Exponentials and discounting

Grow money forward, discount it back, and spread a building cost into equal yearly payments

Math toolbox K3 of Energy Systems, Step by Step: compound growth, present value, a stream of equal payments, and the capital recovery factor that turns a building cost into a yearly cost.
  • Math toolbox
  • 15 min guided · about 15 min practice
  • No prerequisites

Money over time, in six moves

  1. Growth compounds: $100 at 7% a year, interest left in, is $196.72 after 10 years and first passes $200 in year 11 ($210.49). Simple interest gives $170.00 after 10 years.
  2. Discounting runs it backwards: $100 paid in 10 years is worth $100 ÷ 1.07¹⁰ = $50.83 today, its present value.
  3. Far-off money feels the rate most: from 7% to 4%, $100 in 30 years rises from $13.14 to $30.83 today.
  4. $1 paid at the end of each year for 30 years is worth $17.29 today at 4%, not $30: the annuity factor.
  5. Spread the wind farm's building cost over 30 years: $1,712 ÷ 17.292 = $99.01 per kW a year. The capital recovery factor is 1 ÷ 17.292 = 0.05783.
  6. Add upkeep: $99.01 + $34.57 = $133.58 per kW a year; per MW, before rounding, 1,000 × $133.575 = $133,575. At 0% the factor is 1 ÷ 30 = 0.0333.
When $100 today grows at 7% to $100 paid then is worth today, at 7% … at 4%
Today $100.00 $100.00 $100.00
In 10 years $196.72 $50.83 $67.56
In 20 years $386.97 $25.84 $45.64
In 30 years $761.23 $13.14 $30.83
Rate (real) CRF over 30 years Repayment, $ per kW a year Plus upkeep: yearly cost, $ per kW
0% 0.0333 $57.07 $91.64
4% 0.0578 $99.01 $133.58
7% 0.0806 $137.96 $172.53

With JavaScript on, you work through this step by step with sliders, a chart and your own answers.

  • The wind farm's building cost, $1,712 per kW, and fixed upkeep, $34.57 per kW a year: EIA, AEO2026 assumptions, Table 3 (2025 dollars, April 2026), as in lessons 7, 12 and 13.
  • 4% a year after inflation over a 30-year cost recovery period, as in lesson 7 and lesson 13: the 2025 ATB's period, and a round rate inside its range (NLR, September 2026).
  • Each yearly payment falls at the end of its year, as the capital recovery factor assumes.
  • $100 and 7% are illustrative, as in lesson 7's step 3. All dollars are real 2025 dollars, so every rate here is a real rate.

