Discounting and compounding

Discount factor is a redirect to this article. In macroeconomics, the discount factor also denotes the rate of time preference .

Discounting to determine the net present value (sample overview)

Annual discounting with discount factors as a diagram

The discounting (even discounting , Eng. Discounting; often falsely also discounting called) is an arithmetic operation from the financial mathematics . The value of a future payment is calculated here. The present value ( present value ) of a future payment is often determined by means of discounting .

Accordingly, the compounding (also discounting ) is the opposite arithmetic operation. With it, the value that a payment has at a later point in time is determined.

Because of the existence of interest , the sooner it is received, the higher the value of the same amount of money. This relationship is represented by the arithmetic operations of discounting and compounding.

The value of V , the a at the time t ₂ flowing payment of the amount C at the time t ₁ has, calculated as the product of C and the discount rate or discount rate DF which is a function of the time points t ₁ and t ₂ and the relevant rate z is .

${\displaystyle V=C\cdot DF(z,t_{1},t_{2})}$

Since it is a matter of discounting, time t _{1 is} before time t ₂ ( t ₁  <  t ₂ ).

The compounding factor AF is simply the reciprocal of the discounting factor for the same period. He serves z. B. to determine a final value .

Assuming positive interest rates, the discounting factor DF is always less than 1 and greater than 0. Accordingly, the compounding factor is always greater than 1. The exact form of the compounding and the discounting factor depends on the selected interest rate convention .

The interest can be both actual interest ( market interest ) and fictitious, such as imputed or alternative interest (such as in the case of company valuation ).

Determination of the discount factor and the compounding factor

In the following, the form of the discounting factor DF is initially given for a discount to the present (i.e. t ₁  = 0). The discounting factor then only depends on the time of the future payment t ₂ = t and the interest rate z used . The compounding factor AF applies analogously to compounding a current payment at a later point in time t ₂ = t .

Linear interest

The linear interest rate is usually used for periods of less than a year. The factors DF and AF are calculated

${\displaystyle DF={\frac {1}{1+{\frac {n}{m}}z}},AF={1+{\frac {n}{m}}z}}$

where n is the number of interest days up to t ₂ and m is the number of interest days per year according to the selected interest calculation method .

example

• If the interest rate is 5% and t ₂ is 3 months in the future, the discount factor when using the interest calculation method is 30/360 (i.e. 30 interest days per month and 360 interest days per year, so-called German method)
• ${\displaystyle DF={\frac {1}{(1+{\frac {3\cdot 30}{360}}\cdot 0{,}05)}}={\frac {1}{1{,}0125}}=0{,}98765}$.
• This means that a payment of EUR 100 received in 3 months, discounted, would have a current value of EUR 98.77.

Exponential Interest

Exponential interest is typically used for periods longer than a year. It implicitly takes compound interest effects into account. If the interest rate is z and the payment is made in t ₂ years, the discount factor is

${\displaystyle DF={\frac {1}{(1+z)^{t_{2}}}},AF={(1+z)^{t_{2}}}}$.

example

• If the interest rate is again 5% and t ₂ is 4 years in the future, then the factors are
• ${\displaystyle DF={\frac {1}{(1+0{,}05)^{4}}}=0{,}82270}$.

Continuous interest

The steady interest rate is a special case of the exponential rate of return and is often used for theoretical financial mathematical questions. It takes compound interest effects into account. The discount factor for a payment in t ₂ years is here

${\displaystyle DF=e^{-z\cdot t_{2}},\,AF=e^{z\cdot t_{2}}}$.

Here e is Euler's number

example

• Using the same parameters as in the example for exponential interest, the discount factor is
• ${\displaystyle DF=e^{-0{,}05\cdot 4}=2{,}71828^{-0{,}2}=0{,}81873}$,
• is therefore close to that for the exponential interest rate

Discounting to a future point in time

If the time to be discounted is in the future, the calculation is carried out analogously. The interest rate to be used is then an interest rate for a period that starts in the future and thus corresponds to a forward interest rate . If an interest rate of z is assumed, the time of payment t ₂ is in 9 months and it is to be discounted to a time t ₁ in 3 months, then the discount factor is

${\displaystyle DF={\frac {1}{1+{\frac {(9-3)\cdot 30}{360}}z}}}$,

if a linear interest rate and again the German interest rate method is assumed. It is discounted over (9 - 3) months = 6 months because the point in time to which it is discounted is 6 months before the point in time of payment.

See also

• Double discounting
• Social Discount Rate

literature

• Lutz Kruschwitz: Investment calculation . 10th edition. Oldenbourg, Munich 2005. ISBN 3-486-57771-9