Q.234. Perpetual retirement Plan

Question 234.
Express the principal amount P needed for perpetual retirement in terms of x, r and i given the following conditions.
1) x = amount of expenses set aside from P for the first year expenses,
2) r = rate of return on investment, expressed as %
3) i = rate of inflation, expressed as %
4) The amount to be set aside for the second year will be
     x * (1 + i/100).
     [Hint: Find Pn = amount remaining in hands at the 
              end of nth years and P = lim (n → ∞) Pn]

Answer 234.
P = initial amount needed for perpetual retirement
x = first year expenses
r = % return on amount invested (constant for all years)
i = % inflation rate (constant for all years)

1) Amount at start = P

2) Amount invested at the start of first year setting aside x
= P - x

3) Amount at the end of the first year
= (P - x) * (1 + r/100)

4) Amount invested at the start of second year setting aside
x (1 + i/100) for use in second year
= (P - x) * (1 + r/100) - x * (1 + i/100)

5) Amount at the end of the second year
= (P - x) * (1 + r/100)^2 - x * (1 + i/100)(1 + r/100)

6) Amount invested at the start of third year setting aside
x (1 + i/100)^2 for use in third year
= (P - x) * (1 + r/100)^2 - x * (1 + i/100)(1 + r/100) - x * (1 + i/100)^2

7) Amount at the end of the third year
= (P - x) * (1 + r/100)^3 - x * (1 + i/100)(1 + r/100)^2
- x * (1 + i/100)^2 * (1 + x/100)
= P * (1 + r/100)^3 - x * (1 + r/100)^3 * [1 + (1+i/100)/(1+r/100)
+ (1+i/100)^2 / (1+r/100)^2]

8) Amount at the end of nth year, Pn
= P(1+r/100)^n - x(1+r/100)^n [1 + (1+i/100)/(1+r/100)
+ (1+i/100)^2 / (1+r/100)^2 + … + (1+i/100)^n / (1+r/100)^n]

9) For perpetual retirement, lim (n → ∞) Pn = 0
=> lim (n → ∞) (1 + r/100)^n * [P - x * lim (n → ∞) [1 + (1+i/100)/(1+r/100)
+ (1+i/100)^2 / (1+r/100)^2 + … + (1+i/100)^n / (1+r/100)^n] = 0
=> lim (n → ∞) (1 + r/100)^n * [P - x / {1 - (1+i/100)/(1+r/100)} ] = 0
=> P = x * (100+r) / (r - i).

Using the above formula, For x = 50000, r = 8 and i = 4, P = 1350,000.

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