Linear interpolation: Difference between revisions
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<span style="color:#4B0082">'''Example 1'''</span> | __TOC__ | ||
<span style="color:#4B0082">'''Example 1: Interpolation'''</span> | |||
Consider a set of cashflows which has: | Consider a set of cashflows which has: | ||
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Net present value (NPV) of +$4m at a yield of 5%. | Net present value (NPV) of +$4m at a yield of 5%. | ||
NPV of -$4m at a yield of 6%. | |||
Using linear interpolation, the estimated yield at which the cashflows have an NPV of $0 is given by: | Using linear interpolation, the estimated yield at which the cashflows have an NPV of $0 is given by: | ||
IRR estimate = a% + ( A / ( A - B) ) x (b - a)% | |||
Where: | |||
= 5.5%. | a% = first estimated yield = 5% | ||
b% = second estimated yield = 6% | |||
A = NPV of cash flows at yield of a% = +$4m | |||
B = NPV of cash flows at yield of b% = -$4m | |||
IRR estimate: | |||
= 5% + ( +4 / ( +4 - -4) ) x (6 - 5)% | |||
= 5% + ( +4 / +8 ) x 1% | |||
= 5% + 0.5% | |||
= '''5.5%'''. | |||
5.5% is the estimated internal rate of return (IRR) of the cashflows. | 5.5% is the estimated internal rate of return (IRR) of the cashflows. | ||
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Interpolation is often used in conjunction with Iteration. | Interpolation is often used in conjunction with Iteration. | ||
Using iteration the straight-line estimated IRR of 5.5% would then be used, in turn, to recalculate the NPV at the estimated IRR of 5.5%, producing a recalculated NPV even closer to $0. | Using iteration, the straight-line estimated IRR of 5.5% would then be used, in turn, to recalculate the NPV at the estimated IRR of 5.5%, producing a recalculated NPV even closer to $0. | ||
5.5% and the recalculated NPV would then be used with interpolation once again to further refine the estimate of the IRR. | 5.5% and the recalculated NPV would then be used with interpolation once again to further refine the estimate of the IRR. | ||
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This iteration process can be repeated as often as required until the result converges on a sufficiently stable final figure. | This iteration process can be repeated as often as required until the result converges on a sufficiently stable final figure. | ||
==Extrapolation== | |||
Another closely related linear estimation technique is extrapolation. | Another closely related linear estimation technique is extrapolation. | ||
This involves the straight-line estimation of values outside the range of the data used to do the estimation with. | This involves the straight-line estimation of values outside the range of the sample data used to do the estimation with. | ||
<span style="color:#4B0082">'''Example 2'''</span> | <span style="color:#4B0082">'''Example 2: Extrapolation'''</span> | ||
Using the data | Using the following data to estimate net present value (NPV) at a yield of 7%, using extrapolation: | ||
NPV of +$4m at a yield of 5%. | |||
NPV of -$4m at a yield of 6%. | |||
'''''Solution''''' | |||
Based on the sample data, for every 1% increase in the yield, the NPV moved by: | |||
-$4m - $4m = -$8m | |||
Extrapolating this trend to a yield of 7%, this is a further increase in the yield of 7 - 6 = 1%. | |||
The NPV would be modelled to fall from -$4m to: | |||
= -$4m - $8m | = -$4m - $8m | ||
= -$12m. | = -$'''12m'''. | ||
== See also == | == See also == | ||
* [[Internal rate of return]] | * [[Internal rate of return]] | ||
* [[Interpolation]] | * [[Interpolation]] | ||
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* [[Linear]] | * [[Linear]] | ||
* [[Straight line]] | * [[Straight line]] | ||
[[Category:Corporate_financial_management]] | |||
[[Category:Cash_management]] | |||
Latest revision as of 23:12, 5 January 2022
A straight-line estimation method for determining an intermediate value.
Example 1: Interpolation
Consider a set of cashflows which has:
Net present value (NPV) of +$4m at a yield of 5%.
NPV of -$4m at a yield of 6%.
Using linear interpolation, the estimated yield at which the cashflows have an NPV of $0 is given by:
IRR estimate = a% + ( A / ( A - B) ) x (b - a)%
Where:
a% = first estimated yield = 5%
b% = second estimated yield = 6%
A = NPV of cash flows at yield of a% = +$4m
B = NPV of cash flows at yield of b% = -$4m
IRR estimate:
= 5% + ( +4 / ( +4 - -4) ) x (6 - 5)%
= 5% + ( +4 / +8 ) x 1%
= 5% + 0.5%
= 5.5%.
5.5% is the estimated internal rate of return (IRR) of the cashflows.
Interpolation and Iteration
Interpolation is often used in conjunction with Iteration.
Using iteration, the straight-line estimated IRR of 5.5% would then be used, in turn, to recalculate the NPV at the estimated IRR of 5.5%, producing a recalculated NPV even closer to $0.
5.5% and the recalculated NPV would then be used with interpolation once again to further refine the estimate of the IRR.
This iteration process can be repeated as often as required until the result converges on a sufficiently stable final figure.
Extrapolation
Another closely related linear estimation technique is extrapolation.
This involves the straight-line estimation of values outside the range of the sample data used to do the estimation with.
Example 2: Extrapolation
Using the following data to estimate net present value (NPV) at a yield of 7%, using extrapolation:
NPV of +$4m at a yield of 5%.
NPV of -$4m at a yield of 6%.
Solution
Based on the sample data, for every 1% increase in the yield, the NPV moved by:
-$4m - $4m = -$8m
Extrapolating this trend to a yield of 7%, this is a further increase in the yield of 7 - 6 = 1%.
The NPV would be modelled to fall from -$4m to:
= -$4m - $8m
= -$12m.
