riaology
Learning path 2 · Topic 13 of 30

Read between factor-table entries

Linear interpolation estimates a factor between tabulated rates. It does not make the exact factor relationship linear.

Change a scenario. Watch the consequence.
052.167.54.3310P/A factorRate (%)
● Current scenarioExact P/AInterpolated chord · dashed
interpolated P/A factor · factorf(7%)≈f(5%)+7−510−5[f(10%)−f(5%)]=4.114001f(7\%)\approx f(5\%)+\frac{7-5}{10-5}[f(10\%)-f(5\%)]=4.114001

Exact value 4.100197; interpolation error 0.013803. A factor is dimensionless.

Follow the reasoning

Use the same factor and the same N at both table endpoints.

Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.

01

What the model means

Use the same factor, period count, and compounding convention at both endpoints. The target must lie inside the bracket for interpolation; outside it, the calculation is extrapolation and less reliable.

02

Build the calculation

If table factors have four decimal places, carry those values consistently. Final dollar results often use two decimals. This tool displays exact-formula comparison so you can see the interpolation error; use the rounding and table precision appropriate for your calculation.

f(i)≈fL+i−iLiH−iL(fH−fL)f(i)\approx f_L+\frac{i-i_L}{i_H-i_L}(f_H-f_L)
Practice with feedback

Set up. Calculate. Explain.

Answer unit: factor

Rebuild the starting scenario

Desired annual rate: 7 %; Number of years: 5 years. Calculate the interpolated P/A factor.

Setup hint

Use the same factor and the same N at both table endpoints.

Open the visual solutionf(7%)≈f(5%)+7−510−5[f(10%)−f(5%)]=4.114001f(7\%)\approx f(5\%)+\frac{7-5}{10-5}[f(10\%)-f(5\%)]=4.114001
  1. Use the same factor and the same N at both table endpoints.
  2. Locate the target rate inside the bracket; compute its fractional position.
  3. Interpolate factor values linearly. The true factor curve is nonlinear, so this is approximate.
052.167.54.3310P/A factorRate (%)
● Current scenarioExact P/AInterpolated chord · dashed
interpolated P/A factor · factorf(7%)≈f(5%)+7−510−5[f(10%)−f(5%)]=4.114001f(7\%)\approx f(5\%)+\frac{7-5}{10-5}[f(10\%)-f(5\%)]=4.114001

Exact value 4.100197; interpolation error 0.013803. A factor is dimensionless.

Visual solution

Use the same factor and the same N at both table endpoints.

Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.

Checks allow 0.1% numerical rounding (minimum $0.05 or 0.001 in the stated nonmonetary unit). Interpolated factors and rates are approximations; keep table precision consistent.

Answer unit: factor

Transfer the idea to a new scenario

Using the 5% and 10% P/A entries for N=8, interpolate P/A at 7.5%. Report the estimated factor.

Setup hint

Use the same factor and the same N at both table endpoints.

Open the visual solutionf(7.5%)≈f(5%)+7.5−510−5[f(10%)−f(5%)]=5.899069f(7.5\%)\approx f(5\%)+\frac{7.5-5}{10-5}[f(10\%)-f(5\%)]=5.899069
  1. Use the same factor and the same N at both table endpoints.
  2. Locate the target rate inside the bracket; compute its fractional position.
  3. Interpolate factor values linearly. The true factor curve is nonlinear, so this is approximate.
053.237.56.4610P/A factorRate (%)
● Current scenarioExact P/AInterpolated chord · dashed
interpolated P/A factor · factorf(7.5%)≈f(5%)+7.5−510−5[f(10%)−f(5%)]=5.899069f(7.5\%)\approx f(5\%)+\frac{7.5-5}{10-5}[f(10\%)-f(5\%)]=5.899069

Exact value 5.857304; interpolation error 0.041766. A factor is dimensionless.

Visual solution

Use the same factor and the same N at both table endpoints.

Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.

Checks allow 0.1% numerical rounding (minimum $0.05 or 0.001 in the stated nonmonetary unit). Interpolated factors and rates are approximations; keep table precision consistent.

Explain the distinction

Which information must be held fixed when interpolating a factor across rates?

Can you defend your setup?

Can you place each amount on the timeline, match the interest period, calculate the equivalent quantity, and state the decision or limitation?

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