Estimating Bond Yield by Linear Interpolation

Why Interpolate a Bond Yield?

A bond’s yield is the rate that makes the present value of its promised cash flows equal to its observed price. For a conventional bond, the price is a decreasing, nonlinear function of the yield. If two trial yields produce prices on opposite sides of the observed price, linear interpolation gives a quick estimate between them.

Interpolation is useful as a numerical estimate, not as a replacement for the bond’s yield equation. Its quality depends on the width of the trial-rate interval and on how much the price curve bends over that interval. Repricing the cash flows at the estimate provides a direct check of the remaining price error.

Definition: Suppose a bond’s observed price is $P$, and two trial yields $y_L<y_H$ produce prices $P_L>P_H$, with $P_L>P>P_H$. Linear interpolation estimates the yield by treating the price as changing linearly between those trial points: $$ \widehat y = y_L+ \frac{P_L-P}{P_L-P_H}(y_H-y_L). $$ The estimate $\widehat y$ lies between the trial yields. The trial prices must bracket the observed price; otherwise this formula does not provide an interpolation within the two trials.

Set Up the Trial Prices Consistently

Before interpolating, write the bond’s cash flows and establish the yield convention. On a coupon date, if a bond pays coupon $C$ at the end of each of $n$ coupon periods and redemption amount $R$ with the final coupon, its price at a coupon-period yield $j$ is

P(j)=C\sum_{k=1}^{n}(1+j)^{-k}+R(1+j)^{-n}.

This form assumes fixed promised cash flows, a common yield per coupon period, and valuation on a coupon date. It does not by itself describe a between-coupon settlement or irregular payment dates. For a nominal annual yield $i^{(m)}$ convertible $m$ times per year, use the coupon-period rate $j=i^{(m)}/m$ when coupon frequency is $m$. Do not compare an annual nominal quote directly with a price calculated using a per-period rate.

A practical process is to calculate both trial prices from the same cash flows, check that they straddle the observed price, interpolate in the same rate units, and then reprice at the estimate.

Choose trial yields.
Use the same yield convention for both trials and make sure the corresponding prices lie on opposite sides of the observed price.

Interpolate the rate.
Use the fraction of the total price difference represented by the difference between the higher trial price and the observed price.

Reprice and check.
Substitute the estimated yield into the bond’s price equation. Compare the result with the observed price, and report the price residual.

Why the Interpolation Formula Works

The formula is exact for a straight-line relationship between price and yield. Between two calculated points, the straight line changes by a fraction of its total price change. That fraction is $(P_L-P)/(P_L-P_H)$: it is zero at $P=P_L$ and one at $P=P_H$. Applying the same fraction to the yield interval gives the estimate.

\frac{\widehat y-y_L}{y_H-y_L} = \frac{P_L-P}{P_L-P_H} \quad\Longrightarrow\quad \widehat y = y_L+ \frac{P_L-P}{P_L-P_H}(y_H-y_L).

A conventional bond’s actual price function is not a straight line. With positive cash flows $C_k$ at periods $k$, its price is $P(j)=\sum_k C_k(1+j)^{-k}$. For $j>-1$, its second derivative is

P''(j)=\sum_k k(k+1)C_k(1+j)^{-k-2}>0.

Thus the price curve is convex as well as decreasing. The interpolated point is on the straight chord between the trial points, while the actual curve lies below that chord between them. Consequently, for this standard positive-cash-flow model, repricing at an interior interpolated yield will generally give a price below the observed price. The size of the difference depends on the interval and the bond’s cash flows; it should be checked rather than assumed negligible.

Worked Example: Interpolating an Annual-Coupon Bond Yield

A hypothetical bond pays an annual coupon of $70 for three years and then redeems at $1,000 with its final coupon. It is priced on a coupon date for $950. Find an interpolated annual effective yield using trial yields of 8% and 10%, then reprice at the estimate. Assume the promised cash flows are fixed and there is no default.

Method and trial prices. At an annual effective yield $i$, the price is $70(v+v^2+v^3)+1{,}000v^3$, where $v=1/(1+i)$. At 8%, the discount factors are approximately $0.92592593$, $0.85733882$, and $0.79383224$. Therefore,

P_L = 70(0.92592593+0.85733882+0.79383224) +1{,}000(0.79383224) \approx \$974.23.

At 10%, the factors are approximately $0.90909091$, $0.82644628$, and $0.75131480$, giving

P_H = 70(0.90909091+0.82644628+0.75131480) +1{,}000(0.75131480) \approx \$925.39.

The prices bracket $950: $974.23>950>925.39$. Interpolating,

\widehat i = 8\%+ \frac{974.2290-950}{974.2290-925.3944}(10\%-8\%) \approx 8.9923\%.

Repricing check. Round the estimated yield to 8.992% for this check. Then $v=1/1.08992\approx0.91750$, $v^2\approx0.84180$, and $v^3\approx0.77235$. Substitution gives

70(0.91750+0.84180+0.77235)+1{,}000(0.77235) \approx \$949.57.

The residual, repriced amount minus observed price, is approximately $\$949.57-\$950.00=-\$0.43$. The interpolated yield is therefore a close estimate, but it does not reproduce the price exactly. The common mistake this check avoids is assuming that a straight-line estimate must solve the nonlinear yield equation exactly.

Worked Example: Measuring the Error Over a Wider Price Interval

A hypothetical two-year bond pays $50 annually and redeems at $1,000 with its final coupon. Its observed price is $1,000. Trial annual effective yields are 0% and 10%. Assume valuation on a coupon date and fixed promised cash flows.

