Two-step education funding

A degree course does not cost money once. It costs money every year it runs.

So a parent funding one faces several dates, not one.

The workbook's Case 2 makes the point with a two-year post-graduate course. Fees fall due at the end of Year 5 and again at the end of Year 6. The parent wants to set a single amount aside at the end of Year 5 and be done with it.

Doing that takes two steps.

Step one. Work out what each year's fee will actually cost in the year it falls due. Fees inflate, so each year has its own number.

Step two. The Year 6 fee is not needed for another year. So a smaller sum, set aside now and left to earn interest for that year, will grow into it. Only that smaller sum is needed today.

Add the Year 5 fee to that smaller sum. That is the single figure for the end of Year 5.

Two different rates do the two jobs. One rate grows the fee. A different rate grows the money. Using one rate for both gives the wrong answer.

The case, as section 20.2 sets it up. Mr Y is aged 40. His son's post-graduate education is due in Year 5, at a current cost of Rs 15,00,000 per year, to be incurred at the end of each year for 2 years, with likely inflation of 15% p.a. A second goal, his daughter's marriage, falls at the end of Year 7 at a current cost of Rs 1,00,00,000 with inflation of 10% p.a. He has provided a corpus of Rs 2,00,00,000 for both, invested in a mix of debt and equity yielding 8% p.a. Taxation is to be ignored.

Question 1 asks how much must be set apart from the corpus at the end of Year 5 to finance the education, assuming the amount set apart will earn 6% interest.

Step one — inflate each year's fee to its own year.

Six years of 15% inflation more than doubles a Rs 15,00,000 fee. Note that the second fee uses 6 periods, not 5, because it is payable at the end of the following year.

Step two — bring the later fee back to the set-aside date.

The workbook spells out what [B] means: Rs 32,73,199.20 invested for 1 year at 6% p.a. grows to Rs 34,69,591.15. The multiple-choice answer is Rs 62,90,235.

Three rates, three jobs, and only two of them are in this calculation.

Why the naive answer is wrong. Simply adding the two future fees gives Rs 30,17,035.78 + Rs 34,69,591.15 = Rs 64,86,626.93. That over-funds the goal, because it ignores the year of growth available on the second fee. It is not even among the four options offered, which tells you the examiner expected the discounting step.

The 6% is deliberately not the 8%. The corpus earns 8%, but the money set apart is assumed to earn 6% — the question says so. A lower assumed rate means a larger required set-aside, because less growth is available in the waiting year. Using 8% by mistake would give PV(8%, 1) of Rs 34,69,591.15 = Rs 32,12,584.40 and a total of Rs 62,29,620.18, which is not an option either.

The general shape. For a goal spread over several years, each year's cost is inflated to its own year, then discounted back to the funding date at whatever the funded money will earn. Only the instalment due on the funding date itself escapes the discounting.

For a goal payable at the end of years n, n+1, n+2, ... funded by
one amount set aside at the end of year n:

  Cost in year k         =  C0 x (1 + inflation) ^ k

  Amount needed at year n
  for the year-k cost    =  Cost in year k / (1 + set-aside rate) ^ (k - n)

  Total set-aside        =  sum of those amounts over all k

Workbook Case 2, question 1:

  C0 = 15,00,000 per year   inflation = 15%   set-aside rate = 6%   n = 5

  A = 15,00,000 x 1.15^5                     =  30,17,035.78
  B = (15,00,000 x 1.15^6) / 1.06
    = 34,69,591.15 / 1.06                    =  32,73,199.20
                                                ------------
  Set-aside required at the end of Year 5     =  62,90,234.98

The figures below are the workbook's own, from Case 2 in section 20.2.

Mr Y, 40, has to fund a 2-year post-graduate course for his son. The fee today is Rs 15,00,000 a year, payable at the end of each year, and the course starts so that the first fee falls at the end of Year 5. Fee inflation is 15% p.a. Whatever he sets aside at the end of Year 5 will earn 6%.

Step 1 — what each fee will cost.

Year 5 fee  =  15,00,000 x 1.15^5  =  Rs 30,17,035.78
Year 6 fee  =  15,00,000 x 1.15^6  =  Rs 34,69,591.15

Step 2 — what he must hold at the end of Year 5.

For the Year 5 fee                       Rs 30,17,035.78
For the Year 6 fee, 34,69,591.15 / 1.06  Rs 32,73,199.20
                                          ---------------
Set-aside at the end of Year 5           Rs 62,90,234.98

So he moves Rs 62,90,235 out of his Rs 2,00,00,000 corpus into a 6% account at the end of Year 5. It pays the first fee immediately. The remaining Rs 32,73,199.20 grows at 6% for a year to Rs 34,69,591.15, which pays the second fee exactly. Nothing is left over and nothing is short.

The error the question is built to catch.

Extending it to a three-year course. Suppose the course ran for 3 years instead of 2, everything else unchanged, with the third fee at the end of Year 7.

Year 7 fee  =  15,00,000 x 1.15^7           =  Rs 39,90,029.82
Held at Year 5: 39,90,029.82 / 1.06^2       =  Rs 35,51,112.34

Total set-aside = 30,17,035.78 + 32,73,199.20 + 35,51,112.34
                = Rs 98,41,347.31

The pattern does not change. Each fee is inflated to its own year, then discounted back over however many years it waits.

Chapter 20 (Case Studies), section 20.2 Case 2, is the whole of this term. Question 1 asks for the amount to be set apart at the end of Year 5, and the answer table that follows gives every figure: the Rs 15,00,000 current cost payable at the end of each year for 2 years, 15% inflation, the two future costs of Rs 30,17,035.78 and Rs 34,69,591.15, the PV(6%, 1) step producing Rs 32,73,199.20, and the total of Rs 62,90,234.98 against the option Rs 62,90,235.

Expect this arithmetic almost unchanged. The examinable skill is knowing which rate does which job — and that a fee payable at the end of the following year takes one more compounding period, then one discounting period back.

• Two rates, two jobs. 15% inflation grows the fee; 6% grows the money set aside. Using one rate for both is the commonest error here.
• The corpus rate is a third rate and does neither job. The Rs 2,00,00,000 corpus earns 8%, but the question tells you the amount set apart earns 6%. Use 6%.
• A lower set-aside rate means a LARGER set-aside. 6% instead of 8% raises the requirement from Rs 62,29,620.18 to Rs 62,90,234.98, because less growth is available in the waiting year.
• Do not just add the two future fees. Rs 64,86,626.93 over-funds the goal by Rs 1,96,392 and is not among the options.
• The second fee takes 6 compounding periods, not 5. It is payable at the end of the following year, so it inflates for one more year and is then discounted for one year.
• Fees are payable at the END of each year in this case. A payment at the beginning of the year changes both the number of inflation periods and the discounting.
• Ignore taxation, because the case says to. The 6% and 8% are pre-tax and the question does not ask for a post-tax figure.
• This is not the same as funding from a level annual saving. Here a single lump sum is carved out of an existing corpus at a future date, not built up by instalments.