is non-compounding and exists only for one period (i.e. 1, 3, 5, 10, or 30 years). So, "g", the interest rate factor can be written as g = (1 + r)^n as per Interest Rate Equations, where "r" is the interest rate and "n" is the number of periods, i.e. n = 1.

2. For any given treasury bond, Future Value FV is a constant; there's no probabilistic uncertainty, i.e. FV has a 0% chance of changing

The above two points means that the Present Value, the price, of any given bond is always PV = FV / (1 + r) where "FV" is the future payout of the bond and "r" is its interest rate aka effective yield.

e.g. if a bond price pays 3000$ at maturity with a 3% effective yield, its price is 3000$ / (1 + 0.03) = ~2913$

The price of a bond changes over time due to the effective yield changing; The yield to maturity (YTM) naturally shrinks over time as time moves forward towards the maturity date, and r = YTM.

In other words r -> 0 as time gets closer to the maturity date of the bond.

Looking again at our price equation, PV = FV / (1 + r), we can see that PV -> FV as r -> 0.

So it's also implied that PV -> FV as time gets closer to the bond's maturity date.

However, the effective yield becomes more complicated once you introduce free-market mechanisms; If 3-year bonds at 3% were being sold by the Treasury yesterday, and today they're selling 3.002737...-year bonds at 1.5%, the price of yesterday's bonds will jump in the secondary market.

This price jump in turn impacts the effective yield, and that impact can be derived as follows:

1. Take the Future Value equation FV = PV *g and rewrite it describe the growth factor as a function of present value: g = FV / PV

2. If g = FV / PV, then (1 + r) = FV / PV, and therefore the yield to maturity, "r" , is r = (FV / PV) - 1

Therefore the effective yield changes due to price changes in PV that are introduced through market competition.

In a monopolistic non-market scenario, over time, the effective yield shrinks linearly and the price grows logarithmically until it reaches FV and the bond expires.

Side-note: "Future Value" is usually called "Face Value" in the bond market