Premium & Discount Bonds

A bond sells at a premium (P > C) when its coupon rate beats the yield, and at a discount (P < C) when it falls short — and either gap is written off toward C by maturity.

By the end you'll be able to tell whether a bond prices above, at, or below its redemption value from the coupon-vs-yield comparison alone, compute the premium or discount amount, and describe how book value glides from the purchase price to C.

Predict: if the coupon rate is above the yield, does the bond sell above or below its redemption value? Drag the coupon rate through the yield below and check.

This is a 1000 par 10-year bond (F = C = 1000, n = 10). Drag r and i and watch the price cross the redemption line — above it's a premium, below a discount, exactly at it's par. The curve is the book value at every year from purchase to maturity: it bends down toward C for a premium bond and bends up toward C for a discount bond, always landing exactly on C at year 10.

P = 1147.20 — premium of 147.20 over C = 1000

r = 8.0% > i = 6.0% → premium (P > C)

Book value glide path — price at year 0 gliding to C at year 10
book value B_t redemption value C

Rewrite the bond price formula so the redemption value C stands alone, and the sign of one bracket tells you everything about whether the bond costs more or less than C.

Intuitive

If a bond's coupon pays more than what the market currently demands, it's worth more than its redemption value C — so you pay extra today to own those above-market coupons, and that extra (the premium) erodes away year by year until the book value lands exactly on C at maturity. A discount bond is the mirror image: its coupon falls short of the market rate, so you pay less than C up front, and that gap closes the other way — the book value climbs up to C — because the missing coupon income has to be made up somehow, and it's made up by that built-in gain.

Formal

Substituting \(C\cdot v^n = C - Ci\cdot a_n\) into \(P = Fr\cdot a_n + C\cdot v^n\) collapses it to the premium/discount form: \(P = C + (Fr - Ci)\cdot a_n\), where \(a_n\) is always positive. So the sign of \((Fr - Ci)\) decides everything: if the coupon Fr exceeds the interest Ci the market demands on C, the bond sells at a premium (P > C); if it falls short, it sells at a discount (P < C). When C = F, this is just \(r\) against \(i\): \(P - F = F(r-i)\cdot a_n\), so premium ⇔ r > i and discount ⇔ r < i.

Applied

Think of it as prepaying for rich coupons: if a bond's coupons pay more than the going rate, you pay extra today to buy them, and that overpayment must vanish by maturity since only C comes back at the end — so the book value glides down to C. A discount bond is the mirror image: you underpay today, and that built-in gain must be earned back up to C by maturity, so the book value climbs.

Worked example

F = C = 1000, r = 8%, n = 10, i = 6% (the bond from bond pricing): P = 1147.20. Since r = 8% > i = 6%, this is a premium bond, and premium = P − C = 1147.20 − 1000 = 147.20, which the amortization schedule writes off over the ten years so the final book value is exactly 1000.

Your turn

Same shape, different rates: F = C = 1000, r = 5%, n = 10, i = 7%. At 7%, \(a_{10} \approx 7.0236\), so \(P = 1000 + (50 - 70)(7.0236) = 1000 - 140.47 = 859.53\). Since r = 5% < i = 7%, this is a discount bond. Now finish it: discount = C − P = 1000 − 859.53 = ____

Reveal the answer

discount = 1000 − 859.53 = 140.47. Set r = 5% and i = 7% on the slider above and check the readout against this figure, and watch the book-value curve climb from 859.53 up to 1000 over the ten years.

More info — why the glide path bends the way it does

At any point in the bond's life, the book value is just the present value of the remaining cash flows at the original yield i: \(B_t = Fr\cdot a_{n-t} + C\cdot v^{n-t}\). As \(t\) grows, the remaining term \(n-t\) shrinks, so \(B_t\) moves toward \(B_n = C\) — but not at a constant speed. A premium bond's book value falls fastest early on and flattens as it nears C (concave), while a discount bond's book value climbs slowly at first and accelerates near the end (convex) — the same asymmetry you can see by dragging r and i apart or together in the diagram above. See the thismatter.com link in Dive deeper below for more worked amortization tables.

Check your understanding

Question 1 of 4

A 1000 par bond pays 7% annual coupons and is priced to yield 5%. Is it a premium or discount bond, and why?

Question 2 of 4

A 1000 par bond pays 6% annual coupons for 20 years and is priced to yield 8%. Using \(P = C + (Fr - Ci)\cdot a_n\) with \(a_{20}\) at 8% ≈ 9.8181, which is closest to the price?

Question 3 of 4

If a bond's coupon rate equals its yield rate, what is true about its price and book value over time?

Question 4 of 4

A 1000 par bond pays 9% annual coupons for 15 years and is priced to yield 9%. Using \(P = Fr\cdot a_n + C\cdot v^n\), what is the price, and is it a premium or discount bond?

Recap

  • Premium/discount form: \(P = C + (Fr - Ci)\cdot a_n\); the sign of \((Fr-Ci)\) decides everything.
  • Premium (P > C) when the coupon Fr exceeds Ci; discount (P < C) when it falls short; par (P = C) when they're equal.
  • When C = F, this is simply premium ⇔ r > i and discount ⇔ r < i.
  • Book value glides from the purchase price to C at maturity — concave for a premium bond, convex for a discount bond.

Dive deeper

Sources

  • Premium and Discount Bonds