You deposited $160,000 into an education savings plan, hoping to have $420,000 a

ID: 455505 • Letter: Y

Question

You deposited $160,000 into an education savings plan, hoping to have $420,000 available 12 years later when your first child starts university. However, you did not invest very well, and two years later, the account’s balance has dropped to $140,000. Let us look at what you need to do to get the savings plan back on track.

What was the original annual rate of return needed to reach your goal when you started the fund two years ago?(3 marks)

With only $140,000 in the fund and 10 years remaining until your first child starts university, what annual rate of return would the fund have to make for you to reach your $420,000 goal if you add nothing to the account?(3 marks)

Shocked by your experience of the past two years, you feel the education savings fund has invested too much in shares, and you want a low-risk fund in order to ensure you have the necessary $420,000 in 10 years. You are willing to make end-of-the-month deposits to the fund as well. You find you can get a fund that promises to pay a guaranteed annual return of 6% that is compounded monthly. You decide to transfer the $140,000 to this new fund and make the necessary monthly deposits. How large a monthly deposit must you make into this new fund each month to obtain the $420,000 required at the end of 10 years?(6 marks)

After seeing how large the monthly deposit would be (in part (c) of this question), you decide to invest the $140,000 today and $500 at the end of each month for the next 10 years into a fund comprising 50% shares and 50% bonds and hope for the best. What Annual Percentage Rate (APR) would the fund have to earn in order to reach your $420,000 goal?

Explanation / Answer

1. Future value = 420,000. PV = 160,000 n = 12 years. Let the rate be "r".

Thus, FV = PV*(1+r)^n or 420,000 = 160,000*(1+r)^12

or, 420,000/160,000 = (1+r)^12

or, 2.625 = (1+r)^12, or 1+r = 1.0837 or r = 0.0837 or 8.37%

2. Now, PV = 140,000. n = 10 and FV = 420,000. using the same formula as in "1" above

420,000 = 140,000*(1+r)^10

or, (1+r)^10 = 3

or 1+r = 1.1161 or r = 0.1161 or 11.61%

3. Now, the principle of monthly compounding will come into play.

No. of deposits = 10 years*12 = 120 deposits. monthly return = 6%/12 = 0.5%

Let the monthly deposits be = x. This is an annuity and we have to use the future value of annuity for 0.5% and 120 deposits.

Future value = x*[1-(1.005)^120/1-(1.005)] = x*163.8793

Value of 140,000 at the end of 120 months = 140,000*(1+6%/12)^10*12

= 140,000*(1.005)^120 = 254,715.54

Thus, 254,715.54+x*163.8793 = 420,000

x = (420,000-254715.54)/168.8793 = $978.71

4. Let the APR be "r"

amount of 140,000 after 10 years will be = 140,000*(1+r/12)^120

future value annuity factor for n = 10 and APR = r and A = $500 = [(1-(1+r/12)^120/1-(1+r/12]*500 = [(1-(1+r/12)^120/-r/12]*500

Now, 140,000*(1+r/12)^120 + (1+r/12)^120/-r/12]*500 = 420,000

let r/12 be "a"

Thus, 140,000*(1+a)^120+ [(1+a)^120/-a]*500 = 420,000

or a = 0.01214

r/12 = a

or, r = 12* 0.01214 = 0.14568 or 14.568%

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