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Present value of the pension formula

Present value of the pension formula (table of contents)

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What is the present value of the pension formula?

The term “present value of the annuity” refers to the series of equal future payments that are discounted to the present day. However, payment can be received either at the beginning or at the end of any period and accordingly there are two different wordings. If the cash flow is to come in at the beginning, it is called the present value of an annuity due and the formula can be derived based on the periodic payment, the interest rate, the number of years and the frequency of occurrence in a year. Mathematically, it is represented as

Where,

  • PVA = Present value of the pension
  • P. = Periodic payment
  • r = Interest rate
  • t = Number of years
  • n = Frequency of occurrence in one year

If the cash flow is to come in at the end of each period, it is called the present value of the ordinary annuity and the formula differs slightly and is expressed as follows:

Examples of the present value of the pension formula (with Excel template)

Let's take an example to better understand the calculation of the present value of pension.

You can download this Excel template for the present value of the pension formula here - Excel template for the present value of the pension formula

Present value of the pension formula - example 1

Take the example of an annual annuity of $ 5,000 expected to be paid annually for the next three years.Calculate the present value of the annuity if the discount rate is 4% while payment is received at the beginning of each year.

Solution:

The present value of the pension due is calculated using the formula below

PVA due = P * (1 - (1 + r / n) - t * n ) * ((1 + r / n) / (r / n))

  • Present Value of Pension Due = $ 5,000 * (1 - (1 + (4% / 1)) -3 * 1 ) * ((1 + (4% / 1)) / (4% / 1))
  • Present value of the pension due = $ 14,430

Therefore, the present value of the annuity is $ 14,430.

Present value of the pension formula - example 2

Take the example of David, who is expected to receive a series of equal quarterly future deposits of $ 1,000 over the next six years.Calculate the present value of the future cash inflow if the relevant discount rate based on the current market rate is 5% while the payment is received:

  1. At the beginning of each quarter
  2. At the end of each quarter

Solution:

At the beginning of each quarter

The present value of the pension due is calculated using the formula below

PVA due = P * (1 - (1 + r / n) - t * n ) * ((1 + r / n) / (r / n))

  • Present value of annuity due = $ 1,000 * (1 - (1 + (5% / 4)) -6 * 4 ) * ((1 + (5% / 4)) / (5% / 4))
  • Present value of due pension = $ 20,882

At the end of each quarter

The present value of the regular pension is calculated using the formula given below

PVA usually = P * (1 - (1 + r / n) - t * n ) / (r / n)

  • Present value of ordinary pension = 1,000 USD * (1 - (1 + 5% / 4) -6 * 4 ) / (5% / 4)
  • Present value of the ordinary pension = $ 20,624

Therefore, the present value of the inflow of funds to be received from David is $ 20,882 and $ 20,624 if payments are received at the beginning and end of each quarter, respectively.

Explanation

Let's first consider the formula for the present value of a maturing annuity and then the formula for the present value of the ordinary annuity, and each of them can be derived using the following steps:

Step 1: First, calculate the same periodic payment expected at the beginning or end of each period. It is denoted by P.

Step 2: Next, calculate the interest rate based on the current market rates and it will be used to discount each recurring payment to today. It is denoted by r.

Step 3: Next, determine the number of years for which future payments are expected to be received and label this with t.

Step 4: Next, determine the frequency or occurrence of payments in a year and it will be denoted by n. It can be used to calculate the effective interest rate and number of periods as shown below.

Effective interest rate = r / n

Number of periods = t * n

Step 5: If the cash flow is to come in at the beginning of each period, the formula for the present value of the pension due can be derived based on the periodic payment (step 1), the effective interest rate (step 4) and the number of periods (step 4) as shown below .

PVA Due = P * (1 - (1 + r / n) - t * n ) * (1 + r / n) / (r / n)

On the other hand, if the cash flow is to come in at the end of each period, the formula for the present value of an ordinary annuity can be expressed as follows.

PVA usually = P * (1 - (1 + r / n) - t * n ) / (r / n)

Relevance and use of the present value of the pension formula

While the concept of the present value of a pension is just another expression of the time value theory of money, it is an important concept from a retirement savings evaluation perspective. In fact, it is mainly used by accountants, actuaries, and insurance staff to calculate the present value of structured future cash flows. It is also helpful in deciding whether a lump sum payment is better than a series of future payments based on the discount rate. In addition, the above decision is also influenced by the fact that payment is received at the beginning or at the end of each period.

Present value of the pension formula calculator

You can use the pension calculator cash value below


PVA = P x (1 - (1 + r / n) - txn ) X (1 + r / n / r / n)
=0 x (1 - (1 + 0/0 ) -0x0 ) X (1 + 0/0/0/0 ) = 0

Recommended articles

This was a guide to the present value of the pension formula. Here we discuss the calculation of the present value of the pension using practical examples. We also offer the Present Value of Annuity Calculator with a downloadable Excel template. You can also check out the articles below to learn more -

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  4. Discount factor formula (examples with Excel template)