Bracketing Bond Yield with Bisection

Why Bracket a Bond Yield?

A bond’s yield is the rate that makes the present value of its promised cash flows equal its observed price. The yield equation is usually straightforward to construct, but it may not have a convenient algebraic solution. Bisection is a systematic numerical method: find two yields that produce prices on opposite sides of the observed price, then repeatedly test the midpoint of the remaining interval.

The method depends on matching the yield convention to the coupon schedule. For a bond with $n$ remaining coupon periods, coupon payment $C$ each period, redemption value $R$, and yield $j$ per coupon period, the price on a coupon date is

P(j)=C a_{\overline{n}|j}+R(1+j)^{-n} =C\frac{1-(1+j)^{-n}}{j}+R(1+j)^{-n}.

Here $P(j)$ is the calculated price at trial period yield $j$. If the quoted yield is nominal annual $i^{(m)}$, convertible at the coupon frequency $m$, then $j=i^{(m)}/m$. If the yield is an effective annual rate $i$ and there are $m$ coupon periods per year, then $j=(1+i)^{1/m}-1$. Use the conversion appropriate to the quote; do not treat an annual rate as though it were already a coupon-period rate.

Definition: A yield bracket $[y_L,y_U]$ for an observed bond price $P_{\text{obs}}$ is an interval whose endpoint yields produce prices on opposite sides of $P_{\text{obs}}$, or one endpoint produces exactly $P_{\text{obs}}$. For a conventional bond with positive cash flows, price decreases as yield increases. Thus a valid bracket satisfies $P(y_L)\ge P_{\text{obs}}\ge P(y_U)$, where the yields and price calculation use consistent compounding conventions.

Why Bisection Narrows the Yield Interval

For a conventional bond, each promised cash flow is positive and is discounted by a factor that decreases as the yield rises. For any payment $CF_k>0$ at period $k>0$, if $j_2>j_1>-1$, then

\frac{CF_k}{(1+j_2)^k}<\frac{CF_k}{(1+j_1)^k}.

Adding this inequality across the bond’s future payments shows that its calculated price at $j_2$ is lower than its price at $j_1$. The price is also a continuous function of the yield for $j>-1$. Therefore, if the two endpoint prices straddle the observed price, a yield matching that price lies between the endpoints. Each bisection step halves the interval while preserving this property.

Find two trial yields.
Calculate the bond price at a lower yield and a higher yield. Choose endpoints for which one price is at least the observed price and the other is at most the observed price.

Calculate the midpoint.
Set the trial yield equal to the average of the two endpoint yields. Convert it to a coupon-period rate if the quote requires conversion, then calculate the bond price.

Keep the half-interval that still brackets the price.
If the midpoint price is above the observed price, the yield must be higher, so replace the lower endpoint with the midpoint. If it is below, replace the upper endpoint with the midpoint.

Repeat and report the precision.
Continue until the interval is sufficiently narrow or a midpoint price rounds to the observed price. State whether the reported yield is an interval, an approximation, or an exact midpoint match.

If the initial interval has width $W$, then after $N$ bisections its width is $W/2^N$. To guarantee a final width no greater than a chosen tolerance $\varepsilon$, choose an integer $N$ such that

\frac{W}{2^N}\le \varepsilon.

This bound concerns the interval width, not the error in its midpoint. If the yield is known to lie in the final interval, its midpoint is at most half the interval width from the yield.

Worked Example: Starting With a Wide Bracket

A hypothetical bond has face value and redemption value $\$1{,}000$, eight remaining semiannual coupons of $\$35$, and an observed price of $\$1{,}040$ on a coupon date. Find a bracket using nominal annual yields convertible semiannually, then perform two bisection steps. State the method and the common mistake it avoids.

Method and rate convention. The coupon-period rate is half the nominal annual yield. Thus trial nominal yields of 4%, 5%, 5.5%, and 6% correspond to period rates of 2%, 2.5%, 2.75%, and 3%, respectively. For any trial period rate $j$,

P(j)=35\frac{1-(1+j)^{-8}}{j}+1{,}000(1+j)^{-8}.

At a 4% nominal annual yield, $j=0.02$. The discount factor is $(1.02)^{-8}\approx0.853490$, and the annuity factor is $(1-0.853490)/0.02\approx7.325500$. Thus the coupon contribution is $35(7.325500)=\$256.39$, the redemption contribution is $1{,}000(0.853490)=\$853.49$, and the calculated price is $\$1{,}109.88$. This is above $\$1{,}040$.

