Kalpna opened this portfolio 4 years ago.
- A $8500 fund that earns 5.45%, compounded annually
- Monthly deposits of $200 into an account earning 3%, compounded monthly
a) What will be the portfolio's value in 30 years when Kalpna is ready to retire?
b) What will be the portfolio's rate of return?
So I will assume a total time of 34 years or 408 months
amount 30 years from now
= 8500(1.0545)^34 + 200( 1.0025^408 - 1)/.0025
= 51641.968 + 141573.33655
= $ 193215.33
b) is quite complicated.
Let the monthly return be i
then 8500(1+12i)^34 + 200( (1+i)^408 - 1)/i = 193215.33
I entered
solve 8500(1+12x)^34 + 200( (1+x)^408 - 1)/x = 193215.33
into Wolfram and exceeded the standard computation time.
(notice I changed the i to x, since Wolfram interprets i as √-1, the standard definition of i )
ahhh, just tried it again without the "solve"
http://www.wolframalpha.com/input/?i=8500%281%2B12x%29%5E34+%2B+200%28+%281%2Bx%29%5E408+-+1%29%2Fx+%3D+193215.33
and got
x = .0031041
or
i = .0031041
12i = .0372492
So the equivalent annual rate of return for his portfolio is
3.725 %
check:
8500(1.03725)^34 + 200(1.0031041^408 - 1)/.0031041
=193216.61
I am off by $1.28 , due to rounding off my decimals
To calculate the portfolio's value in 30 years, we need to find the future value of both the initial fund and the monthly deposits, taking into account the different compounding periods.
Let's start with the initial fund:
- Principal amount = $8500
- Annual interest rate = 5.45%
- Compounded annually
- Time period = 30 years
Using the compound interest formula:
Future Value = Principal * (1 + (interest rate/100))^time period
Future Value = $8500 * (1 + (5.45/100))^30
Future Value = $8500 * (1.0545)^30
Future Value = $8500 * 3.70411
Future Value = $31,489.35
So, the initial fund will grow to $31,489.35 in 30 years.
Now, let's calculate the future value of monthly deposits:
- Monthly deposit amount = $200
- Monthly interest rate = 3%
- Compounded monthly
- Time period = 30 years
Using the compound interest formula for monthly deposits:
Future Value = Monthly Deposit * (((1 + (interest rate/100))^(12 * time period) - 1) / (interest rate/100))
Future Value = $200 * (((1 + (3/100))^(12 * 30) - 1) / (3/100))
Future Value = $200 * (((1 + 0.03)^(360) - 1) / 0.03)
Future Value = $200 * (((1.03)^(360) - 1) / 0.03)
Future Value = $200 * (3.611 - 1) / 0.03
Future Value = $200 * 2.611 / 0.03
Future Value = $173,733.33
So, the monthly deposits will grow to $173,733.33 in 30 years.
Now, to calculate the portfolio's value in 30 years, we simply add the future value of the initial fund and the future value of the monthly deposits:
Portfolio Value = Future Value of initial fund + Future Value of monthly deposits
Portfolio Value = $31,489.35 + $173,733.33
Portfolio Value = $205,222.68
Therefore, the portfolio's value in 30 years will be $205,222.68.
To calculate the portfolio's rate of return, we need to find the overall increase in value over the 30-year period and then calculate the average annual rate of return.
Initial Investment = $8500
Increase in Value = $205,222.68 - $8500
Increase in Value = $196,722.68
Average Annual Rate of Return = (Increase in Value / Initial Investment)^(1/Number of years) - 1
Average Annual Rate of Return = ($196,722.68 / $8500)^(1/30) - 1
Average Annual Rate of Return = (23.13556)^(1/30) - 1
Average Annual Rate of Return = 1.119139 - 1
Average Annual Rate of Return = 0.119139
Therefore, the portfolio's rate of return is approximately 11.91%.
To calculate the value of Kalpna's portfolio in 30 years and the portfolio's rate of return, we can use the formulas for compound interest.
For the first investment of $8500 with a yearly interest rate of 5.45%, compounded annually, we can use the formula:
Final Value (FV) = Principal Amount (P) * (1 + Interest Rate/Number of Compounding Periods) ^ (Number of Compounding Periods * Number of Years)
FV = $8500 * (1 + 5.45%/1) ^ (1 * 30)
FV = $8500 * (1 + 0.0545) ^ 30
Now, let's calculate the value of the second investment. Kalpna makes monthly deposits of $200 into an account earning 3%, compounded monthly. Here, we will use a slightly different formula:
Final Value (FV) = Monthly Deposit (PMT) * (((1 + Interest Rate/Number of Compounding Periods) ^ (Number of Compounding Periods * Number of Years) - 1) / (Interest Rate/Number of Compounding Periods))
FV = $200 * (((1 + 3%/12) ^ (12 * 30) - 1) / (3%/12))
FV = $200 * (((1 + 0.0025) ^ 360 - 1) / 0.0025)
Finally, we can calculate the total portfolio value by adding the values of both investments:
Total Portfolio Value = Value of First Investment + Value of Second Investment
a) To find the portfolio's value in 30 years, let's calculate the total portfolio value based on the formulas above.
Total Portfolio Value = ($8500 * (1 + 0.0545) ^ 30) + ($200 * (((1 + 0.03/12) ^ (12 * 30) - 1) / (0.03/12)))
b) To find the portfolio's rate of return, we can use the formula:
Rate of Return = ((Total Portfolio Value / Total Deposits) ^ (1/Number of Years)) - 1
Let's plug in the values to calculate the rate of return.