What you just did, as diagrams

Compound growth
Each year adds a percentage of what is there now, so the amount grows faster and faster: × (1 + r)t after t years.
Present value
What an amount paid t years from now is worth today: divide by (1 + r)t.
Annuity factor
What $1 at the end of each year for n years is worth today: the sum of the discounted dollars, 17.29 at 4% over 30 years.
Capital recovery factor
One over the annuity factor: the share of a building cost repaid each year, 0.0578 at 4% over 30 years.
Yearly cost
CRF × building cost + fixed upkeep: $99.01 + $34.57 = $133.58 per kW a year for the wind farm.
Go deeper · AnalystWhere the formula comes from, and real against nominal rates
  • A stream is a geometric series: a sum in which each term is the one before times the same factor, here \(1 \div (1+r)\).
  • Add it up. \(n\) yearly payments \(P\), each at the end of its year: \(\frac{P}{1+r} + \dots + \frac{P}{(1+r)^n} = P\,\frac{1-(1+r)^{-n}}{r}\). The fraction is the annuity factor: 17.29 at 4% over 30 years.
  • Set it equal to the building cost \(C\) ($ per kW): \(P = \frac{r}{1-(1+r)^{-n}}\,C = \mathrm{CRF} \cdot C\), the yearly repayment before upkeep.
  • Two ways to write it. Multiply top and bottom by \((1+r)^n\): \(\frac{r}{1-(1+r)^{-n}} = \frac{r(1+r)^n}{(1+r)^n - 1}\). Lesson 7 writes the first, lesson 13’s code the second.
  • Exact doubling time. Solve \((1+r)^t = 2\) for \(t\): \(t = \ln 2 \div \ln(1+r)\), where \(\ln\) is the natural logarithm (a calculator’s ln key). At 7% that is 10.2 years; the rule of 70 says 10.
  • Real against nominal. A nominal rate includes inflation; a real rate takes it out: \((1 + \text{nominal}) = (1 + \text{real})(1 + \text{inflation})\). Lesson 7’s illustrative blend, 7% nominal with 2.5% inflation, is 4.39% real.
  • Do not mix them. Costs in today’s dollars go with a real rate; cash in the dollars of each future year goes with a nominal rate.
Go deeper · ModelerThe yearly cost inside a planning model (a preview)
  • One year stands for the life. Lessons 12 and 13 minimise one year’s cost, \(1000 \sum_k (\mathrm{CRF}_k\, C_k + \mathrm{FOM}_k)\,K_k\) plus running costs, with \(K_k\) the MW built of plant \(k\).
  • Units. \(C_k\) (building cost) and \(\mathrm{FOM}_k\) (fixed upkeep) are $ per kW and \(K_k\) is in MW, so the 1,000 (kW per MW) makes the units match: one MW of wind adds $133,575 a year.
  • The assumption. Every year of the life looks like the modelled year, and each plant’s cost is recovered over \(n\) years at the same rate: 30 years for plants, 15 for lesson 13’s battery.
  • Where next. A model that spans decades discounts each year’s cash instead, and credits plants that outlive its horizon (Modeler lesson M13); M11 puts a project’s full cash flow inside the optimization.

Check yourself

Myth or fact? “A plant paid off over 30 years costs 1/30 of its building cost each year.”

  • Myth
  • Fact
Answer

Myth, unless the rate is 0%. At 4% the yearly share is 0.0578, not 0.0333: lenders and owners are paid for the wait.

Myth or fact? “Doubling the discount rate halves what $100 paid in 30 years is worth today.”

  • Myth
  • Fact
Answer

Myth. From 3.5% to 7% it falls from $35.63 to $13.14, by 63%: the rate sits inside an exponent.

What is $100 paid in 20 years worth today, at 7% a year?

  • $386.97
  • $41.67
  • $25.84
  • $50.00
Answer

$25.84: 100 ÷ 1.07²⁰. $41.67 divides by 1 + 20 × 0.07 (simple interest); $386.97 grows the $100 forward instead.

Which is the capital recovery factor at 4% over 30 years?

  • 0.0333
  • 0.0400
  • 0.0578
  • 0.0806
Answer

0.0578. 0.0333 is 1 ÷ 30, the 0% case; 0.0400 pays the return but never the cost; 0.0806 is the factor at 7%.

Numbers, sources and what’s next

Numbers to remember

Number What it means Source
70 ÷ rate Years to double at a steady rate in %, roughly: 10 years at 7%. Rule of thumb; the exact time is in Go deeper
0.0578 The capital recovery factor at 4% over 30 years: the share of a building cost repaid each year. This page, from the formula
$1,712 · $34.57 The wind farm's building cost per kW and fixed upkeep per kW a year (2025 dollars). EIA, AEO2026 assumptions, Table 3 (PDF) · April 2026
4% · 30 years The real rate and cost recovery period of lessons 7, 12 and 13. NLR, 2025 ATB · September 2026 (4%: a round value inside its range)
$133.58 The wind farm's yearly cost per kW: $99.01 of repayment plus $34.57 of upkeep. This page, from the formula

Next

Lesson 7 · What does electricity cost to make?

Sources and further reading

Spotted an error, or stuck? Write to hlli@ksu.edu. Corrections are made in place and dated here.

Text licensed CC BY 4.0, code MIT: teachers and students may reuse and adapt it, with credit to Honglin Li.