The following is the problem and the solution:
Before looking at the solution, here is how I approached the problem:
Let $X$ be the amount that each child receives. (i) and (ii) imply that $Xa_{20} = v^{20}\cdot{}2X\cdot\frac{1}{i},$ where $a_{20}$ is the present value of an annuity immediate and $\frac{1}{i}$ is the present value of a perpetuity immediate.
Let $T$ be the value of the estate. Then $Xa_{20} = \frac{1}{3}T$ and $v^{20}\cdot{}2X\cdot\frac{1}{i} = \frac{1}{3}T.$
We also know that $Xa_{20} + Xa_{20} + v^{20}\cdot{}2X\cdot\frac{1}{i} = T$.
At this point I didn't know how to solve for either the interest rate or the value of the estate.
The solution does not even appear to define the value of the estate. At the same time, I don't understand the sentence:
"Payment to each child for 20 years = $\frac{1}{3a_{20}}$." Could someone walk me through the solution and specifically the idea of taking the reciprocal?
Also, is there a way to make my solution attempt work?
First of all it is assumed that john´s estate is 1. It works also with other values. Thus all three, children and charity, receive $\frac{1}{3}$.
Now you use the formula for the present value. The present value have to be 1/3 for both children.
The present value for one child is
$C_0=r\cdot v\cdot \frac{1-v^n}{1-v}\Rightarrow\frac{1}{3}= r \cdot v\cdot\frac{1-v^n}{1-v}$
r are the annual payments.
$v=\frac{1}{1+i}\ $. i is the interest rate.
Solving the equation for r gives:
$r=\frac{1}{3v}\cdot\frac{1-v}{1-v^n}$
Simplifying $\frac{1-v}{v}$
$\frac{1-v}{v}=\frac{1-\frac{1}{1+i}}{\frac{1}{1+i}}=(1+i)\cdot (1-\frac{1}{1+i})=1+i-1=i $
Therefore it is $r=\frac{i}{3\cdot(1-v^n)}$
$n=20$
And for both children it is $\frac{2i}{3\cdot(1-v^{20})}$
The present value for a perpetuity is $C_0=\frac{r}{i}$.
$C_0=\frac{r^c}{i}$
The index c is for charity.
But it has to be discounted for 20 years, because it starts 20 years later.
$\frac{1}{3}=\frac{r^c}{i}\cdot v^{20}$
Solving for $r^c$
$r^c=C_0\cdot i=\frac{1}{3\cdot v^{20}}\cdot i$
Setting both terms equal:
$\frac{1}{3\cdot v^{20}}\cdot i=\frac{2i}{3\cdot(1-v^{20})}$
Multiplying both sides by 3 and dividing both sides by i:
$\frac{1}{ v^{20}}=\frac{2}{(1-v^{20})}$
Taking the reciprocals of both sides
$v^{20}=\frac{(1-v^{20})}{2}$
$v^{20}=\frac{1}{2}-\frac{v^{20}}{2} \quad |+\frac{v^{20}}{2}$
$\frac{v^{20}}{2}+v^{20}=\frac{1}{2}$
$\frac{3}{2}v^{20}=\frac{1}{2}$
$3\cdot v^{20}=1$
$v^{20}=\frac{1}{3}$
$v=\left( \frac{1}{3} \right)^{1/20}$
$v\approx 0.9465=\frac{1}{1+i}$
$\Rightarrow \frac{1}{0.9465}=1+i$
$i=\frac{1}{0.9465}-1\approx 0.0565=5.65\%$