At a 6% nominal annual yield, $j=0.03$. The discount factor is $(1.03)^{-8}\approx0.789409$, and the annuity factor is $(1-0.789409)/0.03\approx7.019700$. The coupon contribution is $35(7.019700)=\$245.69$, and the redemption contribution is $\$789.41$. The price is $\$1{,}035.10$, below $\$1{,}040$. The initial bracket is therefore 4% to 6%.

First bisection. The midpoint nominal yield is 5%, so $j=0.025$. Using $(1.025)^{-8}\approx0.820747$, the annuity factor is $(1-0.820747)/0.025\approx7.170120$. The coupon contribution is $35(7.170120)=\$250.95$, and the redemption contribution is $\$820.75$. The total is $\$1{,}071.70$, above the observed price. The yield must be higher, so the bracket becomes 5% to 6%.

Second bisection. The midpoint of 5% and 6% is 5.5%, so $j=0.0275$. Using $(1.0275)^{-8}\approx0.804907$, the annuity factor is $(1-0.804907)/0.0275\approx7.094291$. The coupon contribution is $35(7.094291)=\$248.30$, and the redemption contribution is $\$804.91$. The total is $\$1{,}053.21$, still above $\$1{,}040$. The resulting bracket is 5.5% to 6%.

After two steps, the interval width is 0.5 percentage points, and its midpoint, 5.75%, is a rough estimate with a maximum bracket-based error of 0.25 percentage points. The calculation avoids the common mistake of using a nominal annual rate directly as a semiannual rate.

Worked Example: Reading the Price Difference in the Correct Direction

A hypothetical three-year bond pays annual coupons of $\$50$, has a redemption value of $\$1{,}000$, and is observed at $\$900$. Its yield is an effective annual rate. Begin with trial yields of 8% and 9%, then bisect twice. Name the method and identify the direction error to avoid.

For annual yield $i$, the price is

P(i)=50\frac{1-(1+i)^{-3}}{i}+1{,}000(1+i)^{-3}.

At 8%, $(1.08)^{-3}\approx0.793832$, and the annuity factor is $(1-0.793832)/0.08\approx2.577100$. The coupon contribution is approximately $\$128.86$, the redemption contribution is $\$793.83$, and the price is $\$922.69$, above $\$900$.

At 9%, $(1.09)^{-3}\approx0.772183$, and the annuity factor is approximately $(1-0.772183)/0.09=2.531300$. The coupon contribution is $\$126.57$, the redemption contribution is $\$772.18$, and the price is $\$898.75$, below $\$900$. Therefore, 8% to 9% is a valid bracket.

First bisection. At 8.5%, the discount factor is $(1.085)^{-3}\approx0.782908$, and the annuity factor is $(1-0.782908)/0.085\approx2.554022$. The coupon contribution is $50(2.554022)=\$127.70$, and the redemption contribution is $\$782.91$. The calculated price is $\$910.61$, above $\$900$; the bracket becomes 8.5% to 9%.

Second bisection. The midpoint is 8.75%. Here $(1.0875)^{-3}\approx0.777521$, and the annuity factor is $(1-0.777521)/0.0875\approx2.542617$. The coupon contribution is $50(2.542617)=\$127.13$, and the redemption contribution is $\$777.52$. The price is $\$904.65$, still above $\$900$. The narrowed bracket is 8.75% to 9%.

The common direction mistake is to lower the trial yield when the calculated price is too high. That is backwards: a higher yield lowers a conventional bond’s price. Keep the higher-yield half of the bracket when the midpoint price exceeds the observed price.

Worked Example: Bisection With a Semiannual Yield Quote

A hypothetical five-year bond has ten remaining semiannual coupons of $\$40$, a redemption value of $\$1{,}000$, and an observed coupon-date price of $\$1{,}050$. Find an initial bracket and perform one bisection step. The yield is quoted as nominal annual, convertible semiannually. State the method and the rate-unit trap it prevents.

The coupon-period rate is $j=i^{(2)}/2$, where $i^{(2)}$ is the nominal annual yield. The price at period rate $j$ is

P(j)=40\frac{1-(1+j)^{-10}}{j}+1{,}000(1+j)^{-10}.

