Howard has deposited RM1000 at the end of each month into a retirement savings plan for the last 10 years in his working life. His deposits earned an interest rate of 2.5% per month for the first 4 years and 3% per month for the rest of the years.
What is the accumulated value of his retirement savings plan?
Think of it as 48 payments of 1000 at 2.5%, then that sum riding along for 72 periods at 3% plus 72 payments of 1000 for the next 72 periods
amount = 1000(1.025^48 - 1)/.025 (1.03)^72 + 1000(1.03^72 - 1)/.03
= 1,009,889.30
can i ask why the first four years need to multiply (1.03)^72
You should get into the habit of making a time graph on a straight line for these kind of questions.
Think of the whole thing as being in two accounts.
1000(1.025^48)/.025 would be amount you would have at the end of 4 years, sitting at time 4
But you want that amount at time 10, with a new interest rate of 3%
So you have to find the future amount of the above value sitting at 4 accumulating at 3% for the next 6 years.
In your second account, my second term in the equation, thing of it as a new account starting at time 4 and going on for 72 months.
okay, i get it. Thanks for helping
To calculate the accumulated value of Howard's retirement savings plan, we need to calculate the future value of each monthly deposit over the 10-year period.
Here's how to do it step by step:
1. Calculate the future value of each monthly deposit during the first 4 years (48 months) using the formula for future value of a series of regular deposits:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = Future Value
P = Monthly deposit amount
r = Monthly interest rate
n = Number of months
For the first 4 years, the monthly deposit is RM1000, and the interest rate is 2.5% per month.
Using the above formula, we can calculate the future value for the first 4 years:
FV1 = 1000 * ((1 + 0.025)^48 - 1) / 0.025
2. Calculate the future value of each monthly deposit for the remaining years (from month 49 to month 120) using the same formula, but with a different interest rate. For the remaining years, the monthly deposit is still RM1000, but the interest rate is 3% per month.
Using the formula again, we can calculate the future value for the remaining years:
FV2 = 1000 * ((1 + 0.03)^(120 - 48) - 1) / 0.03
3. Calculate the total accumulated value of the retirement savings plan by adding the future values from step 1 and step 2:
Accumulated Value = FV1 + FV2
Now, let's calculate it:
FV1 = 1000 * ((1 + 0.025)^48 - 1) / 0.025
≈ 56720.02
FV2 = 1000 * ((1 + 0.03)^(120 - 48) - 1) / 0.03
≈ 96003.95
Accumulated Value = FV1 + FV2
= 56720.02 + 96003.95
≈ RM152,723.97
Therefore, the accumulated value of Howard's retirement savings plan is approximately RM152,723.97.