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Premium, discount and amortisation of premium

The gap between price and redemption value is written off over the bond's life; the write-down each period is a geometric series like a loan's principal repayments.

The formulas

Premium
P−C=(Fr−Ci) an‾∣iP - C = (Fr - Ci)\,a_{\overline{n}|i}
Write-down in coupon t
(Fr−Ci) v n−t+1(Fr - Ci)\,v^{\,n-t+1}
Interest earned in coupon t
It=i Bt−1I_t = i\,B_{t-1}
Book value recursion
Bt=Bt−1(1+i)−FrB_t = B_{t-1}(1+i) - Fr

Where it comes from

  1. Each coupon FrFr splits into interest actually earned, iBt−1iB_{t-1}, and a return of capital.
  2. For a premium bond the coupon exceeds the interest earned, so the excess (Fr−Ci)vn−t+1(Fr - Ci)v^{n-t+1} is written OFF the book value.
  3. For a discount bond the same expression is negative and the book value accumulates UP towards CC.

Worked example

For the 1,000 par 10-year 8% bond yielding 6%, find the amount of premium amortised in the 4th coupon.

  1. Fr−Ci=80−1,000(0.06)=20Fr - Ci = 80 - 1{,}000(0.06) = 20 per period.
  2. Write-down in coupon 4 is (Fr−Ci)v n−t+1=20 v7(Fr-Ci)v^{\,n-t+1} = 20\,v^{7}.
  3. v7=1.06−7=0.665057v^{7} = 1.06^{-7} = 0.665057, so the amortisation is 13.3013.30.
  4. Total premium is P−C=147.20P - C = 147.20, and the ten write-downs sum to exactly that.

Answer: 13.30

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

  • Write-downs grow by (1 + i) each period, exactly like loan principal repayments - same algebra, different name.
  • Premium bond: book value falls to C. Discount bond: book value rises to C. Both end at C.

Traps

  • Signing the write-down wrongly for a discount bond.
  • Using n − t rather than n − t + 1 in the exponent.

Related

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