# Present Value of a Series of Cash Flow

The present value of a series of cash flow is used to obtain the initial/current worth of a series of cash flow to be invested for some years at a prevalent rate of interest. In other words, it is the amount required to be invested today at a certain rate of interest to meet future uniform cash flow needs. E.g. Money to be invested today to meet rent payment liabilities for the next 5 years.

### Uniform series and Non-Uniform series of payments

• The word series means something one after the other in a pattern, like weekly episodes of a daily soap, 5 volumes of a book, etc.
• If the episodes of a daily soap are telecasted weekly, then it is known as uniform series and the duration between two episodes is the same.
• The duration between the volume of two books may or may not be the same, if it is different it is known as non-uniform series. ### Annuity

• An Annuity is a uniform series of payments made at equal intervals. Examples of Annuity include recurring deposit account, loan repayment in fixed EMIs, etc. • If the amount paid is variable at intervals or different amounts is paid at the same intervals, then it is called as uneven cash flow instead of Annuity. #### Example. 1:

• Sanjay invests Rs 25000 every year in mutual funds and Rakesh pays ₹15000 every month as loan EMI are examples of the annuity.

### Types of Annuities (Annuity Due, Ordinary Annuity and Perpetuity)

• Annuity Due: In this Annuity, the amount is paid at the beginning of each period.
• Ordinary Annuity: In this Annuity, the amount is paid at the end of each period.
• Perpetuity: This is an Annuity for the Infinite Period.

#### Example. 2:

• Ruchi pays her LIC premium of Rs 10000 at the beginning of each year is an Annuity Due
• Nisha deposits Rs 500 in her recurring deposit account at the end of every month is an Ordinary Annuity
• Dividend paid by a company on its share for a lifetime, hence it’s a Perpetuity
• In case of an Annuity Due, as the installment is paid at the beginning of the year, hence all installments are considered for interest payments.
• In the case of Ordinary Annuity, the installment is paid at the end of the year, hence the last installment is not considered for interest calculation.

### Examples and Formula for Present Value of Cash Flow

#### Example 3:

• Priyanka wants Rs 10000 after two years and goes to the bank to invest in an FD that offers interest at the rate of 6%. Let us help Priyanka in finding out how much she should invest. In other words, it is the Present Value of Rs 10000 receivable after 2 years.
• Solution: The formula for calculating the present value of cash is PV=  FV (1 + R) -T
• where FV is the maturity value to be obtained in T years at R% interest
• Hence, PV = 10000 (1 + 0.06) -2  = Rs 8899.96
• The Present Value of an ordinary annuity and annuity due is calculated by adding present values of individual installments as shown below:

#### Example 4:

• Consider an ordinary annuity of Rs 10000 for 5 years at the rate of 10% PA compounded annually.
Year Completed 0 1 2 3 4 5
Instalment Due 0 Rs 10000 Rs 10000 Rs 10000 Rs 10000 Rs 10000
Number of years after which the installment is due to be paid (T) 0   1   2 3 4 5
Calculation of Amount (Using PV of single cash flow) 0 Rs 9090.90   Rs 8264.46 Rs 7513.15   Rs 6830.13   Rs 6209.21
• Here, the Present Value will be some of the present values of all payments made, i.e. PV = Rs 37907.85
• Instead of following this lengthy process, there is a direct formula to calculate the present value of a series of cash flow

### Present Value of Ordinary Annuity:

• PV= A {[1 - (1 + r)-N] /r}
• where A is the amount paid per year for a period of N years and r is the rate of interest.

#### Example 5:

• Consider an annuity due to Rs 10000 for 5 years at the rate of 10% PA compounded annually.
Year Completed 0 1 2 3 4 5
Instalment Due Rs 10000 Rs 10000 Rs 10000 Rs 10000 Rs 10000 0
Number of years after which the installment is due to be paid (T) 0 1 2 3 4 5
Calculation of Amount (Using PV of single cash flow) Rs 10000.00   Rs 9090.90   Rs 8264.46   Rs 7513.14   Rs 6830.13   0
• Here, the Present Value will be some of the present values of all payments made, i.e. PV = Rs 41698.63
• Instead of following this lengthy process, there is a direct formula to calculate the present value of a series of cash flows.

### Present Value of Ordinary Annuity:

• PV= A {[1 - (1 + r)-N] /r}  (1+r)
• where A is the amount paid per year for a period of N years and r is the rate of interest.

#### Example 6:

Amit has rented his house at Rs 50000 per year to Sunil for 5 years. The current rate of interest is 5%. Amit wants to know that what is the Present Value of the total rent he will receive from Sunil in 5 years and Sunil is also concerned and wants to know that how much amount should he invest today so that he is able to pay rent for 5 years.
• Solution: Both Amit and Sunil are talking about one and the same thing
• Rent is always paid in advance, so it is a case of Annuity Due.
• A = 50000, r = 5% and N = 5 years.
• PV = 50000 {[1-(1 + 0.05) -5 ] /0.05 } (1+0.05) = Rs 227297.53
• Hence, if Sunil invests Rs 227297.53 at an interest rate of 5% for 5 years, he will be able to pay all his rent for the next 5 years.
• Also, Amit can now understand that the present value of the total rent receivable in the next 5 years is Rs 227297.53

### Important Points

• In case of uneven cash flows, there cannot be any standard formula as the installment amount keeps changing, therefore it is advised to calculate the future value of uneven cash flows by adding future values of individual installments.
• The present value of a perpetuity is A/R, where A is the periodic payment to be received forever and the expected rate of interest is r%. This is very important in calculations like share price.

#### Example 7:

• M/S ABC pays an annual dividend of Rs 10 indefinitely, ten what should be the per-share price of M/S ABC to attract the investors assuming no capital growth in share price and rate of return expected is 5%.
• Solution: PV of all the dividends received is = A / r =10/.1 = Rs 100,
• If M/S ABC will keep share price below Rs 100 then the investor will be attracted as it will be profitable because the net present value of dividends will be more than the investment.
• If the company quotes share price more than Rs 100 then the investors will not get attracted to the scheme. 