Bond Amortization: Book Value and Premium/Discount Schedules
Study bond amortization schedules and book value calculations for Exam FM.
Book Value of a Bond
The book value of a bond at time t (just after the t-th coupon) is B_t = Fr * a-angle-(n-t) + C * v^(n-t), computed at the yield rate i. The book value starts at the purchase price B_0 = P and ends at B_n = C (the redemption value). Alternatively, using the premium/discount form, B_t = C + C(g - i) * a-angle-(n-t).
The book value represents the theoretical value of the bond at time t if the yield rate remains constant. It moves smoothly from P toward C, which is the basis for bond amortization accounting. This smooth progression contrasts with the market value, which fluctuates as market yields change.
Example: A $1,000 par bond with a 5% semiannual coupon rate (Fr = $25 per period) has 10 years remaining and a yield of 4% per period. The book value just after the 6th coupon is B_6 = 25 * a-angle-14 at 4% + 1000 * v^14 at 4% = 25 * 10.5631 + 1000 * 0.5775 = 264.08 + 577.47 = $841.55.
Premium Amortization
For a premium bond (P > C), the book value decreases over time toward the redemption value. The "amortization of premium" in period t is the excess of the coupon over the interest earned on the book value: P_t = Fr - i * B_{t-1}. Since the coupon exceeds the interest earned for a premium bond, the difference reduces the book value.
The premium amortization amounts form a geometric sequence increasing by factor (1+i): P_t = P_1 * (1+i)^(t-1). The first period's amortization is P_1 = C(g - i) * v^n * (1+i)^0 = C(g - i) * v^n. This means later periods have larger write-downs than earlier periods, even though the book value is decreasing. The total premium amortized over the life of the bond equals P - C.
Example: A bond with P = $1,100, C = $1,000, g = 0.05, i = 0.04, and n = 20 has P_1 = 1000(0.05 - 0.04) * v^20 at 4% = 10 * 0.4564 = $4.56. The premium amortization in period 10 is P_10 = 4.56 * (1.04)^9 = $6.50.
Discount Accumulation
For a discount bond (P < C), the book value increases over time toward the redemption value. The "accumulation of discount" in period t is the excess of interest earned over the coupon: A_t = i * B_{t-1} - Fr. The book value increases by A_t each period.
The discount accumulation amounts also form a geometric sequence: A_t = A_1 * (1+i)^(t-1), where A_1 = C(i - g) * v^n. The total discount accumulated over the life of the bond equals C - P. Notice that the formulas for premium amortization and discount accumulation are essentially the same, with the sign reversed.
Income Reporting and Accounting
Bond amortization matters for accounting and tax purposes. Under the scientific (or effective interest) method used on Exam FM, the investment income reported in period t is i * B_{t-1}, regardless of whether the bond was purchased at a premium or discount. For a premium bond, income = Fr - P_t (the coupon minus the premium write-down). For a discount bond, income = Fr + A_t (the coupon plus the discount write-up).
This approach ensures that the investment income reflects the actual yield earned on the carrying amount. The sum of all investment income over the bond's life equals the total coupons received plus (C - P), which is n*Fr + (C - P). If the bond was purchased at a discount, C - P is positive, so total income exceeds total coupons. If purchased at a premium, C - P is negative, so total income is less than total coupons.
Constructing a Bond Amortization Schedule
A complete bond amortization schedule has columns for: period, coupon payment, interest earned, premium amortization (or discount accumulation), and book value. Each row is computed sequentially: interest earned = i * previous book value, amortization = coupon minus interest earned, new book value = previous book value minus amortization.
For a premium bond with P = $1,050, C = $1,000, Fr = $40, i = 3.5%, n = 5: Period 1 has interest = 0.035 * 1050 = $36.75, amortization = 40 - 36.75 = $3.25, and B_1 = 1050 - 3.25 = $1,046.75. Period 2 has interest = 0.035 * 1046.75 = $36.64, amortization = 40 - 36.64 = $3.36, and B_2 = 1046.75 - 3.36 = $1,043.39. The amortization increases slightly each period by the factor (1+i) = 1.035.
Study Tips and Common Pitfalls
The most common mistake is confusing the coupon amount with the interest earned. The coupon Fr is a fixed cash flow; the interest earned i * B_{t-1} changes each period as the book value changes. Another frequent error is computing the book value at the wrong time. B_t is the book value just after coupon t is paid, not before.
On Exam FM, you rarely need to build the entire schedule. Instead, use the direct formulas: B_t = Fr * a-angle-(n-t) + C * v^(n-t) for book value, and P_t = C(g-i) * v^(n-t+1) * (1+i)^(t-1) for the premium amortization in a specific period. Practice finding individual entries without constructing the full table, as this saves considerable time on the exam.