## Formula to Calculate PV of Ordinary Annuity

Ordinary Annuity Formularefers to the formula that is used in order to calculate present value of the series of equal amount of payments that are made either at the beginning or end of period over specified length of time and as per the formula, present value of ordinary annuity is calculated by dividing the Periodic Payment by 1 minus 1 divided by 1 plus interest rate (1+r) raise to the power frequency in period (in case of payments made at the end of period) or raise to the power frequency in period minus one (in case of payments made at the beginning of period) and then multiplying the resultant with rate of interest.

The formula to calculate the present value of the ordinary annuity is given below

**Present Value of Ordinary Annuity (Beg) = r * P / {1 – (1+r) ^{-(n-1)}}**

**Present Value of Ordinary Annuity (End) = r * P / {1 – (1+r) ^{-(n)}}**

Where,

- P is the Periodic Payment
- r is the interest rate for that period
- n will be a frequency in that period
- Beg is Annuity due at the beginning of the period
- The end is Annuity due at the end of the period

### Explanation

The present value of ordinary annuity takes into account the three major components in its formula. PMT which is nothing but r*P which is the cash payment then we have r which is nothing, but prevailing market interest rate P is the present value of initial cash flow, and finally, n is the frequency or the total number of periods. Then there two types of payment one annuity which due at the beginning of the period and the second one is which due at the end of the period.

Both the formulas have a slight difference that is in one we compound by n and in another, we compound by n-1 that’s because the payment 1^{st} that is made will be made today and hence no discounting is applied to the 1^{st} payment for the beginning annuity.

### Examples

#### Example #1

Keshav has inherited $500,000 as per the agreement. However, the agreement stated that the payment will be received in equal installments as an annuity for the next 25 years. You are required to calculate the amount that shall be received by Keshav assuming interest rate prevailing in the market is 7%. You can assume that annuity is paid at the end of the year.

**Solution**

Use the following data for calculation of ordinary annuity due at the end period

Therefore, the calculation of the ordinary annuity (end) is as follows

- =500,000* 7% /{1-(1+7%)
^{-25}}

**Ordinary Annuity Value (end) will be –**

**Ordinary Annuity Value (End) = 858105.17**

** **Hence, the present value of the ordinary annuity will be 858105.17.

#### Example #2

Mr. Vikram Sharma has just settled in his life. He got married to a girl he was wishing for and also got the job he was looking for a long time. He has done his graduation from London and he has also inherited $400,000 from his father which are his current savings.

He and his wife are looking to buy a house in the town which is worth $2,000,000. Since they don’t own that much funds, they have decided to take a bank loan whereby they will be required to pay 20% from their own pocket and rest would be taken care of by the loan.

The Bank charges an interest rate of 9% and the installments need to pay monthly. They decide to go for 10 years loan and have confidence that they shall repay the same sooner than the estimated 10 years.

You are required to calculate the present value of the installments that they will be paying monthly starting at the month.

**Solution**

Use the following data for calculation of ordinary annuity due at a beginning period

- Here, Mr. Vikram Sharma and family have taken a housing loan which equals $2,000,000 * (1 – 20%) to $1,600,000.
- Now we know the present value of the lump sum amount that shall be paid and now we need to calculate the present value of monthly installments using the below start of the period formula.
- The interest rate per annum is 9%, therefore monthly rate shall be 9%/12 is 0.75%.

Therefore, the calculation of the ordinary annuity (Beg) is as follows

- = 0.75%*1,600,000/{1-(1+0.75%)
^{-119}}

**Ordinary Annuity Value (Beg) will be –**

**Ordinary Annuity Value (Beg) = 20,373.40**

Hence, the present value of the ordinary annuity or the installment will be $20,373.40.

#### Example #3

Motor XP has been recently made available in the market and in order to promote their vehicle the same has been offered a rate of 5% for the initial three months of launch.

John who is aging 60 years now is eligible for an annuity that he purchased 20 years ago. Wherein he made the lump sum amount of 500,000 and the annuity will be paid yearly till 80 years of age and the current market rate of interest is 8%.

He is interested in buying the model XP motor and wants to know whether the same would be affordable for the next 10 years if he takes it on EMI payable yearly? Assume that the price of the bike is the same as the amount he invested in the annuity plan.

You are required to advise John where his annuity will meet the EMI expenses?

** **Assume both are incurred at the end of the year only.

**Solution**

In this case, we need to calculate two annuities one is a normal one and another one is loan annuity.

**Annuity**

Therefore, the calculation of the ordinary annuity (end) is as follows

- = 500,000 * 8%/{1-(1+8%)
^{-20}}

**Ordinary Annuity Value (end) will be –**

**Ordinary Annuity Value (end) = 50,926.10**

**Motor XP**

Therefore, the calculation of the ordinary annuity (end) is as follows

- = 5%*500,000/{1-(1+5%)
^{-10}}

**Ordinary Annuity Value (end) will be –**

**Ordinary Annuity Value (end) = 64,752.29**

There is a gap of 13,826.18 between Annuity payment and Loan payment and hence either John should be able to take out of from pockets or he should extend the EMI till 20 years which is the same as an annuity.

### Relevance and Uses

Ordinary annuities real-life examples could be interest payments from issuers of the bond, and those payments are generally paid monthly, quarterly or semi-annually and further dividends that are paid quarterly by a firm that has maintained payout which is stable for years. PV of an ordinary annuity will be majorly dependent upon the current market interest rate. Due to the time value of money, in case of rising interest rates, the present value of an ordinary annuity will decrease, while in the scenario of declining interest rates it shall lead to an increase in the annuities present value.

### Recommended Articles

This has been a guide to the Ordinary Annuity Formula. Here we discuss the formula to calculate the present value of ordinary annuity along with practical examples and downloadable excel template. You can learn more about financing from the following articles –

- Formula to Calculate the Velocity of Money
- Formula of Annuity Due
- Present Value of an Annuity Calculation
- Future Value of Annuity Due Calculation
- Present Value of an Annuity Formula

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