Pension present value factor

The pension present value is the calculated cash capital that would be required to pay money in the form of a pension in a specific amount at a given interest rate over a selected period.

equation

The present value of the pension is the sum of the discounted payments:

${\ displaystyle {\ text {Present value of pension}} = \ sum _ {t = 1} ^ {T} {Z (t) \ over \ left (1 + i \ right) ^ {t}}}$

in which

• ${\ displaystyle Z}$ Payments
• ${\ displaystyle T}$ Number of periods
• ${\ displaystyle i}$ Interest rate for each such period.

In the special case of constant payments, the pension cash value factor (RBF) can be derived, which multiplied by the constant rate gives the pension cash value:

${\ displaystyle {\ text {Rentenbarwert}} = {{\ text {RBF}} \ left (i, T \ right)} \ cdot {\ text {Pension}}}$

The pension present value factor for an additional pension is calculated as follows: ${\ displaystyle {{\ text {RBF}} \ left (i, T \ right)} = {\ frac {(1 + i) ^ {T} -1} {(1 + i) ^ {T} \ cdot i}} = {\ frac {1- (1 + i) ^ {- T}} {i}} = {\ frac {1} {i}} - {\ frac {1} {i {\ big (} 1 + i {\ big)} ^ {T}}}}$

The financial mathematical formula makes it possible to determine the present value of a uniform series of payments (pension payment). The pension present value factor is part of the annuity method of the classic, dynamic investment calculation .

The present value factor for annuities is also referred to variously as the total discount factor, present value annuity factor and total discount factor.

Derivation of the pension present value of the subsequent pension

The equation:

${\ displaystyle {\ text {Rentenbarwert}} = {{\ text {Pension}} \ cdot {\ text {RBF}} \ left (i, T \ right)} = {Z_ {0} \ cdot {\ frac { (1 + i) ^ {T} -1} {(1 + i) ^ {T} \ cdot i}}}}$

With

• ${\ displaystyle T}$ Number of periods
• ${\ displaystyle Z_ {0} = Z (1) = Z (2) = ... = Z (T)}$ Payment for each such period
• ${\ displaystyle i}$ Interest rate for each such period

can be derived as follows:

{\ displaystyle {\ begin {aligned} {\ text {Pension cash value}} & = \ sum _ {t = 1} ^ {T} {Z (t) \ over \ left (1 + i \ right) ^ {t} } = Z_ {0} \ sum _ {t = 1} ^ {T} {1 \ over \ left (1 + i \ right) ^ {t}} = Z_ {0} \ sum _ {t = 0} ^ {T-1} {1 \ over \ left (1 + i \ right) ^ {t + 1}} \ end {aligned}}}

Substitution: ${\ displaystyle \ quad q = {1 \ over 1 + i} \ quad {\ text {with}} | q | <1}$

{\ displaystyle {\ begin {aligned} \ quad \ quad \ quad \ quad \ quad \ quad \ & = Z_ {0} \ sum _ {t = 0} ^ {T-1} q ^ {t + 1} \ \ & = Z_ {0} q \ sum _ {t = 0} ^ {T-1} q ^ {t} \\ & = Z_ {0} q \ left ({\ frac {1-q ^ {T} } {1-q}} \ right) \\ & = Z_ {0} {\ frac {q} {1-q}} (1-q ^ {T}) \ end {aligned}}}

Consider : ${\ displaystyle {\ frac {q} {1-q}}}$

{\ displaystyle {\ begin {aligned} {\ frac {q} {1-q}} = {\ frac {\ frac {1} {1 + i}} {1 - {\ frac {1} {1 + i }}}} = {\ frac {1} {1 + i-1}} = {\ frac {1} {i}} \ end {aligned}}}

Resubstitution:

{\ displaystyle {\ begin {aligned} \ quad \ quad \ quad \ quad \ quad \ quad & = Z_ {0} {\ frac {1} {i}} \ left ({\ frac {(1 + i) ^ {T}} {(1 + i) ^ {T}}} - {\ frac {1} {(1 + i) ^ {T}}} \ right) \\ & = Z_ {0} \ cdot {\ frac {(1 + i) ^ {T} -1} {(1 + i) ^ {T} \ cdot i}} \ end {aligned}}}

The use of the (partial sum) formula of the geometric series must be taken into account when deriving it.

special cases

If the interest rate is zero, the following applies: ${\ displaystyle i}$

${\ displaystyle {{\ text {RBF}} \ left (0, T \ right)} = T}$

If the period approaches infinity, the result is: ${\ displaystyle T}$

${\ displaystyle {{\ text {RBF}} \ left (i, \ infty \ right)} = {\ frac {1} {i}}}$

If the point in time at which the first of the constant payments flows is not , but rather , then the pension present value factor for calculating the present value of the payments at the point in time is determined using: ${\ displaystyle t = 1}$${\ displaystyle t = n}$${\ displaystyle t = 0}$

${\ displaystyle {{\ text {RBF}} \ left (i, T, n \ right)} = {\ frac {(1 + i) ^ {T} - (1 + i) ^ {n-1}} {(1 + i) ^ {T + n-1} \ cdot i}}}$

The reciprocal value (reciprocal value) of the pension present value factor results in the annuity factor (ANF): ${\ displaystyle {\ text {ANF}} \ left (i, T \ right) = {\ frac {1} {{\ text {RBF}} \ left (i, T \ right)}}}$

The annuity factor is also referred to as the recovery factor or capital recovery factor.

Examples

For a pension that is to be paid annually over a period of 10 years, at an interest rate of 5%

${\ displaystyle {{\ text {RBF}} \ left (5 \, \%, 10 \ right)} = 7 {,} 722}$

For a pension that is to be paid annually over a period of 10 years in 5 years (from 01.01. Of the 6th year), this results in an interest rate of 5%

${\ displaystyle {{\ text {RBF}} \ left (5 \, \%, 15.6 \ right)} = 6 {,} 050}$