Present Value of Annuity
Present value, also known as present discounted value, is the value on a committed date of a payment or series of payments made at other times.
If the payments are in the future, they are discounted to speculate the time value of money and other elements such as investment risk. If they are in the past, their value is correspondingly increased to speculate that those payments could have been earning interest in the intervening time. Present value computations are widely employed in business and economics to render a means to compare cash flows at different times on a significant like to like basis.
If put forward ed a choice between 100 today or 100 in one year and there is a positive real interest rate during the year ceteris paribus, a rational person will choose 100 today. This is discovered by economists as time preference. Time preference can be evaluated by auctioning off a risk free security - like a US Treasury bill. If a 100 note, payable in one year, sells for 80, then the present value of 100 one year in the future is 80. This is because money can be put in a bank account or any other investment that will bring back interest in the future.
An investor who has some money has two options either to spend it or to save it. But the financial compensation for saving it (and not spending it) is that the money value will accrue through the compound interest that he will get from a borrower (the bank account on which he has the money deposited).
Thus, to assess the real value of an amount of money today after a devoted period of time, economic agents compound the amount of money at a given interest rate. Most actuarial computations employ the risk-free interest rate which corresponds the minimum guaranteed rate rendered by a bank's saving account for example. To compare the alteration in purchasing power, the real interest rate (nominal interest rate minus inflation rate) should be employed.
The operation of evaluating a present value into the future value is known a capitalization (how much will 100 today be worth in 5 years?). The reverse operation-evaluating the present value of a future amount of money-is known a discounting (how much will 100experienced in 5 years-at a lottery for example-be worth today?).
It follows that if one has to choose between receiving 100 today and 100 in one year, the rational determination is to choose the 100 today. If the money is to be experienced in one year and excessively forward the savings account interest rate is 5%, the person has to be put forward ed at least 105 in one year so that two alternatives are equivalent (either receiving 100 today or receiving 105 in one year). This is since if 100 is deposited in a savings account, the value will be 105 after one year.
Technical details
Present value is linear. The present value of a bunch of cash flows is the sum of each one's present value.
In fact, the present value of a cash flow at a constant interest rate is mathematically one point in the Laplace transform of that cash flow, evaluated with the transform variable in general denoted as s equal to the interest rate. The full Laplace transform is the curve of all present values, plotted as a function of interest rate. For discrete time, where payments are separated by large time periods, the transform reduces to a sum, but when payments are ongoing on continuous basis, the mathematics of continuous functions can be employed as an estimation.
There are basically two types of Present Value. Whenever there will be uncertainties in both timing and amount of the cash flows, the anticipated present value approach will often be the appropriate technique.
Traditional Present Value Approach - in this approach a single set of estimated cash flows and a single interest rate (commensurate with the risk, typically a weighted average of cost components) will be employed to estimate the fair value.
Expected Present Value Approach - in this approach multiple cash flows scenarios with different/anticipated probabilities and a credit-adjusted risk free rate are employed to estimate the fair value.
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