Question

Course: Theory of Interest (Actuarial Science)

Chapter: Yield Rates

Problem: This is a Multi-Part Question

Joe's retirement scheme at work pays $500 at the end of each month. Joe puts his money in an account which earns a nominal 12% converted monthly, the interest is reinvested at a nominal 4% converted monthly. Carol's account also pays $500 at the end of each month, but she earns nominal 12% convertible monthly (principal and interest both earn 12%). After 20 years, Joe and Carol retire.

a) How much money will Carol have? Answer: $184,465.8246

b) How long will it be before Carol's account exceeds Joe's by $1,000,000? Answer: 355.0933299 months or 29.59111082 years

Answer 1

Below is he formula for an annuity where the reinvested at different rate is given below:

Accumulated Future Value = A * (n + i% * ((((1+r%)^n-1)/(r%)) - n)/(i%))

Where i% = coupon interest earned by investor.

A = annuity amount

n =number of periods

r% = reinvestment rate

By using formula, substituting the values, Joe's accumulated value after 20 years

Future value(Joe) = =500*(240+1%*(( ((1+0.333%)^(240)-1)/(0.3333%))-(240))/0.3333%)

Where i and r are given in terms of monthly

Future value (Joe) = $310,161.94

Future value (Carol) = usual annuity with 1% monthly interest compounding

Future value (carol) = 500 * ((1+1%)^240-1)/(1%) = $494,627.68

After 20 years, Future value (Carol)- Future value (Joe) = $494,627.68 - $310,161.94 = $184,465.74 Answer

Similar to the above, we can now find the period n at which difference is $1,000,000

Future value (Carol)- Future value (Joe) = $1,000,000

500 * (n + 1% * ((((1+4%/12)^n-1)/(4%/12)) - n)/(1%)) - 500 * ((1+1%)^n-1)/(1%) = 1000,000

For 300 months, the difference can be calculated by using above basis as $468,229

For 340 months, it is $818,051.59

For 350 months, the difference is 934,966.24

For 360 months, the difference is 1,066,407.96

Hence, the required months should be between 350 months and 360 months. The approximate months can now be calculated using linear interpolation.

A difference of (1,066,407.96-934,966.24) involves a reduction of 10 months,

so, a difference of (1066,407.96 - 1,000,000) will include a reduction of

Difference = (10/(1,066,407.96-934,966.24) *(1066,407.96 - 1,000,000) = 5.05

So, a diff of 1,000,000 will involve 360-5.05 months = 354.95 months

Hence the answer using linear interpolation formula is 355 months or 29.58 years.

(Note the entire above calculation can be easily done using Microsoft excel sheet and excel's goal seek function by tabulating the values and using formulas. The above example is done to explain the concept

Using goal seek feature of excel, we can find that for a period of 355.09

Future value (Carol)= $1,661,831.72

Future value (Joe) = $ 661,831.72

So , it will take 355.09 months or 29.59 years for Carol's accumulated future value to exceed Joe's by $1,000,000.