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Chapter 4 of 6 · 10 min

Discounting and the mid-year convention

Why a future dollar is worth less today, and why DCFs assume cash arrives halfway through the year.

By the end of this chapter you can
  • Turn a future cash flow into a present value with a discount factor
  • Apply the end-of-year and mid-year conventions, and say which gives more
  • Back out the discount rate from a present value
  • Know which cash flows the mid-year convention should not touch
1

The intuition

Would you rather have $100 today or $100 in a year? Today, because you could invest it: at 10% it becomes $110 by next year. Turn that around, and $100 arriving next year is only worth $90.91 today, because $90.91 invested at 10% grows to $100.

$100 received later, valued today at 10%
In 1 year$90.91
In 2 years$82.64
In 3 years$75.13
In 4 years$68.30
In 5 years$62.09

Each extra year of waiting divides by 1.10 once more.

The key idea

Discounting is compounding run backwards. The discount factor for year n is 1 ÷ (1 + rate)^n: what one future dollar is worth today.

2

Why it works

A DCF adds up cash flows from different years, and dollars from different years cannot be added directly. Discounting turns each of them into today's dollars first, at the return investors require: WACC, for unlevered cash flow.

The end-of-year convention discounts year 1's cash by one full year, year 2's by two, and so on, as if every dollar arrived on 31 December. But a business collects cash all year round. On average it arrives halfway through the year, so the mid-year convention discounts year n by n − 0.5 years instead.

  • Every cash flow is discounted half a year less, so every present value rises by the same factor: √(1 + WACC). At 10% that is about 4.9% more.
  • Mid-year gives the higher value, and for a business with steady cash coming in it is also the more realistic one.
  • It applies to the forecast cash flows. For the terminal value it depends on the method: a perpetuity built from those mid-year cash flows takes the mid-year factor for its year, but an exit multiple is a sale at the end of the year and takes the full N years.
$100 in year 1 and in year 2, at 10%
Year 1, end-of-year100 ÷ 1.10 = $90.91
Year 1, mid-year100 ÷ 1.10^0.5 = $95.35
Year 2, end-of-year100 ÷ 1.10^2 = $82.64
Year 2, mid-year100 ÷ 1.10^1.5 = $86.68

Both mid-year values are 4.88% higher than their end-of-year twins: √1.10 − 1.

3

The formulas

DF (end of year n) = 1 ÷ (1 + WACC)^n

What one dollar received at the end of year n is worth today.

DF (mid-year) = 1 ÷ (1 + WACC)^(n − 0.5)

The same dollar, assumed to arrive halfway through the year: half a year less discounting.

PV = cash flow × DF

Present value: the future amount in today's dollars.

Mid-year uplift = (1 + WACC)^0.5 − 1

How much higher every mid-year value is. It is the same for every year.

Implied WACC = (cash flow ÷ PV)^(1 ÷ (n − 0.5)) − 1

Run it backwards: how much the money grew, spread evenly over the years of waiting.

4

Worked example

One cash flow, both conventions. The two answers should differ by the same percentage whatever the year.

Drawing the numbers…
5

See it move

Same company, now with five years of growing cash flow. Change the rate and the growth and watch what each year is worth today.

Drawing the numbers…
Try this
  • Raise WACC. Every filled bar shrinks, and the later years shrink the most, because they are divided by (1 + WACC) more times.
  • Compare the two filled bars in any year. The mid-year bar is higher by the same percentage every year: the uplift readout.
  • Raise growth. The later outlines get taller, but a bigger share of each one is lost to discounting.
6

Run it backwards

Same company, reversed: a model shows a present value. What discount rate turns the cash flow into that number?

Drawing the numbers…

Present value = cash flow ÷ (1 + r)^(n − 0.5). Divide the cash flow by the present value to see how much the money grew. That growth happened over n − 0.5 years, so take that root and subtract one.

Always check the convention before reading a rate out of a model. The same present value spread over a full n years implies a lower rate than over n − 0.5 years.

7

Traps

Multiplying by (1 + r)^n instead of dividing.
Future to present is division. With a positive rate, a present value is always smaller than the cash flow.
Using mid-year for an exit multiple terminal value.
An exit multiple assumes a sale at the end of year N, so discount it N full years.
Calling mid-year the aggressive choice.
It gives a higher value, but it matches how cash really arrives. The aggressive mistake is applying it to amounts that do not arrive mid-year.
Using the wrong exponent in year 1.
Under mid-year, year 1 is discounted half a year: 1 ÷ (1 + r)^0.5.
Adding cash flows before discounting them.
Only present values can be added together. Discount each year first, then sum.
8

Say it in the interview

The interviewer asks

“What is the mid-year convention, and why use it?”

Say yours out loud first, then compare.
9

Check yourself

4 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.

Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.

0 of 4
Drawing your questions…
Remember
  • DF = 1 ÷ (1 + r)^n; present value = cash flow × DF.
  • Mid-year: discount year n by n − 0.5 years.
  • Mid-year raises every present value by √(1 + r) − 1.
  • An exit multiple terminal value takes the full N years.