AJ Designer

Capital Recovery Calculator

Annual payment equals present value times i times one plus i to the n divided by one plus i to the n minus one

Solution

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How It Works

The capital recovery factor converts a present amount P into an equivalent series of equal periodic payments A. The formula A = P · i(1 + i)^n / [(1 + i)^n − 1] is the basis of loan amortization — every fixed-payment loan (mortgage, car loan, student loan) uses this equation to compute the level payment that recovers the principal plus interest over n periods at rate i per period. Invert it to find the present value (principal) supported by a given annual payment.

Example Problem

A $10,000 loan must be repaid in 10 equal annual payments at 10% interest per year. What is the annual payment?

  1. Choose Solve For = Annual Payment (A). The formula is A = P · i(1 + i)^n / [(1 + i)^n − 1].
  2. Substitute P = 10,000, i = 0.10, n = 10.
  3. Compute (1.10)^10 = 2.593742.
  4. Numerator: 0.10 × 2.593742 = 0.2593742. Denominator: 2.593742 − 1 = 1.593742.
  5. A = 10,000 × 0.2593742 / 1.593742 = 1,627.45 per year.

Total interest paid is 10 × 1,627.45 − 10,000 = 6,274.54 over the life of the loan.

Key Concepts

The capital recovery factor A/P is the inverse of the uniform-series present-worth factor P/A — together they convert between a present principal and an equivalent stream of equal payments. The factor depends only on the rate and number of periods, not the dollar amount, so the same factor amortizes a $10,000 loan and a $1,000,000 loan at the same terms. Every standard mortgage, car loan, and equipment loan uses this formula under the hood; the only thing changing on each loan is the values of P, i, and n.

Applications

  • Mortgage payment — calculate the monthly principal-and-interest payment from the loan amount, rate, and term
  • Car loan / auto loan payment — determine the monthly payment to amortize a vehicle loan
  • Equipment leasing — find the level lease payment that recovers the equipment cost plus a lessor's required return
  • Engineering project annual cost — convert a one-time capital outlay into an equivalent annualized cost (EAC)
  • Bond coupon analysis — back out the implied capital recovery from a callable bond's principal

Common Mistakes

  • Entering an annual rate but expecting monthly payments — convert to a monthly rate (annual ÷ 12) and use total months for n
  • Entering interest rate as a percent instead of a decimal — use 0.10 for 10%, not 10
  • Confusing capital recovery (P → A) with sinking fund (F → A) — capital recovery starts from a present amount, sinking fund targets a future amount
  • Forgetting that the first payment is one period after the loan origination — this is an ordinary annuity, not an annuity due
  • Adding upfront fees to the principal incorrectly — financed fees do increase P, but cash-paid fees do not

Frequently Asked Questions

How do you calculate capital recovery?

Use A = P × i(1 + i)^n / [(1 + i)^n − 1] where P is the present value (loan amount), i is the periodic interest rate, and n is the number of periods. Example: a $10,000 loan at 10% per year for 10 years gives A = 10,000 × 0.10 × (1.10)^10 / [(1.10)^10 − 1] = $1,627.45 per year.

What is the formula for capital recovery factor?

The capital recovery factor is A/P = i(1 + i)^n / [(1 + i)^n − 1]. Multiply the factor by the present amount P to get the equal periodic payment A. The factor depends only on i and n — the dollar amount of P is separate.

Is capital recovery the same as loan amortization?

Yes. Every fixed-payment amortizing loan uses the capital recovery formula to compute the level payment. The split between principal and interest changes each period (more interest early, more principal late), but the total payment stays constant.

What's the difference between capital recovery and sinking fund?

Capital recovery (A/P) converts a present amount into equal annual payments — you start with P today and pay A per period. Sinking fund (A/F) converts a future target into equal annual deposits — you save A per period to reach F in the future. They are different factors with different formulas.

How do I use this for monthly mortgage payments?

Convert the annual rate to a monthly rate (annual ÷ 12) and use the total number of months for n. A $250,000 mortgage at 6% annual interest over 30 years uses i = 0.005 per month and n = 360: A = $1,498.88 per month.

Why does the same loan have different payments at different rates?

The capital recovery factor is very sensitive to rate, especially at longer terms. A $200,000 30-year mortgage at 4% is $954.83 per month; at 7% it's $1,330.60 — 39% higher even though the principal is the same. The interest compound effect over 360 periods amplifies small rate changes.

Reference: Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.

Worked Examples

Mortgage Payment

What is the monthly payment on a $250,000 30-year mortgage at 6% APR?

  • Convert to monthly: i = 0.06 / 12 = 0.005, n = 360
  • A = P × i(1 + i)^n / [(1 + i)^n − 1]
  • A = 250,000 × 0.005 × (1.005)^360 / [(1.005)^360 − 1]
  • A ≈ $1,498.88 per month

Auto Loan

What is the monthly payment on a $25,000 5-year car loan at 6% APR?

  • i = 0.06 / 12 = 0.005, n = 60
  • A = 25,000 × 0.005 × (1.005)^60 / [(1.005)^60 − 1]
  • A = 25,000 × 0.006745 / 0.348850
  • A ≈ $483.32 per month

Max Loan Affordability

If you can afford $1,200 per month, how much can you borrow over 30 years at 7%?

  • P = A × [(1 + i)^n − 1] / [i(1 + i)^n]
  • i = 0.07 / 12 ≈ 0.005833, n = 360
  • P = 1,200 × [(1.005833)^360 − 1] / [0.005833 × (1.005833)^360]
  • P ≈ $180,365

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