Discounted (present value) calculation

Question

I had a homework problem regarding calculating a discounted present value. My solution is:

15000 * (1+.08)^1 + 15000 * (1+.08)^2 = $33 696

The question is this:

Laura has an undergraduate degree in economics and has been working as a utility rate analyst at the local electricity utility. If she continues on her present career path, the present value of her lifetime earnings is \$250,000. If she takes two years off and gets an MA degree in economics, the present value of her lifetime earnings is \$275,000. The annual cost of an MA degree in economics is \$15,000 and the interest rate is 8%. Assume that school fees are paid at the beginning of the year.

a.) Calculate the discounted (present value) cost of Laura's graduate degree in economics. (2 points)

The given solution is this:

Answer: $28, 889

I don't understand the logic of this question. To me, it seems the present value cost would be the cost of what I would pay if I was accounting for the interest of the tuition.

Answer 1

The calculation that you gave doesn't make sense. To answer this question, you should revise your definition of "present value." The definition you gave isn't specific enough to be operational:

To me, it seems the present value cost would be the cost of what I would pay if I was accounting for the interest of the tuition.

What does present value mean? Think about it like this. How much money do you need today to pay the future costs. Well you need $\$15,000$ to to pay for the first year and $\$ X$ to pay for the second year. Why did I write $X$? Because we want to know how much money we need today to pay for the second year and that amount is less than $\$15,000$. This is because we can invest $X$ amount of dollars at $8\%$ interest so that we have $\$15,000$ in a year. So, we want to solve $$ X * 1.08 = 15000. $$ Thus, the proper present value calculation is $$ 28,889 \approx 15,000 + \frac{15,000}{1.08}. $$

Answer 2

"Your solution" is the Future Value (FV) of the payments schedule as though you were saving money on $8\%$ interest calculated yearly, and made no withdrawals at all. So this value exists two years in the future, and not now. Your solution sends the payment schedule to a single point in the future. Laura says to the school "instead of paying $15,000$ now and $15,000$ one year from now to receive services up until the end of the 2nd year, it is the same to pay $33,696$ at the end of the second year" (assuming that the school operates indeed under an $8\%$ interest rate, in the sense that the school will have to borrow the money and at the end of the 2nd year repay them with interest. You will have save the money at the same interest and you will have the nominal amount needed).

But the exercise asks for the Present Value (PV) of this payment schedule. It asks to bring the payment schedule to the single point that it is "now". Here, Laura says to the school "instead of the payment schedule of nominal value $30,000$ that you propose it is the same if I pay you now $28,889$" -because the school will receive $15,000$ one year earlier than planned and it will have the opportunity to save it at a $8\%$ interest rate.

Both are legitimate concepts of course, they both calculate the value of a payment schedule spread out in time, if it were to be compacted into a single payment -but each calculates this value for a different point in time.