Let
v0 = initial value
vn = value after n periods
n = number of periods
r = annual rate of return
y = required annual net income
The one period discount factor is:
j = 1/(1+r)
The present value of the investment is:
pv = - v0 + j*y + j^2*y + j^3*y +..+ j^n*y + j^n*vn
= - v0 + y*(j + j^2 + j^3 +..+ j^n) + j^n*vn
= - v0 + y*sn + j^n*vn
where
sn = j + j^2 + j^3 + j^4 +..+ j^n
We can calulate sn
as follows:
sn = j + j^2 + j^3 + j^4 +..+ j^n
j*sn = j^2 + j^3 + j^4 +..+ j^n + j^(n+1)
sn -j*sn = j*(1 - j^n)
sn = j*(1 - j^n)/(1-j)
= (1 - j^n)/[(1+r)(r/(1+r)]
= (1 - j^n)/r
Set pv = 0
and solve for y
:
y*sn = v0 - vn * j^n
y = (v0 - vn * j^n)/sn
= r * (v0 - vn * j^n)/(1 - j^n)
Our Ruby method:
def ann_ret(v0, vn, n, r)
j = 1/(1+r)
(r * (v0 - vn * j**n)/(1 - j**n)).round(2)
end
With annual compounding:
ann_ret(18000000, 13000000, 5, 0.1) # => 2618987.4
With semi-annual compounding:
2 * ann_ret(18000000, 13000000, 10, 0.05) # => 2595045.75
With daily compounding:
365 * ann_ret(18000000, 13000000, 5*365, 0.10/365) # => 2570881.20
These values differ slightly from the required annual return you calculate. You should be able to explain the difference by comparing present value formulae.