Overall difference in percent

Question

I want to calculate the total difference in % between two investments {A,B} in the following scenario:

In year t=0 revenue A is 70 % smaller than revenue B. Every year the revenue from A further decreases by 50 % while revenue B decreases by 65% each year.

What is the overall difference in % between investment A and B after 10 years?

Answer 1

In year $t=0$ the revenue $A$ is $70 \%$ smaller than the revenue $B$:

$$t=0: A_0=B_0-0.7B_0=0.3B_0$$

$$$$

Every year the revenue $A$ decreases by $50 \%$:

After $1$ year: $A_1=A_0-0.5A_0 =0.5A_0 =0.5 \cdot 0.3B_0$

After $x$ years: $A_x =0.5^x A_0 =0.5^x \cdot 0.3B_0$

$$$$

The revenue $B$ decreases by $65 \%$ each year:

After $1$ year: $B_1=B_0-0.65B_0 =0.35B_0 $

After $x$ years: $B_x=0.35^x B_0 $

$$$$

Can you calculate now the overall difference in $\%$ between investment $A$ and $B$ after $10$ years ??

Answer 2

If A is 70% smaller than B, then $A=(1-0.7)B=0.3B$

If A decreases 50% every year, then A is after 1 years: $A\cdot (1-0.5)^1=0.3B\cdot 0.5^1$. And after 10 years the sum of all revenues is $\sum_{i=0}^{9} 0.3B\cdot (0.5)^i=0.3B \cdot \frac{1-0.5^{10}}{0.5} $

For the last transformation the formula for partial sum of a geometric series has been used.

If B decreases 65% every year, then B is after 1 years: $B\cdot (1-0.65)^1=B\cdot (0.35)^1$. And after 10 years the sum of all revenues $\sum_{i=0}^{9} B\cdot (0.35)^i=B \cdot \frac{1-0.35^{10}}{0.65} $

Now calculate the two terms and compare them.