Value of Investment in the Past

Question

An amount of $1000$ is to be accumulated at a compound rate of discount of $9$% per year. (a) Find the present value $3$ years before (b) Find the value of i corresponding to d.

For a) i have done the following: $1000=X(1-0.09)^{-3}$

$X=0.19$

I am not sure on how to go on about (b).

Answer 1

EDIT: This answer is probably incorrect. It is left up in order for OP to provide clarification.

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If you have \$1000 now and you want to know the value three years prior, \$1000 is your initial value. The relevant computation is therefore $$ 1000(1 - 0.09)^{-3} \approx 1327.01. $$

An alternative approach is to say I have an initial value of $X$ that will be worth \$1000 in three years. In this case, you would solve $$ 1000 = X(1 - 0.09)^3 $$ and find $X \approx 1327.01$.

Answer 2

For (b), it would seem to me that what you seek is a value of interest $i$ such that

$$1+i = \frac{1}{1-d}$$

where $d$ is the depreciation. This then means that

$$i=\frac{d}{1-d} = \frac{0.09}{0.91} \approx 0.0989$$