What it means
Markets quote yields only for certain maturities, such as 1, 2, 5, 10 and 30 years. A bond that matures in 3 years, or a loan that ends in 7 and a half, falls between those points, so its rate has to be estimated.
Interpolation is the standard way of doing that. The simplest method is linear interpolation, which draws a straight line between the two closest observed points and reads the value off that line.
More advanced methods use smooth curves, which avoid sharp corners and can better match the typical shape of the market. Interpolated curves are used constantly in practice.
Banks use them to price loans and derivatives, companies use them to discount future payments to a present value, and accountants use them when valuing liabilities such as pensions and leases. Risk systems also rely on them to calculate profit and loss each day.
The choice of method matters for large exposures. A small difference in the estimated rate can change the present value of a long-dated stream of payments by a meaningful amount, so firms document which method they use and apply it consistently.
Data quality also matters, because the curve is only as reliable as the points it connects. If the quoted yields are from thinly traded bonds, the interpolated rate may reflect noise rather than a true market price.
The nuance is that interpolation is an estimate, not an observation. Between two actual points the real market rate might sit above or below the line, particularly when the curve bends sharply or the market is under stress.
In practice
Real-world examples.
Example
A bank is pricing a 7-year fixed-rate loan. The market provides yields for 5 and 10 years, so the bank interpolates between them to estimate a 7-year benchmark. It then adds a margin for credit risk.
Example
A pension fund values liabilities that fall due at many different dates. Its actuaries interpolate the yield curve to find a discount rate for each date. The total present value of those liabilities appears in the accounts, so the method has a direct effect on reported figures.
Example
A corporate treasurer is valuing an interest rate swap with a maturity of 4 years. Quoted swap rates exist for 3 and 5 years. The treasurer uses interpolation to estimate the 4-year rate and checks that the valuation system uses the same method. A mismatch could produce a valuation difference that looks like a real gain or loss.
Formula
Calculation
Interpolated yield = Yield at lower maturity + (Yield at higher maturity - Yield at lower maturity) x (Target maturity - Lower maturity) / (Higher maturity - Lower maturity)
Suppose the 2-year yield is 3.0% and the 5-year yield is 3.9%, and a company needs a 3-year rate. The yield difference is 3.9 - 3.0 = 0.9 percentage points. The fraction of the way from 2 to 5 years is (3 - 2) / (5 - 2) = 1 / 3. The interpolated 3-year yield is 3.0 + 0.9 x (1 / 3) = 3.0 + 0.3 = 3.3%. The company would then discount a $1,000,000 payment due in 3 years at 3.3%, giving 1,000,000 / (1.033)^3, which is about $907,200.Case study
Seen in the real world.
This is an illustrative story about a fictional company, Redstone Logistics, which signed a 6-year equipment lease. The accounting team needed a discount rate to record the lease liability and had only 5-year and 10-year benchmark yields to work from.
The finance manager used linear interpolation between the two points. With a 5-year yield of 4.0% and a 10-year yield of 4.5%, the 6-year benchmark came out at 4.0 + 0.5 x (1 / 5) = 4.1%, to which the company added its credit margin.
The auditors asked for documentation of the method and the source of the yields, and the team recorded both in the lease file. The same note was reused for the next lease, which saved time at the following year-end. The illustrative case shows that interpolation is a practical tool, but the method, the data and the date should be written down so that results can be repeated and checked.
Watch out
Common mistakes.
- Treating an interpolated rate as an observed market rate. It is an estimate based on nearby points.
- Interpolating across a very wide gap. The further apart the known points, the less reliable the estimate, especially where the curve bends between them.
- Changing methods from one period to the next without saying so. Inconsistent methods make results hard to compare.
Questions
People also ask.
What is the difference between interpolation and extrapolation?
Interpolation estimates a value between known points, while extrapolation extends the curve beyond the last known point and is riskier.
Is linear interpolation always good enough?
It is often adequate for short gaps, but smoother methods may be better for large exposures or sharply curved yield curves.
Why do companies need rates for every maturity?
Because cash flows fall due on many dates, and each needs the right discount rate to calculate present value.
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