Consider a savings plan with an initial deposit of $3500, an annual interest rat
ID: 3144537 • Letter: C
Question
Consider a savings plan with an initial deposit of $3500, an annual interest rate 8.5% compounded monthly and a monthly deposit of $125.
Please note that in this assignment you should round the interest rate to no less than 5 or 6 decimal places. Rounding to less than 5 or 6 decimal places will give incorrect answers. When entering the interest rate in any formulas within your spreadsheet you will get more accurate answers if you (for example) enter 10.25% over a period of 12 months as 0.1025/12 instead of 0.00854.
a. State the finite difference model and its initial condition.
y0 =
(Symbolic) Note: Use ONLY the variable yn. Use numbers in place of all other variables.
yn + 1 =
b. Write the formula derived by the Algebraic Method with the numbers appropriate for this problem.
c. Suppose you wanted to have exactly $20000 in your savings account after 48 months at an annual interest rate of 8.5% compounded monthly and you made an initial deposit of $3500. What would your monthly savings have to be (how much would you have to deposit into the account each month) to achieve this goal? You may use trial-and-error in Excel or find the algebraic solution with Maple or by hand to obtain your answer.
$
d. Using a spreadsheet, recursively generate a table showing the amount in the savings plan over two years. Use this spreadsheet to answer the following questions. Note: Use the numbers from part (a).
How much is in the saving account after...
e. Repeat part (d) if the annual interest rate changes from 8.5% to 9.5%.
How much is in the saving account after...
Explanation / Answer
Assume that deposit is made every month after adding interest.
a) y(0) = 3500, i = 8.5%, let r =1+i=108.5%, d = 125
b) The recurrence will be
y(k+1) = y(k)r+d
y(1) = y(0)r+d
y(2) = y(1)r+d = (y(0)r+d)r+d = y(0)r2 + dr + d
y(3) = y(2)r+d = (y(0)r2 + dr + d)r + d = y(0)r3 + dr2 + dr + d = y(0)r3 + d(1+r+r2)
Hence,
y(n) = y(0)rn + d(1+r+r2+r3+...+rn-1)
Applying the geometric sum formula on the part in the brackets
y(n) = y(0)rn + d(rn-1)/(r-1)
y(n) = (3500)*(1.085)^n + 125{(1.085)^n -1)}/(0.085)
Hence, savings after 4 years i.e. 48 months, y(48) = 176236.58009 Dollars
c) Given that y(48) = 20000, r = 1.085, y(0) = 3500
using the above derived equation: y(n) = y(0)rn + d(rn-1)/(r-1)
20000 = 3500(1.085)^48 + d{(1.085)^48 -1} / (0.085)
d = - 268.98879
D is negative implying that you have to spend $ 268.98879 per month to achieve your goal.