# Present value

The present value , and present value called ( English present value ), is a term used in financial mathematics . The present value is the value that future payments have in the present. It is determined by discounting the future payments and then adding them up. There is also the term actuarial present value, which is a generalization of the present value of financial mathematics.

## Present value of a single payment

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

• the amount of the future payment C ,
• the time T at which payment C flows, calculated from today (usually in years),
• the interest rate for which the payment discounted is

The present value is BW then

${\ displaystyle BW = C \ cdot DF (z, T)}$,

whereby the exact form of the discounting factor depends on the selected interest rate convention . For the simple case that denotes a whole number of years, the present value is ${\ displaystyle DF (z, T)}$${\ displaystyle T}$

${\ displaystyle BW = {\ frac {C} {(1 + z) ^ {T}}}}$

and thus

${\ displaystyle DF (z, T) = {\ frac {1} {(1 + z) ^ {T}}}}$.

## Present value of a bond

A common use of the present value formula is to calculate the price using the yield on a fixed income bond . If the bond with a nominal value (repayment amount) N has a term of T whole years and it pays a coupon of c annually , the present value is calculated from the sum of the present values ​​of the interest payments and the present value of the repayment:

${\ displaystyle BW = \ sum _ {t = 1} ^ {T} {\ frac {c} {(1 + z) ^ {t}}} + {\ frac {N} {(1 + z) ^ { T}}} = {\ frac {c} {(1 + z)}} + {\ frac {c} {(1 + z) ^ {2}}} + \ dots + {\ frac {c} {( 1 + z) ^ {T}}} + {\ frac {N} {(1 + z) ^ {T}}}}$.

If the time until the first coupon payment is less than a year, the present value includes pro-rata interest for the first coupon and is referred to as the “dirty price”. If you subtract the accrued accrued interest from the “dirty price”, you get the so-called “clean price”. The “clean price” is the price that is used for stock market quotations, price lists, etc. is listed. The "dirty price" is the price that is actually paid for a sale.

## Present value of an annuity

In financial mathematics, an annuity (or pension ) is a constant, regular payment. If this payment is not limited to a period of time, but flows in for an unlimited period of time, one speaks of a perpetual annuity (also called perpetuity ).

The present value of the amount C , which accrues once a year for an unlimited period ( z = interest rate: 100, e.g. 5: 100 = 0.05), is:

${\ displaystyle BW _ {\ text {forever}} = {\ frac {C} {z}}}$.

For the perpetual annuity, the very simple connection that the present value is greater than the payment by a factor equal to the reciprocal of the interest rate applies. The perpetual annuity can be viewed as regular interest on an investment in the amount of the present value. A perpetual annuity (with positive interest) has a finite present value, although the total of all payments is infinite.

If the (additional) pension only flows for years, the present value is: ${\ displaystyle N}$

${\ displaystyle BW = BW _ {\ text {forever}} \ cdot \ left (1 - {\ frac {1} {{\ big (} 1 + z {\ big)} ^ {N}}} \ right) = {\ frac {C} {z}} - C {\ frac {1} {z {\ big (} 1 + z {\ big)} ^ {N}}} = C \ cdot \ left ({\ frac { 1} {z}} - {\ frac {1} {z {\ big (} 1 + z {\ big)} ^ {N}}} \ right)}$.

The larger and the larger , the more the result approaches that of a perpetual annuity. The factor between payment and present value is called the pension present value factor (= BW / C), its reciprocal value is called the annuity present value factor (= C / BW). ${\ displaystyle N}$${\ displaystyle z}$

Examples: With an interest rate of 5%, the cash value of the perpetual annuity is 20 times the annual payment, the pension cash value factor is 20. The cash value of a 30-year pension is 15.4 times the annual payment, the pension cash value factor is 15. 4th The present value of a one-year pension is 1 / 1.05 times as high as the distribution, the pension present value factor is 1 / 1.05, which is slightly greater than 0.95.

## Actuarial present value

The actuarial present value is a generalization of the financial mathematical present value. Where the latter represents the value that future payments have in the present (only) taking into account discounting , the actuarial present value also includes statistical or stochastic variables such as the probability of death and the like.

The actuarial present value of an annuity, for example, is the sum of all possible future pension payments (including possible survivor pension payments after the death of the pension recipient), each weighted with the probability of their occurrence and discounted to the calculation date.