TVM practice problem

CFA Level I Quant
#1

Hi everyone, there is a question from the curriculum that I don’t know how to do.

Q: A couple plans to pay their child’s college tuition for 4 years starting 18 years from now. The current annual cost of college is C$7,000, and they expect this cost to rise at an annual rate of 5 percent. In their planning, they assume that they can earn 6 percent annually. How much must they put aside each year, starting next year, if they plan to make 17 equal payments?

The solution is as follow:

Step 1 : Draw a timeline and calculate the future value of the college tuition from t= 18 to t = 21 which is the sum of the following cash flows: ( I get this part)

7000(1.05)^18 = 16846
7000(1.05)^19 = 17689
7000(1.05)^20 = 18573
7000(1.05)^21 = 19502

Step 2 : Using the formula for the present value of a lump sum (r = 6%), equate the four college payments to single payments as of t = 17 and add them together. C$16,846(1.06)−1 + C$17,689(1.06)−2 + C$18,573(1.06)−3 + C$19,502(1.06)−4 = C$62,677*

Step 3 : Equate the sum of C$62,677 at t = 17 to the 17 payments of X , using the formula for the future value of an annuity (Equation 7). Then solve for X .
C$62,677=X[(1.06)17−10.06]=28.21288X
X=C$2,221.58

I am currently stuck in step 2. Can someone please explain why an interest rate of 6% is used instead of 5%?Isn’t 5% the annual growth rate of the tuition cost ? I know we need to calculate the present value of the lump sum at t= 17 but I don’t get why 6%is used instead of 5%.

Thanks in advance!!!

#2

6% is what any money actually invested in a savings account will earn. :moneybag:

If you search for this question, you will find lots of discussion around it with helpful calculator hints!!

#3

Thank you for your reply breadmaker, I understand that 6% is interest rate that the couple can earn in a saving account. That’s why Step 3 totally makes sense to me.

However, for calculating the present value of the lump sum of the tuition fee at t= 17 , shouldn’t we use the rate of 5% to find the present value as 5% is the annual growth rate of cost?

#4

given this description, the 6% rate acts as the risk-free interest rate in the annuity calculation. step 2 simply discounts all the payments needed to time t = 17 using that “risk-free” rate.