Doubts

Ex Set 2 - 5c

Ex Set 2 - 5c

by Eric Halmes -
Number of replies: 2

Hello Julio,

I have difficulties understanding the logic of the formula used in 5c.
As I understand it, we start saving "now" which would mean that we put aside 200€ excluding growth and then from period t=1 until t=12 we count for growth.
The growth annuity formula used assumes that he starts saving one month from now (at t=1), with the first contribution already growing at rate gg until t=13 then discounting it 13 periods (and compounding it with *(1+r) at the end).
Hence I dont understand why this solution is correct since the very first CF of "now" is equal to 200*1.002 while the exercise states that he earned 2000€ at t=0 which would be 200€.

Thanks in advance!
Eric 


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In reply to Eric Halmes

Re: Ex Set 2 - 5c

by Julio Crego -
Let's write it down. If we start saving from now, then the present value is:

\(200 + \dfrac{200(1+g)}{1+r} + \dfrac{200(1+g)^2}{(1+r)^2} + ... + \dfrac{200(1+g)^{12}}{(1+r)^{12}}\)

We can multiply and divide everything by (1+r)

\( (1+r)\left(  \dfrac{200}{1+r} + \dfrac{200(1+g)}{(1+r)^2} + \dfrac{200(1+g)^2}{(1+r)^3} + ... + \dfrac{200(1+g)^{12}}{(1+r)^{13}} \right) \)

The part inside the parenthesis is indeed the annuity formula. Note that in the annuity formula, the first payment is in the next period, but the growth is from periods 1 to 2 and not 0 to 1.