Refine IRR in two interpolation steps
A table-driven IRR estimate begins with trial rates that give opposite signs of PW. The rate is unknown; the cash-flow amounts and dates are fixed.
Period 0: −$50,000.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.
Exact root 15.238237%. Replacing an endpoint based on the residual retains the sign-change bracket.
PW(10%) = $6,861.80 > 0; PW(20%) = $-5,140.82 < 0.
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.
What the model means
Compute PW at a lower rate and an upper rate, then estimate the zero by linear interpolation. Next evaluate PW at that estimate. Replace whichever endpoint has the same sign as the new residual and interpolate again.
Build the calculation
This is a secant approximation within a bracket. A narrower valid bracket usually improves accuracy. Never assume the interpolated point is an exact root; report the residual and use required rate precision.
Set up. Calculate. Explain.
Rebuild the starting scenario
Low trial rate: 10 %; High trial rate: 20 %. Calculate the second interpolated IRR estimate in percent.
Setup hint
PW(10%) = $6,861.80 > 0; PW(20%) = $-5,140.82 < 0.
Open the visual solution
- PW(10%) = $6,861.80 > 0; PW(20%) = $-5,140.82 < 0.
- Interpolate i₁ = i_L + PW_L(i_H−i_L)/(PW_L−PW_H).
- Evaluate PW(i₁); retain the endpoints of opposite sign and interpolate again: 15.285908%.
Period 0: −$50,000.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.
Exact root 15.238237%. Replacing an endpoint based on the residual retains the sign-change bracket.
PW(10%) = $6,861.80 > 0; PW(20%) = $-5,140.82 < 0.
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.
Transfer the idea to a new scenario
For −$50,000 now and +$15,000 annually for five years, start at 12% and 18%. Perform two interpolation steps and report the second rate estimate.
Setup hint
PW(12%) = $4,071.64 > 0; PW(18%) = $-3,092.43 < 0.
Open the visual solution
- PW(12%) = $4,071.64 > 0; PW(18%) = $-3,092.43 < 0.
- Interpolate i₁ = i_L + PW_L(i_H−i_L)/(PW_L−PW_H).
- Evaluate PW(i₁); retain the endpoints of opposite sign and interpolate again: 15.248863%.
Period 0: −$50,000.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.
Exact root 15.238237%. Replacing an endpoint based on the residual retains the sign-change bracket.
PW(12%) = $4,071.64 > 0; PW(18%) = $-3,092.43 < 0.
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.
After evaluating a trial with positive PW, how do you update a monotone investment bracket?
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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