Accumulation function: Difference between revisions

Revision as of 10:44, 27 June 2018

Revision as of 00:40, 28 June 2018

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:<math>A(t) = A(0) \cdot a(t)</math>.

where the initial investment is <math>A(0).</math>

'''A'''(a,b) = A(b)÷A(a) where 0 < a < b

For various interest-accumulation protocols, the accumulation function is as follows (with ''i'' denoting the [[interest rate]] and ''d'' denoting the [[annual effective discount rate|discount rate]]):

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In the case of a positive [[rate of return]], as in the case of interest, the accumulation function is an [[increasing function]].

===Variable rate of return==

Derivation of Compounded Interest rate function:

Assume an investment of 1 unit at time T₀ .

At time T₁ the invest ment increases 1 × i , thus the value at T₁ =1 + i.

At time T₂ the invest ment increases with (1 + i) i , thus the value at T₂ = (1 +i ) + (1+i)i = (1+i) (1+i) = (1+i)²

We can continue with this pattern up until time T_k thus the value at time T_k = (1 + i )^k

We can then define a function that finds the value of an investment 1 at time t as the following a(t) = (1 + i)^t where i is the fixed compounded interest rate.

===Variable rate of return==

The [[Rate_of_return#Logarithmic_or_continuously_compounded_return|logarithmic or continuously compounded return]], sometimes called [[Compound interest#Force of interest|force of interest]], is a function of time defined as follows: