Pension bill

The pension calculation is a classic method of financial mathematics .

definition

A pension is a periodic series of payments. If the payments agreed in advance are only made if one or more specific persons are still alive on the relevant payment date, this is called an annuity . These are the subject of actuarial mathematics . If the agreed payments are made regardless of the life of the people involved in the contract, this is called a time pension. This article deals only with term annuities.

Basic concepts

The time unit is a year. In addition, the same pension amount r has to be paid annually . A pension is in arrears or arrears pension if the payments are made at the end of each contract year; if they occur at the beginning of the contract years, one speaks of an advance or a prenumerando pension.

If someone has invested n amounts of r euros with compound interest at annual intervals , the capital can be calculated that is available at the end of the n th year. It is called the final value of the pension. In the case of an additional pension, this is the value of the pension immediately after the last payment, whereas in the case of an advanced pension, it is the value one year after the last payment.

Another question is about the capital that must be available when the contract is signed so that one can use it and its interest to make future individual payments of r euros. It is called the present value of the pension.

Another point of view: the final value and present value replace the result of the annuity payments with a one-time payment - taking into account compound interest.

Both values depend on the amount r and the number n of pension payments as well as the rate of interest p> 0 .

Basic formulas

In the following formulas denotes the interest factor , if the interest rate is. ${\ displaystyle q}$ ${\ displaystyle q = 1 + i}$ ${\ displaystyle i}$

In the literature with or not quite correctly is referred to as . ${\ displaystyle i}$ ${\ displaystyle p \ \%}$ ${\ displaystyle p}$

Example for an interest rate of 5%:

{\ displaystyle i = 5 \ \% \ \ Rightarrow \ q = 1 + i = 1 + 0 {,} 05 = 1 {,} 05}

	Advance	Follow-up
Present value	${\ displaystyle B _ {\ text {vor}} = r \ cdot {\ frac {q ^ {n} -1} {q ^ {n-1} (q-1)}}}$	${\ displaystyle B _ {\ text {to}} = r \ cdot {\ frac {q ^ {n} -1} {q ^ {n} (q-1)}}}$
Final value	${\ displaystyle E _ {\ text {vor}} = r \ cdot {\ frac {q (q ^ {n} -1)} {q-1}}}$	${\ displaystyle E _ {\ text {to}} = r \ cdot {\ frac {q ^ {n} -1} {q-1}}}$

Note: ${\ displaystyle q-1 = (1 + i) -1 = i}$

Graphic illustration of the advance and subsequent pension formulas:

Legend for the picture below:

${\ displaystyle B _ {\ text {to}}}$ : cash value in arrears at the time ${\ displaystyle t = 0}$

${\ displaystyle E _ {\ text {to}}}$ : subsequent final value at the time ${\ displaystyle t = n}$

${\ displaystyle B _ {\ text {before}}}$ : advance cash value at the time ${\ displaystyle t = 1}$

${\ displaystyle E _ {\ text {before}}}$ : advance final value at the time ${\ displaystyle t = n + 1}$

The following definitions apply:

The final value of an additional pension is the current value on the day of the last installment payment.

The final value of an advance pension is the current value of an interest period after the last installment payment.

The present value of an additional pension is the current value of an interest period before the first installment payment.

The present value of an advance pension is the current value on the day of the first installment payment.

Payment duration

The number of pension payments, after which a capital is used up, results from the formula (with advance payment)

{\ displaystyle n = {\ frac {\ ln {\ left ({\ frac {r} {q \ cdot rB \ cdot (q-1)}} \ right)}} {\ ln (q)}} + 1 }

.

Here, B is the originally available capital (the present value), q the interest factor with which this capital is invested and earns interest, and r the amount of the annuity regularly paid from it.

Hints:

Of course, this calculation assumes that the interest rate remains the same over the entire duration of the annuity payment and also does not change due to the fact that the capital becomes smaller over time.

If the annual interest rate is used to calculate q , then the annual pension must also be inserted for r . In the case of advance payment, the monthly pension is slightly higher than a twelfth of the annual pension (because the monthly payments that have not yet been paid are still subject to interest). If you want to use months as payment periods instead , you can use one twelfth of the annual interest as the monthly interest rate if the interest credit is only paid annually. If the interest credit is also paid monthly, the monthly interest factor is the 12th root of the annual interest factor. These inaccuracies are insignificant for a rough calculation.

height

The amount of the pension that can be paid out of a lump sum is derived from the formula (with advance payment)

{\ displaystyle r = {\ frac {B \ cdot q ^ {n-1} \ left (q-1 \ right)} {q ^ {n} -1}}.}

Again, B is the originally available capital (present value) and q is the interest factor. n is the number of annuity payments to be made.

The same notes apply as in the previous section.

Math background

The following results for the final value of the advance pension: The first contribution earns interest n times, the second contribution (n − 1) times interest and so on up to the last ( nth ) contribution, which is exactly once (i.e. for one year ) earns interest. The following applies to the final value E of the advance pension:

{\ displaystyle {\ begin {matrix} E & = & r \ cdot q ^ {n} + r \ cdot q ^ {n-1} + \ dotsb + r \ cdot q \\ & = & r \ cdot (q ^ {n } + q ^ {n-1} + \ dotsb + q) \ end {matrix}}}

Because of

{\ displaystyle {\ begin {matrix} (q ^ {n} + q ^ {n-1} + \ dotsb + q) \ cdot (q-1) & = & (q ^ {n + 1} + q ^ {n} + \ dotsb + q ^ {2}) - (q ^ {n} + q ^ {n-1} + \ dotsb + q ^ {2} + q) \\ & = & q ^ {n + 1 } -q \\ & = & q (q ^ {n} -1) \ end {matrix}}}

can be replaced by and you get the above formula. The other basic formulas can be derived analogously. ${\ displaystyle q ^ {n} + q ^ {n-1} + \ dotsb + q}$ ${\ displaystyle {\ frac {q \ cdot (q ^ {n} -1)} {q-1}}}$

Eternal Rent and Eternal Loan

A pension, in which the number of pensions from unlimited payments, called "perpetuity": This is only the current interest income from paid, the capital itself, however remains. Counterpart of "perpetuity" the (in Germany rather ungebräuchlichen) "eternal bonds," are thus (Engl. Perpetuals ), which reversed only serves the current interest , d. H. a must be paid, the loan debt itself, however, remains unpaid .

literature

Jürgen Tietze: Introduction to financial mathematics. Vieweg, Wiesbaden 2006, ISBN 3-8348-0093-7 .

Web links

Direct calculation of present value and final value as well as interest rate and term.
Pension calculation tutorial with examples and solutions. ( Memento from April 14, 2015 in the Internet Archive )

Individual evidence

↑ Arne Storn: Please be patient ! ; DIE ZEIT No. 15/2015, April 9, 2015 , last accessed August 20, 2016.

[1] Arne Storn: Please be patient ! ; DIE ZEIT No. 15/2015, April 9, 2015 , last accessed August 20, 2016.