# Present value

The present value (partly present value, from the English: present value) is a term used in financial mathematics. The present value is the value to have the future payments in the present. It is calculated by discounting the future payments and then summing. There is also the concept of the actuarial present value, which is a generalization of the actuarial present value.

## Present value of a single payment

In the simplest case, the present value of a single payment is to be determined. The following information must be given:

- The amount of future cash C,
- The time T at which the payment flows C, starting from today (usually in years),
- The rate of interest z, with which the payment is discounted

The present value PV is then

The exact form of the discount depends on the chosen interest rate convention. For the simple case that a whole number of years referred to, is the present value

And thus

## Present value of a bond

A common application of the present value formula to calculate the price using the rate of return for a fixed rate bond. Has the bond with a nominal value ( redemption value ) N is a term of T throughout the years and she pays an annual coupon of c, then calculates the present value of the sum of the present values of the interest payments and the repayment of:

If the time to the first coupon payment less than a year, the present value contains rata accrued interest for the first coupon and is referred to as " dirty price ". If we take the " dirty price " from the pro-rata accrued interest, one obtains the so-called " clean price ".

## Present value of an annuity

As an annuity (or pension ) is called in financial mathematics a constant periodic payment. If such payment is not limited to a period, but flows indefinitely, it is called a perpetuity (also perpetuity ).

The present value of the amount of C that flow once a year for an unlimited duration (z = rate of return: 100, for example, 5: 100 = 0.05) is:

So apply the very simple relation for perpetuity, that the present value by a factor of the inverse of the interest rate is equal to is greater than the payment. One can consider the perpetuity as periodic interest on an investment in the cash value. An eternal bond has ( with a positive interest rate ) has a finite present value, although the total of all payments is infinite.

The flows ( annuity ) pension only years, so is the present value:

The bigger and bigger, the more the result approaches to the perpetuity. The factor between payment and the present value is called annuity value factor ( = PV / C), its inverse is called Annuitätenbarwertfaktor (= C / PV).

Examples: At an interest rate of say 5 % of the present value of the perpetuity is 20 times as large as the annual payment of the annuity value factor is 20, the present value of a 30 -year annuity that 15.4 times the annual payment, the annuity value factor is 15, 4, the present value of an annual pension is 1/1, 05 times as high as the distribution, the annuity value factor is 1/1, 05, which is slightly larger than 0.95.

## Actuarial present value

The actuarial present value is a generalization of the actuarial present value. Where the latter (only) representing the value, have the future payments resulting in the present, discounting, flow in the actuarial present value a as well as statistical and stochastic variables such as mortality rates and the like.

The actuarial present value of an annuity, for example, is the sum of all possible future pension payments (including any survivor's pension payments after the death of the pensioner ), each weighted by the probability of its occurrence and discounted to the time of calculation.