At 5% nominal annual, $j=0.025$. The discount factor is $(1.025)^{-10}\approx0.781198$; the annuity factor is approximately $(1-0.781198)/0.025=8.752080$. The coupon contribution is $\$350.08$, the redemption contribution is $\$781.20$, and the price is $\$1{,}131.28$, above the observed price.

At 7% nominal annual, $j=0.035$. The discount factor is approximately $(1.035)^{-10}=0.708918$, and the annuity factor is $(1-0.708918)/0.035\approx8.316629$. The coupon contribution is $\$332.67$, and the redemption contribution is $\$708.92$, giving a price of $\$1{,}041.58$, below the observed price. The initial bracket is 5% to 7%.

The midpoint is 6% nominal annual, or $j=0.03$. Since $(1.03)^{-10}\approx0.744094$, the annuity factor is $(1-0.744094)/0.03\approx8.530200$. The coupon contribution is $\$341.21$, and the redemption contribution is $\$744.09$. The price is $\$1{,}085.30$, above $\$1{,}050$, so the narrowed bracket is 6% to 7%.

Bisection is applied to the quoted annual yield interval, but each price calculation uses the corresponding semiannual rate. Dividing by two is correct here because the quote is nominal annual convertible twice per year; it would not be the correct conversion for an effective annual yield.

Precision, Reporting, and Common Traps

Bisection gives a guaranteed narrowing of the interval when the price function is continuous and the initial endpoints bracket the observed price. It does not make the displayed midpoint exact unless its calculated price matches the observed price under the chosen precision. A clear report gives the final bracket and, if a single estimate is requested, identifies the midpoint and its resolution.

Situation	Correct interpretation
Midpoint price is above observed price	Increase the yield; replace the lower yield endpoint with the midpoint.
Midpoint price is below observed price	Decrease the yield; replace the upper yield endpoint with the midpoint.
Quoted yield is nominal annual, convertible $m$ times yearly	Use $i^{(m)}/m$ as the period rate when coupon frequency is $m$.
Quoted yield is effective annual, with $m$ coupon periods per year	Use $(1+i)^{1/m}-1$ as the period rate.
Midpoint price rounds to the observed price	Check the unrounded price or report the rate as an approximation at the stated precision.

A frequent exam trap is to compare prices correctly but update the wrong endpoint. Remember the inverse relationship: a price above the target calls for a higher yield; a price below the target calls for a lower yield. Another trap is to stop after reporting a midpoint without saying how wide the final interval is. The interval width provides a direct statement of the numerical resolution.

Key takeaway: Establish the price straddle first, calculate each midpoint using the correct coupon-period rate, and preserve the half-interval whose endpoint prices still straddle the observed price. The bracket, not an unsupported guess, is the central result of bisection.

Check Your Understanding

For each problem, keep the yield convention and coupon frequency consistent. Answers are given after the questions.

A hypothetical two-year bond pays annual coupons of $\$50$, redeems at $\$1{,}000$, and has an observed price of $\$1{,}000$. Starting with effective annual trial yields of 4% and 6%, calculate the midpoint price and state the resulting yield. (Answer: At 5%, the price is $\$1{,}000$; the yield is 5%.)
A bond’s nominal annual yield is convertible semiannually. The initial yield bracket is 4% to 12%. How many bisection steps are required to make the interval width at most 0.25 percentage points? (Answer: 5 steps, since $8/2^5=0.25$ percentage points.)
A hypothetical four-year bond pays annual coupons of $\$70$, redeems at $\$1{,}000$, and is priced at $\$970$. Starting with effective annual yields of 6% and 10%, use the first midpoint calculation to update the bracket. (Answer: At 8%, the price is about $\$966.88$, below $\$970$; the bracket becomes 6% to 8%.)
A hypothetical bond has six remaining semiannual coupons of $\$30$, redemption value $\$1{,}000$, and an observed price of $\$1{,}000$. Its yield quote is nominal annual, convertible semiannually, and the initial bracket is 4% to 8%. Calculate the midpoint price and state what happens to the bracket. (Answer: The midpoint is 6% nominal annual, so the period rate is 3%; the price is $\$1{,}000$, and the yield is exactly 6%, so the bracket collapses to that rate.)

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