Carlos has worked very hard through his career as a teacher and is contemplating an early...
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Carlos has worked very hard through his career as a teacher and is contemplating an early retirement. He already has a pension in place commencing at retirement age but wants to cut back on his teaching duties immediately. To supplement his income, Carlos has a retirement savings plan which will allow him to make withdrawals from a savings account from 10 years prior to the retirement age. Carlos needs to work out the minimum amount required in the savings account with the following specifications. Interest Credit - Interest on Savings will be paid at a rate of 6% p.a. Withdrawals - starting with $50 000 at the end of the first year decreasing by 10% each year (end) as Carlos wants to travel and experience the good life before retiring. The savings account will be depleted at the end of the 10 years (nil balance) Let S to be the minimum required savings account balance at time t. (t=0 represents today, t=10 represents the time of retirement). (a) Explain why the following equation is a suitable finite difference model for this scenario: St+1 = 1.06 St 50 000 x 0.9t (1) (b) Considering the savings account will be depleted at the end of the 10 years, what is a suitable boundary condition that will be used to find a solution to this problem? (c) Calculate the complementary function for the retirement savings model. (d) Because the term 50 000 x 0.9 in Equation (1) is a function in t and not a constant, an adjustment needs to be made. Divide Equation (1) throughout by 0.9 and redefine a new variable Y = St 0.9 Rewrite Equation (1) again in the form of Y++1 = AY+ + B for suitable A and B values. (1a) (e) Compute the particular solution for (1a), first in terms of Y, but ultimately in terms of SE for the solution to Equation (1). (f) Calculate So, the amount of money needed today to finance Carlos' pre-retirement plan. Carlos has worked very hard through his career as a teacher and is contemplating an early retirement. He already has a pension in place commencing at retirement age but wants to cut back on his teaching duties immediately. To supplement his income, Carlos has a retirement savings plan which will allow him to make withdrawals from a savings account from 10 years prior to the retirement age. Carlos needs to work out the minimum amount required in the savings account with the following specifications. Interest Credit - Interest on Savings will be paid at a rate of 6% p.a. Withdrawals - starting with $50 000 at the end of the first year decreasing by 10% each year (end) as Carlos wants to travel and experience the good life before retiring. The savings account will be depleted at the end of the 10 years (nil balance) Let S to be the minimum required savings account balance at time t. (t=0 represents today, t=10 represents the time of retirement). (a) Explain why the following equation is a suitable finite difference model for this scenario: St+1 = 1.06 St 50 000 x 0.9t (1) (b) Considering the savings account will be depleted at the end of the 10 years, what is a suitable boundary condition that will be used to find a solution to this problem? (c) Calculate the complementary function for the retirement savings model. (d) Because the term 50 000 x 0.9 in Equation (1) is a function in t and not a constant, an adjustment needs to be made. Divide Equation (1) throughout by 0.9 and redefine a new variable Y = St 0.9 Rewrite Equation (1) again in the form of Y++1 = AY+ + B for suitable A and B values. (1a) (e) Compute the particular solution for (1a), first in terms of Y, but ultimately in terms of SE for the solution to Equation (1). (f) Calculate So, the amount of money needed today to finance Carlos' pre-retirement plan.
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a The equation St1 106 St 5000009t is a suitable finite difference model because it takes into accou... View the full answer
Related Book For
Foundations Of Finance
ISBN: 9780135160619
10th Edition
Authors: Arthur J. Keown, John H. Martin, J. William Petty
Posted Date:
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