Calculate the trial prices. At 0%, the discount factor is 1, so $P_L=50+50+1{,}000=\$1{,}100$. At 10%, $v=1/1.10$, $v^2=1/1.21$, and

P_H = 50\left(\frac{1}{1.10}+\frac{1}{1.21}\right) +\frac{1{,}000}{1.21} \approx \$913.22.

The observed price is bracketed. The fraction of the price interval from the 0% price down to $1,000 is $100/(1{,}100-913.2231)\approx0.53540$. Thus,

\widehat i=0\%+0.53540(10\%-0\%)\approx5.354\%.

Reprice. At 5.354%, $v=1/1.05354\approx0.9491808$ and $v^2\approx0.9009442$. The price is

50(0.9491808)+1{,}050(0.9009442) \approx \$993.45.

The residual is approximately $-\$6.55$. This larger error than in the preceding example is consistent with using a wide trial interval: the straight chord is only an approximation to the curved price function. The estimate is between the two trial yields, but the repricing check shows it is not sufficiently accurate if a price close to $1,000 is required.

Worked Example: Interpolating a Nominal Semiannual Yield

A hypothetical bond has three remaining semiannual periods. It pays $40 each period and redeems at $1,000 with its final coupon. Its observed price is $1,042.9819. Trial quotes are nominal annual yields convertible semiannually of 4% and 6%. Assume settlement on a coupon date and fixed promised cash flows.

Convert the rates before pricing. The corresponding per-period rates are 2% and 3%. At 2%, the discount factors for periods one through three are approximately $0.98039216$, $0.96116878$, and $0.94232233$. Thus,

P_L = 40(0.98039216+0.96116878+0.94232233) +1{,}000(0.94232233) \approx \$1{,}057.6777.

At 3%, the factors are approximately $0.97087379$, $0.94259591$, and $0.91514166$, so

P_H = 40(0.97087379+0.94259591+0.91514166) +1{,}000(0.91514166) \approx \$1{,}028.2861.

The observed price is the midpoint of these trial prices to the displayed precision. Therefore the interpolation fraction is approximately $0.5$, and the estimated nominal annual yield is

\widehat{i}^{(2)} = 4\%+0.5(6\%-4\%) = 5.000\%.

Reprice using the matching period rate. A 5.000% nominal annual yield convertible semiannually gives $j=0.05/2=0.025$. At 2.5%, the period discount factors are approximately $0.97560976$, $0.95181440$, and $0.92859941$. Hence,

P(0.025) = 40(0.97560976+0.95181440+0.92859941) +1{,}000(0.92859941) \approx \$1{,}042.84.

The residual is approximately $-\$0.14$. Rates are quoted here to three decimal places, and prices to the nearest cent; the displayed trial prices and target are rounded. The key trap is to insert 5% as the rate for each half-year. The correct half-year rate is 2.5%.

Common Mistakes and Practical Checks

Reversing the price differences. For a conventional bond, the lower yield has the higher price. With $P_L>P>P_H$, the interpolation fraction is $(P_L-P)/(P_L-P_H)$, not $(P-P_H)/(P_L-P_H)$ unless the rate endpoint and corresponding expression are adjusted consistently.
Using trial rates that do not bracket the price. If both trial prices are above or both below the observed price, the formula is extrapolating, not interpolating. Choose a wider interval that brackets the target, or use a different numerical approach.
Mixing yield units. For a semiannual bond, a nominal annual yield convertible semiannually must be converted to a half-year rate for discounting. Interpolate in the quoted nominal annual rates if both trial rates and the desired answer use that convention.
Reporting the estimate without checking it. A plausible rate between the trials may still produce a material price residual. Reprice the original cash flows at the estimated rate, using sufficient precision in the intermediate factors.
Rounding too early. Keep trial prices and the interpolation fraction at several decimal places during calculation. Round the reported yield and repriced amount only at the end, and recognize that rounded displayed factors can affect the last cent.

Key takeaway: Linear interpolation uses the fraction of the bracketed price change to estimate the corresponding fraction of the yield interval. It is a quick estimate for a nonlinear bond price equation, so verify it by repricing with the correct coupon-period rate and checking the price residual.

Check Your Understanding

For each problem, identify the rate convention and use the supplied trial prices consistently. Treat all bonds as hypothetical, with fixed promised cash flows and valuation on a coupon date unless stated otherwise.

A bond’s price is $1,028. Trial annual effective yields of 4% and 6% give prices of $1,048 and $1,008, respectively. Estimate the yield by interpolation. (Answer: 5.000%)
Trial annual effective yields of 5% and 7% give prices of $1,012 and $972. The observed price is $990. Estimate the yield. (Answer: 6.100%)
A bond quoted on a nominal annual basis convertible semiannually has trial yields of 4% and 6%, with prices of $1,057.6777 and $1,028.2861. Its price is $1,042.9819. Estimate the nominal annual yield and state its corresponding half-year rate. (Answer: Approximately 5.000% nominal, convertible semiannually; the half-year rate is approximately 2.500%)
Trial yields of 3% and 9% give prices of $1,100 and $980. The observed price is $1,040. What is the interpolated yield, and does that guarantee that repricing at the estimate gives exactly $1,040? (Answer: 6.000%; no, interpolation estimates the yield from a straight line and need not solve the nonlinear price equation exactly.)
A three-year annual-coupon bond pays $50 annually and redeems at $1,000 with its final coupon. Its observed price is $1,000. The trial annual effective yields are 0% and 10%; their prices are $1,150 and approximately $875.6576. Find the interpolated yield to three decimals. (Answer: Approximately 5.467%)

What You'll Learn