Present Value of an Annuity Formula

Formula to Calculate PV of an Annuity

The present value of annuity formula is calculated by determining present value which is calculated by annuity payments over the time period divided by one plus discount rate and the present value of the annuity is determined by multiplying equated monthly payments by one minus present value divided by discounting rate.

PV of an Annuity = C x [ (1 – (1+i)-n) / i ]

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For eg:
Source: Present Value of an Annuity Formula (wallstreetmojo.com)

Where,

  • C is the cash flow per period
  • i is the rate of interest
  • n is the frequency of payments

Explanation

The PV formula will determine at a given period, the present value of several future timely interval payments. The PV of annuity formula can be seen from the formula that it depends upon the time value of money conceptTime Value Of Money ConceptThe Time Value of Money (TVM) principle states that money received in the present is of higher worth than money received in the future because money received now can be invested and used to generate cash flows to the enterprise in the future in the form of interest or from future investment appreciation and reinvestment.read more, in which a one-dollar amount of money in the current day is more worthy than the same dollar that shall be due at a date which is going to happen in future. Also, the PV of the annuity formula takes care of the frequency of payment, whether it’s annual, semi-annual, monthly, etc. and accordingly does calculation or say compoundingCompoundingCompounding is a method of investing in which the income generated by an investment is reinvested, and the new principal amount is increased by the amount of income reinvested. Depending on the time period of deposit, interest is added to the principal amount.read more.

Examples

You can download this Present Value of Annuity Formula Excel Template here – Present Value of Annuity Formula Excel Template

Example #1

Suppose that there is an annuity payment of $1,000 for the next 25 years beginning at every end of the year. You are required to compute the present value of the annuity, assuming a rate of interest is 5%.

Solution:

Here the annuities begin at the end of the year, and therefore, n will be 25, C is $1,000 for the next 25 years, and i is 5%.

Use the following data for the calculation of the PV of an annuity.

  • Cash flow per Period (C): 1000.00
  • Number of Period (n): 25.00
  • Rate of Interest (i): 5.00%

So, the calculation of the PV of an annuity can be done as follows –

Present Value of Annuity Formula Example 1.1png

Present Value of the Annuity will be – 

Present Value of Annuity Formula Example 1.3png

=  $1,000 x [ (1 – (1+5%)-25) / 0.05 ]

Present Value of an Annuity = 14,093.94

Example #2

John is currently working in an MNC where he is paid $10,000 annually. In his compensation, there is a 25% portion, which is will be paid an annuity by the company. This money is deposited twice in a year, starting 1st July and second is due on the 1st of January and will continue till the next 30 years, and at the time of redemption, it would be tax-exempt.

He was also given an option at the time of joining to take $60,000 at once, but that would be subject to tax at the rate of 40%. You are required to assess whether John should take the money now or wait until 30 years to receive the same, assuming he is not in the requirement of funds, and the risk-free rate in the market is 6%.

Solution

Here, the annuities begin at the end of the semi-annually and therefore n will be 60 (30*2), C is $1,250 ($10,000 * 25% / 2) for next 30 years and i is 2.5% (5%/2).

Use the following data for the calculation of the present value of an annuity.

  • Cash flow per Period (C): 1250.00
  • Number of Period (n): 60.00
  • Rate of Interest (i): 2.5%

So, the calculation of the (PV) present value of an annuity formula can be done as follows –

Present Value of Annuity Formula Example 2.1png

Present Value of the Annuity will be – 

Example 2.2png

= $1,250 x [ (1 – (1+2.5%)-60) / 0.025 ]

Present Value of an Annuity = $38,635.82

Hence, if John opts for an annuity, then he would receive $38,635.82.

The second option is he opts for $60,000, which is before tax, and if we deduct a tax of 40%, then the amount in hand will be $36,000.

 Therefore, John should opt for annuity since there is a benefit of $2,635.82

Example #3

Two different retirement products are being offered to Mrs. Carmella as she is nearing retirement. Both of the products will start their cash flow at the age of 60 years and continue annuity till 80 years of age. Below are more details of the products. You are required to compute the present value of the annuity and advise, which is the better product for Mrs. Carmella?

Assume Rate of interest 7%.

1) Product X

Annuity Amount = $2,500 per period. Payment frequency =Quarterly.Payment will be at the beginning of the period.

2) Product Y

Annuity Amount  = 5,150 per period. Payment frequency =Semi-Annually. Payment will be at the end of the period

Given,

ParticularsProduct XProduct Y
Cash flow per Period (C)2500.005150.00
Number of Periods (n)79.0040.00
Rate of Interest (i)1.75%3.50%

Solution:

Here, the annuities for product x begins at the beginning of the quarter, and therefore, n will be 79  as the payment is made at the beginning of the annuity (20*4 less 1), C is $2,500 for the next 20 years, and i is 1.75% (7%/4).

So, the calculation of the present value of an annuity for a product X can be done as follows –

Example 3.1png

Present Value of an Annuity for Product X will be –

Present Value of Annuity Formula Example 3.2png

=$2,500 x [ (1 – (1+1.75%)-79) / 0.0175 ]

Present Value of  Annuity = $106,575.83

Now we need to add $2,500 to above present valuePresent ValuePresent value factor is factor which is used to indicate the present value of cash to be received in future and is based on time value of money. This PV factor is a number which is always less than one and is calculated by one divided by one plus the rate of interest to the power, i.e. number of periods over which payments are to be made.read more since that was received at the start of the period and hence total amount will be 1,09,075.83

The 2nd option is paying semi-annually. Hence n will be 40 (20*2), i will be 3.50% (7%/2), and C is $5,150.

So, the calculation of the PV of an annuity for a product Y can be done as follows –

Present Value of Annuity Formula Example 3.3png

Present Value of  Annuity for Product Y will be –  

Example 3.4png

= $5,150 x [ (1 – (1+3.50%)-40) / 0.035 ]

Present Value of Annuity =  $ 109,978.62

There is only $902.79 excess when opted for option 2. Hence Mrs. Carmella should select opt 2.

Relevance and Uses

 The formula is quite important not only in calculating the retirement options, but this can also be used for cash outflows in case of capital budgetingCapital BudgetingCapital budgeting is the planning process for the long-term investment that determines whether the projects are fruitful for the business and will provide the required returns in the future years or not. It is essential because capital expenditure requires a considerable amount of funds.read more, where there could be an example of rent or periodic interest paid, which are mostly static; hence those can be discounted back by using this annuity formula. Also, one has to be cautious while using the formula as one needs to determine if the payments are made at the beginning of the period or at the end of the period, as the same can affect the values of cash flows due to compounding effects.

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