Suppose that S2,400 is set aside each year and invested in a savings account tha
ID: 1155700 • Letter: S
Question
Suppose that S2,400 is set aside each year and invested in a savings account that pays 8% nterest pe yer, co ponded aranaus a. Determine the acoumulated savings in this aopount at the end of year 29 b. Suppose that an annuity will be withdrawn from savings that have been interest rate and compounding frequency in Part (a) do not change? acoumulated at the end of year 29. The annuity vill xtnd eill extend from the and of year 30 to the end of year 39. What is the value of this annuity if the nding when 1.8%per year. gack i on to view the interest and annuity table for coinuas compou .The acoumulated savings amount at the and of 29 years well be Round to the neareset dolar) ?. The value ofthematy wd be S0(Rond to tho nearest dolar)Explanation / Answer
a) When the interest is compounded continuously, the formula of the Amount (A) is:
A = P*ert where r is the interest rate and t is the time in years.
Here, P = $2400, r = 8%p.a. or 0.08 and t = 29 years.
Therefore amount accumulated in savings account is:
A = 2400*e^(0.08*29)= $24421.62
Thus, the value of accumulated savings at the end of 29 years = $24421.62
b) It has been found that the future value of the savings account that is to be accumulated in the future is $24421.62. The rate of interest is still 8%p.a compounded continuously. The number of payments made per year = 1. In this example since withdrawal is being made from end of the year 30 to 39, the time in years = 9.
The formula future value of annuity with continuous compounding is given as:
A = R[{e^(kt)-1}/(e^(k/p)-1}]
Where R is the withdrawal per period,
k = r which is the annual continuous compounding rate
t: time in years
And p: number of payments per year
Therefore, 24421.62 = R[{e^(0.08*9)-1}/(e^(0.08/1)-1}]
Or 24421.62 = R (1.05/0.08)
Or R = $1860. 69
Thus, the value of the annuity to be withdrawn is $1860.69.
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