Half a year of discounting looks like a detail, but applied to the terminal value it is worth more than most arguments about the cost of capital.
The mid-year convention discounts each year's cash flow half a year earlier, so it raises every present value by the square root of (1 + WACC), minus one: 4.40 per cent at a 9 per cent cost of capital. Whether the whole enterprise value moves by that much depends on the terminal value. On Cawdrey Instruments, the case in Business Valuation, shifting only the forecast years adds 875,510, or 1.20 per cent; shifting the perpetuity as well adds 3,216,905.
Worked in full in Business Valuation by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A year-end DCF assumes that a whole year's free cash flow arrives on 31 December. A business that collects and pays through the year produces its cash, on average, around mid-year. The convention corrects for that, and it is standard in banking and fairness opinion models. It is also a small adjustment applied to a large number, which is why it is worth knowing exactly what it adds and where.
| Year | Free cash flow | Year-end factor | Mid-year factor | PV year-end | PV mid-year |
|---|---|---|---|---|---|
| 1 | 4,664,000 | 0.9174 | 0.9578 | 4,278,899 | 4,467,302 |
| 2 | 4,920,520 | 0.8417 | 0.8787 | 4,141,503 | 4,323,856 |
| 3 | 5,166,546 | 0.7722 | 0.8062 | 3,989,521 | 4,165,183 |
| 4 | 5,399,041 | 0.7084 | 0.7396 | 3,824,816 | 3,993,226 |
| 5 | 5,615,002 | 0.6499 | 0.6785 | 3,649,366 | 3,810,050 |
| Sum | 19,884,106 | 20,759,617 |
Net debt is 11,200,000 and there are 4,000,000 shares. The terminal value, on the Gordon growth formula, is 81,818,603 at the end of year five.
Year-end factor = 1 ÷ (1 + WACC)t; mid-year factor = 1 ÷ (1 + WACC)t − 0.5
Uplift on any flow = (1 + WACC)0.5 − 1
In Excel, with the year number in row 3: =1/(1+WACC)^(C3-0.5). For the perpetuity: =TV/(1+WACC)^4.5 if it is shifted, =TV/(1+WACC)^5 if not.
Step 1, the forecast years. Every factor rises by 1.0440, the square root of 1.09. The five present values sum to 20,759,617 instead of 19,884,106: +875,510.
Step 2, the terminal value. Discounted over 4.5 years instead of five, the same 81,818,603 is worth 55,517,873 today instead of 53,176,478: +2,341,395, more than twice the forecast-year effect, because the terminal value is 72.78 per cent of the answer.
| Treatment | Enterprise value | Uplift | Per cent | Per share |
|---|---|---|---|---|
| Year-end discounting | 73,060,585 | 0 | 0 | 15.47 |
| Mid-year, forecast years only | 73,936,095 | 875,510 | 1.20% | 15.68 |
| Mid-year, forecast and perpetuity | 76,277,490 | 3,216,905 | 4.40% | 16.27 |
Applied in full, the convention is worth 4.40 per cent of enterprise value and 5.20 per cent of equity value per share, because net debt does not move. Expressed another way, it is the same as cutting a year-end cost of capital from 9.0 to 8.71 per cent, about 29 basis points, which is a larger change than most cost of capital debates end up making.
The convention also has to be adapted when the valuation date falls part-way through a financial year. The first, partial period carries only the cash flow still to come, and its mid-point sits halfway through the stub, not halfway through the year; every later period is then discounted from that shorter first step. Leaving the half-year exponents unchanged on a stub period quietly adds or removes months of discounting, and on a business of this size each month is worth several hundred thousand.
It depends on what the terminal value represents.
The common error is to mix the two: compute the terminal value on an exit multiple and then discount it at 4.5 years because the rest of the model is mid-year. That adds 2,341,395 to Cawdrey for a timing assumption that does not apply to a sale. The reverse error, a Gordon terminal value left at five years in a mid-year model, understates value by the same amount. A model that runs both methods side by side should state the discount period beside each one.
| WACC | Uplift on every present value |
|---|---|
| 8% | 3.92% |
| 9% | 4.40% |
| 10% | 4.88% |
| 12% | 5.83% |
When the terminal value is shifted, the uplift is the same on every line and therefore on the total. When it is not, the uplift on enterprise value is the full rate multiplied by the forecast years' share of value: 4.40 per cent times 27.22 per cent, roughly 1.20 per cent on Cawdrey. The longer and more front-loaded the forecast, the more the partial convention matters.
Seasonality can break the convention. Mid-year assumes cash arrives evenly. A retailer that earns most of its cash in the fourth quarter, or a contractor paid on milestones, should be discounted on its actual cash timing, or at least on quarterly flows.
Use the mid-year convention when cash arrives through the year, shift a Gordon growth terminal value with it, and leave an exit multiple terminal value at year end. Then say which you did: on Cawdrey the choice is worth 3,216,905, or 0.80 a share. The model, with both bounds computed on its terminal value sheet, is in the free workbook for this case, and what percentage of a DCF the terminal value should be explains why the perpetuity carries most of the effect.
Yes for a Gordon growth terminal value, because the perpetual flows also arrive through each year; no for an exit multiple, which values a sale at the end of the final year. On Cawdrey Instruments, shifting the 81,818,603 terminal value to 4.5 years adds 2,341,395, more than twice the 875,510 the forecast years add.
1 / (1 + WACC)^(t - 0.5), where t is the year number. In Excel: =1/(1+WACC)^(C3-0.5). At 9 per cent the year-one factor rises from 0.9174 to 0.9578, and every factor rises by the same 1.0440, the square root of 1.09.
Yes, in percentage terms, because net debt does not change. On Cawdrey the full convention adds 4.40 per cent to enterprise value and 5.20 per cent to equity value, taking the value per share from 15.47 to 16.27.
This article is one calculation from Business Valuation. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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