Annuity Calculation - FinAlgebra

Annuity calculation determines the value of long term regular payments. The two main use cases are calculating the height of regular payments to be funded by a current amount of savings, and converting back regular payments into lump sums at some point in time.

Many financial contracts, like pension entitlements, car loans, mortgages or retirement savings plans, can be modeled as annuities. There are however two major types of such contracts: fixed and floating interest rate annuities. Unfortunately, floating rate annuities are way more complicated to value and consequently, this introduction is limited to fixed rate annuities.

Fixed Interest Annuity Formulas

The first two annuity calculation formulas in this section determine the present and future value. The present value PV is what a series of regular payments to be received from now until some future end date would be worth right now. In contrast, the future value FV is what those payments would be worth at their scheduled end time.

\begin{align} PV &= C \cdot \frac{1 - (1 + i)^{-N}}{i}\\ FV &= C \cdot \frac{ (1 + i)^{N} - 1}{i}\\ i &= r / m \end{align}

\begin{alignat*}{2} PV &&\space:\space &\mathrm{present \space value}\\ FV &&\space:\space &\mathrm{future \space value}\\ C &&\space:\space &\mathrm{regularly \space paid \space amount}\\ r &&\space:\space &\mathrm{annual \space interest \space rate}\\ i &&\space:\space &\mathrm{period \space interest \space rate}\\ m &&\space:\space &\mathrm{payments \space per \space year}\\ N &&\space:\space &\mathrm{total \space number \space of}\\ && &\mathrm{payments}\\ \end{alignat*}

Equations (1) and (2) assume a total number of periods N with payments made at the end of each period. These are deferred annuities matching the terms of popular types of consumer credits. Examples are cars or TV sets bought on credit. In such contracts, the first payment is typically due at the end of the first month.

Mortgages and pensions, on the other hand, schedule payments at the beginning of each month. Because of the earlier payment, such advance annuities require an adjustment factor.

\begin{align} PV_a = PV \cdot (1 + i)\\ FV_a = FV \cdot (1 + i)\\ \end{align}

\begin{alignat*}{2} PV_a \space&& : \space &\mathrm{present \space value \space of}\\ && &\mathrm{advance \space annuity}\\ FV_a \space&& : \space &\mathrm{future \space value \space of}\\ && & \mathrm{advance \space annuity}\\ \end{alignat*}

Quite intuitively, payments moved to the beginning of periods earn another period interest i. Since this happens for all of the payments, one simply needs to multiply the expressions for deferred annuities by a factor (1 + i) to get valuation formulas for advance annuities.

Also note that the height of regular payments C that can be generated from lump sums results from simple rearrangements of Equations (1) and (2).

Usage Examples of Annuity Calculation Formulas

Because of the variety of use cases, some examples calculations for advance and deferred annuities will help understanding the formulas.

Installment loan with full repayment

Problem 1: an installment loan at 3% nominal interest fully pays off in 24 monthly installments of $100 each. What is the loan amount that the borrower receives at the start of the contract?

Solution 1: Since the loan amount is the valuation of the installments at the beginning of the contract, we need to use the formula for the present value of a deferred annuity. First, use Equation (3) for the period interest with m=12:

i = r/m = 3%/12 = 0.25%

Now, use Equation (2) with N=24 and C=100:

PV = 100 * (1 - (1 + 0.25 / 100)^(-24))
     / 0.25 * 100
   = 2326.60

So the loan amount that the borrower receives is $2326.60. A web calculator filled with the values for the problem is here.

Problem 2: a loan of $1000 fully pays of in 36 monthly installments at 3% nominal interest. What is the amount of the installments?

Solution 2: Because of the loan paying out at start, it’s again a use case for Equation (2). However, the formula needs to be rearranged to solve for the installment C.

C = 1000 * 0.25 / 100 
    / (1 - (1 + 0.25/100)^(-36))
  = 29.08

So the required monthly installment is $29.08. Again, follow this link for a web calculator with these values.

Installment loan with partial repayment

Problem 3: $5.000 of a car loan totaling $25.000 pay off in 60 monthly installments. The contract states 4% of nominal interest. What is the height of installments?

Solution 3: this computation requires the equ ation for compound interest in addition to the annuity formulas. First, invest the loan balance at 4% interest with monthly compounding. Inserting into the compound interest formula gets:

L = 25000 * (1 + 4 / 12 / 100)^60
  = 30524.915

In 60 months time, the problem states that the loan balance shall be $20.000. Therefore, the future value of the installments must reduce the value of the invested loan L to that target.

L - FV = 20.000
FV = 30524.915 - 20000 = 10524.915

So the solution is to rearrange Equation (2) for installment C:

C = 10524.915 * 4 / 12 / 100
    / ((1 + 4 / 12 / 100)^60 -1) 
  = 158.75

The required height of installments is $158.75 and the prefilled web calculator for this is here.

Savings Plan

Problem 4: A savings account pays 3% of interest with monthly compounding. What will be its balance in 10 years time if a saver deposits $100 at the beginning of each month?

Solution 4: This is a use case for Equation (5), the future value of an advance annuity:

FVa = 100 (1 + 3 / 12 / 100)
      * ((1 + 3 / 12 / 100)^(10*12) - 1)
      / 3 * 12 * 100
    = 14,009.08

So the final balance of the savings plan will be $14,009.08 and the web form with these values is here.

Pension Insurance

Problem 5: at an age of 35 years a saver starts paying $100 per month into a private pension insurance until retirement, that is, over a period of 30 years. The accumulating savings earn 3% of interest with monthly compounding. After retirement at age 65, the insurance expects the client to live for another 20 years. Assuming a fair contract, where the insurance doesn’t pocket any fees, what would be the height of the monthly pension?

Solution 5: the solution to this annuity calculation problem follows from a combination of an advance annuities according to Equation (5) for the savings and Equation (4) for the retirement phase. Since the question asks the height of pension payments, it is also necessary to rearrange Equation (4). First, compute the savings balance at age 65:

FVa = 100 * (1 + 3 / 12 / 100)
      * ((1 = 3 / 12 / 100)^(30*12)-1)
      / 3 * 12 * 100
    = 58419.37

So the insurance holds $58,419.37 in savings on behalf of the client at start of retirement. Web calculator here.

Whilst the client keeps receiving his pension, the remaining savings still earn interest. Therefore, rearrange Equation (4) to convert savings into a pension payment.

C = 58419.37 / (1 + 3 / 12 / 100)
    / (1 - (1 + 3 / 12 / 100)^(-20*12))
    * 3 / 12 / 100
  = 323.18

Given savings of $58,419.37, the insurance can fund a pension of $323.18 over a course of 20 years. Web calculator producing a minor rounding error is here.

References

Finance in a Nutshell, Javier Estrada, 2005, Prentice Hall

Annuity, Wikipedia.org