19  Bond Amortization

For taxation and other accounting purposes, it may be necessary to determine the amount of interest received and principal returned in a bond coupon or redemption payment. This can be done by viewing the bond as a standard amortized loan.

Once a bond is purchased, we can use its original yield to keep track of its value. This yield is called the book yield, and the bond values based on it are book values.

The market yield of a bond can change during the life of the bond, and therefore the market value of the bond can change as well. This section deals with book values, not market values. The book values are useful for tracking the value of a bond within an accounting framework.

19.1 General Formulae

The initial book value is equal to the price at which the bond is purchased: \[BV_0 = P = Fr \times a_{\overline{n|}i} + Cv^n\] The book value at time \(t\) is equal the remaining cash flows, discounted at the book yield \(i\). \[\begin{align}BV_t &= Fr\times a_{\overline{n-t|}i} + Cv^{n-t} \qquad &\text{(Prospective)}\\ &= P\times (1+i)^t - Fr\times s_{\overline{t|}i} \qquad &\text{(Retrospective)}\end{align}\] If we know the book value at time \(t\) and want to find the book value at time \(t+k\), then \[BV_{t+k} = BV_t\times (1+i)^k - Fr\times s_{\overline{k|}i}\] The investment income, which can be reported on an income statement, is equal to the book value at the beginning of a period times the book yield: \[InvInc_t = BV_{t-1}\times i\]

Another way to calculate investment income is to add the coupon payment to the decrease in book value: \[InvInc_t = Fr + (BV_t - BV_{t - 1}) = Fr - PA_t \] where the decrease in book value \(PA_t\) is also called the amortization of premium: \[PA_t = BV_{t-1} - BV_t\] On the contrary, the increase in the bond’s book value is known as the accumulation of discount, denoted \(DA_t\) and:\[DA_t = BV_t - BV_{t-1} = - PA_t\]

19.2 Bond Amortization

Example: A 3-year bond makes semiannual coupon payments at an annual rate of \(6\%\) per year. The par value is \(\$100,000\). The yield is \(4\%\) per year, compounded semiannually. Prepare an amortization schedule for the bond.

Solution: Since the payments are made semiannually, the unit of time is 6 months: \[n = 3 \times 2 = 6 \qquad i = 4\% / 2 = 2\% \qquad r = 6\% / 2 = 3\%\] The price of the bond is: \[P = 100,000\times 3\% \times a_{\overline{12|}2\%} = 105,601.43\] Each row of the amortization table can be found with the following formula: \[\begin{matrix}& InvInc_t = BV_{t-1} \times i & \\ PA_t = Fr - InvInc_t & \text{or} & DA_t = InvInc_t - Fr \\ BV_t = BV_{t-1} - PA_t & \text{or} & BV_t = BV_{t-1} + DA_t\end{matrix}\] Since the bond is purchased at a premium, we have the following amortization schedule:

Year	Period \(t\)	Payment \(Fr\)	Interest Due \(I_t\)	Premium Amortized \(PA_t\)	End-of-period Book Value \(BV_t\)
0	0				105,601.43
0.5	1	3,000.00	2,112.03	887.97	104,713.46
1	2	3,000.00	2,094.27	905.73	103,807.73
1.5	3	3,000.00	2,076.15	923.85	102,883.88
2	4	3,000.00	2,057.68	942.32	101,941.56
2.5	5	3,000.00	2,038.83	961.17	100,980.39
3	6	3,000.00	2,019.61	980.39	100,000.00

The BA II can be used to find the amortization table:
6 [N] 2 [I/Y] 3,000 [PMT] 100,000 [FV] [CPT] [PV]
Result is -105,601.43. Price is 105,601.43.
[2nd] [AMORT] 1 [ENTER] ↓ 1 [ENTER] ↓
Pressing ↓ to see balance (Book Value), Interest (Investment Income), Principal (Premium Amortized). To get the next range of payments, press CPT while the worksheet is on P1 to make P1 and P2 automatically update.

Example: A 2-year bond makes semiannual coupon payments at an annual rate of \(2\%\) per year. The par value is \(\$1,000\). The yield is \(4\%\) per year, compounded semiannually. Prepare an amortization schedule for the bond.

Solution: The price of the bond is: \[P = BV_0 = 10\times a_{\overline{4|}2\%} + 1000 (1.02)^{-4} = 961.92\]

Since the bond is purchased at a discount (\(r < i\)), we have the following amortization schedule:

Year	Period \(t\)	Payment \(Fr\)	Interest Due \(I_t\)	Discount Amortized \(DA_t\)	End-of-period Book Value \(BV_t\)
0	0				961.92
0.5	1	10.00	19.24	9.24	971.16
1	2	10.00	19.42	9.42	980.58
1.5	3	10.00	19.61	9.61	990.20
2	4	10.00	19.80	9.80	1000.00