Bond Calculations

Given "$1,000 par, 7% semi-annual, 15 years, book value $1,042", find the yield, future book value, and next write-down. Each is a different rearrangement of the same pricing equation.

HIGH-FREQUENCY: Given partial bond information, computing the rest is one of the most common FM question types.

$$P=Fr\cdot a_{\overline{n|i}}+C\cdot v^n=C+(Fr-Ci)\cdot a_{\overline{n|i}}$$

No closed form. The bond salesman's approximation:

$$i\approx \frac{Fr+(C-P)/n}{(P+C)/2}$$

This is "average income over average investment." Gets within 10 basis points, enough to pick the answer choice.

$$r=\frac{P-C\cdot v^n}{F\cdot a_{\overline{n|i}}}$$

From the premium/discount formula:

$$a_{\overline{n|i}}=\frac{P-C}{Fr-Ci}$$

Then solve for $n$ using $v^n=1-i\cdot a_{\overline{n|}}$.

The syllabus also asks for the point in time a bond reaches a stated book value. Book value obeys the same premium/discount formula, just with the remaining term $n-t$:

Common mistakes

• Confusing $g$ with $r$ in Makeham's. $g=Fr/C$, not $r$. When $C\ne F$: $g\ne r$. Trap: using $r$ in Makeham's.
• Wrong exponent for write-down. Period $t$ uses $v^{n-t+1}$, not $v^t$. For period 9 of a 20-coupon bond: exponent = 12, not 9. Trap: 8.01 instead of 7.44.
• Using market yield for book value. BV uses original purchase yield. Current market yield gives market value. Different questions.

Bottom line

• Six parameters $P,F,C,r,i,n$ linked by the price formula. Yield needs iteration; $r$, $n$, $C$ have closed forms.
• Yield: no closed form. Bond salesman's formula $i\approx (Fr+(C-P)/n)/((P+C)/2)$, "average income over average investment," then verify.
• Price linear in coupon: each 1% of coupon adds $F(0.01)a_{\overline{n|}}$; a coupon equal to the yield prices at par $(P=C)$.
• Book value: $B{V}_t=C+(Fr-Ci)\cdot a_{\overline{n-t|i}}$. Uses the original purchase yield, not the current market yield.

Exam shortcut

Compute the bond salesman's approximation first. 15 seconds, usually identifies the answer. Then verify by pricing at that rate. Two close choices? Bracket by pricing at both. "Price = Annuity + Bullet." "Premium Pays Down, Discount Digs Up." Closed-form amortization has the same geometric structure as loans: $(Fr-Ci)\cdot v^{n-t+1}$.

The full lesson (about 2,651 words, 18 min read) adds 3 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 4b

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