You will find many resources explaining the concept of mid-year discounting. But just like conventional discounting method, which takes overly simplistic view of real life problems, knowing basic mid-year convention too will not suffice. What you need is to understand how to apply this mid-year convention when you are sitting in say Q2 or Q3 of fiscal year and not the beginning of year (as most will explain). This article will help you in exactly that!
Why Mid-Year Discounting (and what’s with STUBS !?!) ?
If you are here understanding mid-year conventions, chances are you already know why we would use mid-year convention, over traditional text-book taught end-of-year discounting. But a quick refresher doesn’t hurt anybody, no ?
So to answer the question of why we would prefer to use mid-year convention to discount the cash flows, let us understand what really is our objective when we are discounting these cash flows. In the most basic sense, we are trying to determine what will be the today’s value of say $100 received at the end of the year. So we have 2 time periods here: Day we expect to receive the amount and the present day – the day on which the value is ascertained.
Conventional method assumes these time period to be the beginning and end of the year. Thus, there is an implicit assumption in end-of-year discounting that entire cash flows for the year are received at the end of the year. Does this really happen in real world? Answer is simple NO! Companies earn such cash flows on daily basis, but since it will be too tedious to calculate each penny earned basis the timings, we arrive at a compromise – we take an average. This is where the mid-year convention comes into the picture.
We use mid-year convention to represent the fact that company’s cash flows do not come all at once ie., at the end of each year, rather it comes evenly throughout the year
Example: End-of-year vs Mid-Year
Example 1: Assume the company expects $100 cash flow for four years with 12% discount rate.
| Year 1 | Year 2 | Year 3 | Year 4 | |
| Cash Flow | $100 | $100 | $100 | $100 |
| Discount Period – Conventional Method | 1 | 2 | 3 | 4 |
| PV of Cash Flow – Conventional Method | $89 | $80 | $71 | $64 |
| Discount Period – Mid-Year | 0.5 | 1.5 | 2.5 | 3.5 |
| PV of Cash Flow – Mid-Year | $94 | $84 | $75 | $67 |
Explanation: Few things that are important to repeat before explaining the calculations above:
- Conventional method assumes that cash flows are received at the last day of each year, hence the discounting by full year
- Mid-year convention assumes that the cash flows are received in the middle of the year
Conventional Method:
Year 1: Since the method assumes that $100 is received at the period end, the first year cash flows are discounted for full year to get the present value. Thus, ($100)/(1+12%)^1 gives us $89
Year 2: Again, for next year’s $100, the assumption is same that it arrives at the end of year 2. Thus, we need to discount it by full 2 years to get the present value. Thus, ($100)/(1+12%)^2 gives us $80 and so on for other years
Mid-Year Method:
Year 1: Now, this method assumes that the $100 in year 1 is received at the mid-year. Thus, to get the present value of this $100 we need to discount only by half a year. ($100)/(1+12%)^0.5 gives us $94
Year 2: This is where the calculations and discount period gets interesting! The second year $100 is again assumed to be received in the middle of that year. However, be little mindful of howto arrive at the discount period for year 2:-
– Let’s divide the time period into 2 parts, year 1 and year 2. Now the cash flow comes in the middle of year 2. Thus when we discount the same by 0.5, we will arrive at the beginning of year 2 or the end of year 1
– To come to present value, we need to discount the above cash flow further so as to know the value today. Since the above cash flow is one year forward, we will discount it by one more year. Hence, the total years by which $100 of Year 2 gets discounted is (0.5 + 1 = 1.5)
Year 3: Let us do another year, $100 received in the mid of Year 3, needs to get discounted by 0.5 to come at the beginning of year 2. But to know its value today, we need to discount it further by 2 years. Thus discount period is (0.5 + 2 = 2.5)
Let us bring in “Stubs” now …
Ok, so what are stubs and why we have been emphasizing on it since the beginning ? In the real world, it is highly unlikely that the company valuations are done right at the start of the year only. Until now, all our examples have been with the assumption that there is a full one year between the cash flow receipt and valuation date. What if we are already a quarter down and now are being asked for this valuation. The remaining period between now and year end is what we call as stub. Thus in this case, we have a stub of 9 months or 0.75 years.
Example 2: Company A has a year end on December 31st of each year. You have been asked to determine the discount periods for cash flows coming at the end of current period and next 3 years into the future.
| Stub (Q2 – Q4) | Year 1 | Year 3 | Year 3 | |
|---|---|---|---|---|
| Discount Period – Conventional Method | 0.75 | 1.75 | 2.75 | 3.75 |
| Discount Period – Mid-Year | 0.375 | 1.25 | 2.25 | 3.25 |
Explanation:
Do not let the above numbers confuse you. If you understood the concept of mid-year convention above, this is just a simple add-on to that. Let’s discuss it step-by-step.
Conventional Method:
– Stub: The approach here is pretty straightforward. There is 9 months between end of the year (when cash flows are received) and today, hence first year cash flow (i.e., stub period) is discounted by 9 months or 0.75 years
– Year 1: This build on one year out. Hence, second cash flow is received after 1 year and 9 months. This translates to 1.75 years for discounting and so on
Mid-Year Convention:
– Stub: Use similar logic as above. If there are 9 months between today and end of period and we are using mid-year convention, then by the very assumption of mid-year rule, the cash flows are received mid of this 9 months period. Thus, the stub cash flows are discounted by (0.75/2 = 0.375 year)
– Year 1: Again, lets divide this into 2 parts – Stub + 1 Year. We are receiving cash flow in the middle of Year 1, thus we discount the cash flow for that year by 0.5. But on doing just 0.5 years, we have brought this cash flow to the end of stub period. Thus, we need to discount further by 9 months to bring that cash flow to today. Thus, for Year 1 we get (0.5 + 0.75 = 1.25 yrs)
Key Note: Mid-Year discounting is about when you receive cash in the year. In Year 1, the stub of 9 months is not halved i.e., we did not do 0.5 + (0.75/2), because we do not receive any Year 1 cash flow in this stub. The stub discounting is purely to get the mid-year discounted cash flow which represents the value at the start of that year to the present day.
Terminal Value With Mid-Year
I will very briefly touch upon the terminal value concept when using mid-year convention. Discounting changes basis the method used:
- Multiple Method: Here add 0.5 to the final year discount number to reflect that you are assuming that the company gets sold at the end of the year (not mid)
- Gordon Method: Use the final year discount number as is since you are assuming that the cash flows grow into perpetuity and are still received throughout the year rather than just at the end
Key Takeaways
- With same cash flows and discount rate, mid-year convention results in higher present value due to lower discount period
- Higher the discount rate, more pronounced the difference between the end-of-year present value and that from mid-year convention
- Longer the period more larger the difference between end-of-year and mid-year present value
Financial Fudge 