Jane invests 500 at t=0 at a nominal annual interest rate of 6%, compounded
quarterly. What additional amount will she need to invest at t=2 in order to have a total of 1,000 at t=5?
500(1.015)^20 + x(1.015)^16 = 1000
1.268985... = 326.572..
x = $257.35
I miscounted on my "time line"
should have been:
500(1.015)^20 + x(1.015)^12 = 1000
...
x = $273.14
amount at end of t = 2
= 500(1.015)^8
= 563.25
add additional payment of 273.14
= 836.39
amount of that after 3 more years
= 836.39(1.015)^12
=999.9986..
not bad, eh?
To solve this problem, we can break it down into two parts:
1. Calculate the future value of Jane's initial investment of $500 at t=5.
2. Determine the additional amount Jane needs to invest at t=2 in order to reach a total of $1,000 at t=5.
Let's start with the first part:
1. Calculate the future value of Jane's initial investment of $500 at t=5.
- Since the nominal annual interest rate is 6% and it is compounded quarterly, we need to adjust the interest rate per quarter. We can do this by dividing the annual interest rate by 4: 6% / 4 = 1.5% per quarter.
- The compound interest formula for calculating the future value is: FV = PV * (1 + r)^n, where FV is the future value, PV is the present value (initial investment), r is the interest rate per period, and n is the number of periods.
- In this case, PV = $500, r = 0.015 (1.5% in decimal form), and n = 5 (as Jane wants to calculate the future value at t=5).
- Plugging these values into the formula, we get:
FV = $500 * (1 + 0.015)^5
Now, let's move on to the second part:
2. Determine the additional amount Jane needs to invest at t=2 in order to reach a total of $1,000 at t=5.
- To find the additional amount, we need to calculate the future value of the investment at t=2 and then subtract it from the desired total of $1,000.
- We can use the same compound interest formula: FV = PV * (1 + r)^n.
- In this case, PV will be the additional amount Jane needs to invest at t=2, r will be 0.015 (1.5% per quarter), and n will be 3 (as Jane wants to calculate the future value at t=5, considering she has already invested for 2 quarters).
- Setting up the equation, we have:
FV2 = PV2 * (1 + 0.015)^3
PV2 = (1,000 - FV) / (1 + 0.015)^3
To get the final answer, we need to substitute FV into the equation for PV2 and calculate the additional amount Jane needs to invest at t=2.