Question

6) a)Nana just retired at the age of 62 and expects to live until she is 85 years old. She has $402,000 in her retirement savings account. She is somewhat conservative with her money and expects to earn 6 percent during her retirement years. How much can she withdraw from her retirement savings at the end of each month if she plans to spend her last penny on the morning of her death? b)Nana has some extra cash on hand and decided to give you S3,000 today. You have decided to save this money so that you can gift it to your grandchildren 50 years from now. How much additional money will you have to gift to your grandchildren if you can earn an average of 8.5 percent annually instead of just 8 percent annually on your savings? c) Nana wants to use the rest of her extra cash to buy herself a new sports car for $41,750, and the finance office at the dealership has quoted her an 8.6 percent APR loan compound 48 months to buy the car. What is the effective interest rate on this loan?

Please refrain from using excel thanks!

Answer 1

a)

retirement age = r1 = 62

expected age = e = 85

amount in retirement savings account = A = 402,000

interest rate = r = 6% = 0.06

years for withdrawal of money , t = e-r1 = 85-62 = 23

let the amount that can be withdrawn from account per year = x

Present value interest rate factor of annuity (PVIFA) = ((1+r)^t -1)/(((1+r)^t )*r) = ((1+0.06)^t -1)/(((1+0.06)^t )*0.06)

= 12.30338

x = A/PVIFA = 402000/12.30338 = 32673.958

amount that can be withdrawn monthly from the account = x/12 = 32673.958/12 = $2722.8292 or $2722.83 ( rounding off to 2 decimal places)

b)

amount saved today = A = 3000

period of investment (years) = 50

interest rate , r1 = 8.5% = 0.085

interest rate , r2 = 8% = 0.080

amount earned in 50 years if interest rate is r1 = x1 = A*(1+r1)⁵⁰ = 3000*(1.085)⁵⁰ = 177258.95

amount earned in 50 years if interest rate is r2 = x2 = A*(1+r2)⁵⁰ = 3000*(1.08)⁵⁰ = 140704.84

additional money if interest rate earned is 8.5% = x1-x2 = 177258.95 - 140704.84 = $36554.11

c)

amount required for sports car = A = 41,750

quoted rate = r = 8.6% = 0.086

compounding frequency = m = 12    (since compounding is done monthly)

interest rate per month = i = r/m = 0.086/12 = 0.0071667

period of investment (months) = n = 48

effective interest rate = ((1+i)^m) - 1 = (1.0071667)¹² -1 = 0.0894721 or 8.94721% or 8.95% ( rounding off to 2 decimal places)