Premium and Discount Amortization Schedules

A bond bought at $1,074 must be worth exactly $1,000 at redemption. The amortization schedule is the bridge: each coupon period writes the book value a little closer to par, on a path fixed entirely by the yield.

Each row splits the coupon into interest earned and a book-value adjustment:

$${\text{Interest}}_t=i\cdot B{V}_{t-1},{\text{Adjustment}}_t=Fr-i\cdot B{V}_{t-1}=(Fr-Ci)v^{n-t+1}$$

For a premium bond $Fr>Ci$, the adjustment is a positive write-down and book value falls. For a discount bond $Fr<Ci$, the coupon falls short of the required interest, so the shortfall is a write-up and book value rises. Either way the adjustments form a geometric sequence with ratio $(1+i)$ and sum to $|P-C|$, the total premium or discount.

HIGH-FREQUENCY: Book value uses the ORIGINAL purchase yield for the whole schedule, never the current market yield.

The exam often gives one row and asks for a parameter. Because the adjustment obeys $(Fr-Ci)v^{n-t+1}$, a single known write-down pins $Fr-Ci$, and from there the coupon, the price, or...

Common mistakes

• Using the current market yield for book value. The schedule runs on the ORIGINAL purchase yield throughout. The market yield gives market value, a different number for a different question.
• Reversing the premium and discount direction. Premium book value FALLS to par (write-down); discount book value RISES to par (write-up). Getting the sign backwards sends book value the wrong way.
• Wrong exponent in the adjustment. Period $t$ uses $v^{n-t+1}$, not $v^t$. For year 6 of a 10-year bond the exponent is 5, not 6. Trap: an adjustment off by a factor of $(1+i)$.

Bottom line

• Premium writes down, discount writes up. Book value marches from the purchase price to the redemption value $C$.
• Each period: interest earned $=i\cdot B{V}_{t-1}$; amortization $=Fr-i\cdot B{V}_{t-1}=(Fr-Ci)v^{n-t+1}$.
• The adjustments are geometric with ratio $(1+i)$, and they sum to $|P-C|$.
• Book value any time: $B{V}_t=C+(Fr-Ci)a_{\overline{n-t|i}}$, at the original yield.

Exam shortcut

You rarely need the whole schedule. For one period, use $(Fr-Ci)v^{n-t+1}$ directly. For a partial-information problem, one known adjustment unlocks $Fr-Ci$, and everything else follows. "Premium pays down, discount digs up." "Original yield, always." "One row unlocks the bond" via the geometric adjustment.

The full lesson (about 1,969 words, 13 min read) adds 3 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 4a
• 4b
• 4c

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