Derivation of formulas of capital recovery factor, Financial Accounting

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Derivation of Formulas

i) Future Value of an Annuity

Future value of an annuity is

FVAn = A(1 + k)n -1 + A (1 + k)n - 2  + .......A (1 + k) + A     ...............(a1)

Multiplying both sides of the equation a1 by (1 + k) gives.

 (FVAn) (1 + k) = A (1 + k)n  +A(1 +k)n -1 +... A (1 +k)2 +A (1 +k)   .......(a2)

 Subtracting eq. (a1) from eq. (a2) yields

FVAnk = A[((1 + k)n - 1)/k]    ......................................(a3)

Dividing both sides of eq. (a3) by k yields

FVAn = A[((1 + k)n - 1)/k]

ii)                  Present Value of an Annuity

The present value of an annuity as:

PVAnk = A (1 + k)-1 + A(1 + k)-2 + .... + A(1 + k)- n     ............(a 4)

Multiplying both sides of Eq (a 4) by (1+ k) provides:

 PVAn (1 + k) = A + A (1 + k)-1 + ...... + A (1 + k)-n +1    .....................(a5)

Subtracting eq (a4) by (a5) yields:

PVAnk = A[1 - (1 + k)-n]

= A [((1 + k)]n - 1)/(k (1 + k)n)            .....................(a6)

Dividing both the sides of Eq (a6) with k outcomes in as:

PVAn = A [((1 + k)]n - 1)/(k (1 + k)n)


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