Bond pricing: premium, discount and book value

The basic formula

A bond is a level annuity of coupons plus a lump sum at redemption, so its price is the present value of the coupons plus the present value of the redemption, both at the yield rate. That is the whole model, and every other bond formula is an algebraic rearrangement of it.

Take a bond with face 1000, an annual coupon rate of 4 percent, twenty years to maturity and redemption at par. At a yield of 3 percent the price is 40 times the twenty-year annuity factor plus 1000 times the twenty-year discount factor, which comes to 1148.77. At a yield of 5 percent the same bond is worth 875.38. At a yield of 4 percent it is worth exactly 1000.

Those three numbers contain the entire premium-and-discount story. A bond sells above par when its coupon rate exceeds the yield the market demands, below par when it falls short, and at par when the two coincide.

Why a bond trades away from par

The coupon is fixed at issue and the yield is whatever the market currently requires. If the coupon pays more than the market rate, buyers compete for that surplus and bid the price above par; the premium is exactly the present value of the surplus coupon.

That gives the premium-and-discount formula: the price equals the redemption value plus the excess of the coupon over the yield-on-redemption, valued as an annuity. For the premium bond, the coupon of 40 exceeds 3 percent of 1000 by 10 a year, and 10 times the twenty-year annuity factor at 3 percent is 148.77 - the same premium the basic formula produced.

Read the other way, the discount bond pays 40 against a required 50, a shortfall of 10 a year, whose present value at 5 percent over twenty years is 124.62. The price is 1000 less that, or 875.38.

The two formulas answer slightly different questions, which is why both are worth having. The basic formula asks what the cash flows are worth; the premium-and-discount formula asks how far the price sits from redemption and why. When a question gives the premium and asks for the coupon rate, or gives the price and asks how much of it is premium, the second form answers directly what the first would require rearranging.

Book value and amortisation

Book value is the price of the same bond with the remaining coupons, computed at the original yield. It is the prospective balance of the bond in exactly the sense that outstanding balance is the prospective balance of a loan, and the two are the same computation.

For the premium bond, the book value falls from 1148.77 at purchase to 1143.24 after one coupon, 1085.30 after ten, 1009.71 after nineteen, and exactly 1000.00 at maturity. The amounts written off are 5.54, then 5.70, and by the twentieth coupon 9.71 - the write-off grows by a factor of one plus the yield each period, which is the same geometric pattern that governs the principal portion of a loan payment.

The discount bond runs the other way. Its book value accretes from 875.38 to 879.15 after one coupon, 922.78 after ten, 990.48 after nineteen, and 1000.00 at maturity. The accretion is 3.77 in the first period and 9.52 in the last, growing by the same factor.

It is worth saying plainly why the book value must land on the redemption value. At the last coupon there is exactly one payment left, so the book value is the redemption plus the final coupon, discounted one period at the yield; after that coupon is paid, only the redemption remains and its present value at time n is itself. Premium and discount are therefore not written off by convention but by arithmetic, and a schedule that misses par has an error in it.

Makeham and the callable rule

Makeham's formula prices the bond as the present value of the redemption plus the modified coupon rate over the yield, times the difference between redemption and that present value. It is worth knowing because it prices a bond without an annuity factor, which occasionally matters, and because it makes the premium visible as a ratio of rates.

For callable bonds, the rule follows directly from the two pictures above. If the bond is at a premium, the investor loses premium every period, so the worst case is the earliest call date; if it is at a discount, the investor accretes toward par, so the worst case is the latest. Price to the date that gives the lowest price and the answer is safe against whatever the issuer chooses.

For the premium bond above, calling after ten years gives a price of 1085.30, which is its ten-year book value and lower than the twenty-year price of 1148.77 - so a cautious buyer pays 1085.30 and is protected either way.

The checks that catch errors

Bond questions are arithmetic-heavy and the errors are almost always mechanical rather than conceptual. Four checks catch nearly all of them and take seconds.

• Compare the coupon rate with the yield before computing anything. If the coupon is higher the answer must exceed the redemption value, and if it is lower the answer must fall short. A premium price on a discount bond means a rate went into the wrong slot.
• Price the bond a second way. The basic formula and the premium-and-discount formula are algebraically identical, so agreement to the cent confirms the arithmetic and disagreement localises the slip.
• Check that the book value at maturity is the redemption value exactly. A schedule that does not land on it has an error in the yield or the coupon count.
• Match the period to the coupon frequency. A semiannual bond has twice the periods at half the nominal rates, and mixing an annual yield with semiannual coupons is the most common way to produce a plausible wrong answer.