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Present Value of an ordinary annuity (Monthly/Annual investments are made to recover a large some of money at the end of n number of periods) can be found using the following formula. [Note:In ordinary Annuity payments are made at the end of the month/year]

**PV = R ****[(1 – (1 / (1 + i) ^{n})) / i]**

where

PV = Present value of the annuity

R = Annual/Monthly cashflow

i = Nominal Interest Rate

n = Number of periods where investments are occurred

To find out the Present value of an annuity due (Where the annuity payments are made at the beginning of the period) the following additional step has to be performed.

Find out the PV of the annuity assuming it is an ordinary annuity using the above mentioned formula.

Use the following formula to find out the PV of the annuity due.

**PV of the annuity due =PV of the ordinary annuity X (1+i)**

Where

i = discount rate per period

**Illustration**

Oct 1st X purchased Lab Equipment (7 year tax life) from “Photonics”.

Photonics offered a zero interest loan for $20,000 (total).

The note is due in 5 years.

X must make monthly payments beginning from Oct. 31st.

similar situation would bear an 8% compounding interest.

Figure out the actual cost of lab equipments.

This is an annuity due as annuity payments are made at the beginning of the period.

As the first step we need to find out the PV of the ordinary annuity ignoring the annuity due for the moment.

R = $ 20000/n = $ 20000/60 = $ 333

Assuming annual interest rate is stated we need to find out the monthly interest rate

i = 8% / 12 = 0.67% n = 5 years X 12 = 60PV = R [(1 - (1 / (1 + i)= $ 333 [ (1-(1/(1+0.0067)^60))/0.0067] = $ 16408 Now convert the PV of the ordinary annuity into annuity due.^{n})) / i]PV of the annuity due =PV of the ordinary annuity X (1+i)= $16408 X (1+0.0067) = $ 16518 Therefore actual cost of the lab equipments is $16518.

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