Present Value of Annuity
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Present Value of Annuity is a series of constant cash Flows (CCF) over limited period of time say monthly rent, installment payments, lease rental. There are two types of Annuity:
Ordinary Annuity or Deferred Annuity
If constant cash flow occur at the end of each period/year. Payment of car loan, mortgage loan and student loan are examples of ordinary annuity
Present Value for Ordinary Annuity (Annual Discounting)
The value of annuity at present time evaluated at a given interest rate assuming that discounting take place one time in a year (Annually). Two methods for calculation:
Formula
Time Value of Money Formula Sheet Download
Example # 1:
You want to buy an ordinary annuity that will pay you $1,000 a year for the next 5 year. You expect annual interest rates will be 7 percent over that time period. What is maximum price you would be willing to pay today? 
Present Value of Annuity Table Download
Present Value for Ordinary Annuity (Intra Year Discounting)
The value of annuity at present time evaluated at a given interest rate assuming that discounting take place more than one time in a year (Intra Year). Interest rate reduced while periods of time increase by frequency of compounding (m) i.e. i/m and n*m. Two methods for calculation:
Formula
Example # 2:
Mr. Bengish has signed a contract which pays him $100,000 per year for five years. Using a discount rate of 8% and discounting take place semi-annually, what is the present value of his contract? 
Present Value of Annuity Table Download
>> Practice Present Value of Annuity MCQs.
Annuity Due
If payments or receipts are made at the beginning of each year/period, the annuity is an annuity due. Rental payment for apartment and life insurance payments are typical example of this annuity.
Present Value of Annuity Due (Annual Discounting)
Present value of annuity due is the value of annuity due at present time evaluated at a given interest rate assuming that discounting take place one time in a year (Annually)
Two methods for calculation 
Formula
Example # 3:
What is the value today of a 5-year annuity due that pays $1,000 a year assuming that rate is 7%?
Present Value of Annuity Table Download
Present Value for Annuity Due (Intra-year Discounting)
The value of annuity due at present time evaluated at a given interest rate assuming that discounting take place more than one time in a year (Intra-year). Interest rate reduced while periods of time increase by frequency of compounding (m) i.e. i/m and n*m. Two methods for calculation
Formula
Time Value of Money Formula Sheet Download
Example # 4:
Find the present value of due annuity with periodic payments of $3,000, for a period of 5 years at an interest rate of 16%, discounted quarterly? 
Present Value of Annuity Table Download
>>> Practice Present Value of Annuity MCQs.
When time n is unknown (Annually)
In situation if we have future value and present value of lump sum with interest rate, than we can find time
Example # 5:
You need to accumulate $10,000. To do so, you plan to make deposits of $1,250 per year, with the first payment being made a year from today, in a bank account which pays 12 % interest, compounded annually. Your last deposit will be less than $1,250 if less is needed to round out to $10,000. How many years will it take you to reach your $10,000 goal, and how large will the last deposit be? 
>> Practice Present Value of Annuity Quiz 1.
Further Readings
References
Financial Management: Theory and Practice, Dr Eugene F Brigham & C Micheal Ehrhardt
Fundamentals of Financial Management: Concise Edition, Brigham Houston
The Economist Guide to Financial Management, John Tennet
Financial Management: Core Concepts, Raymond M Brooks
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