PV of Terminal Value – Full Year Discounting vs Mid-Year Discounting

This is the second post in the Mid-Year discounting series. In the first post, I discussed in detail the workings and logic behind the mid-year convention and touched very briefly upon the terminal year calculations as well. It is strongly recommended to read the first post in conjunction with this one.

Now, unlike the free cash flow discounting during the explicit period, wherein the discounting period varies basis if mid-year convention is adopted or not along with taking note of any stubs therein, the Terminal Value discounting only gets affected by the stubs and not the adoption of mid-year convention.

Let me explain the above with an example:

Assume we are valuing the cash flows for Company A that has its FYE in December. For the sake of simplicity lets have explicit forecast period as 5 years. Also, lets assume WACC as 6.25%.

First lets assume simple discounting convention, where we assume cash flows are received at the end of each period. There are further be 2 cases assumed:

1. Analysis is done on January 1: As can be seen below, while assuming no stub period and that cash flows are received at the end of year, we discount the cash flows by full number of years to bring it to the present.

2. Analysis is done on April 30: Here note that, we are still assuming that the cash flows are received at the end of the year but the only difference being that in the current period the company has already finished with one quarter. Thus, we need to adjust the cash flows as well as the discounting period to only account for the remaining 9 months left for the year end and not include the full year financials.

Year 1 discounting above takes the remaining number of days in the year and divided by total number of days (in this case 365). Years 2 onwards its simply the full year + 9 months of current year to bring the cash flows to the present date and so on.

This was simple case where cash flows were assumed at the year end. As noted in previous post, this assumption is not widely used and rather assuming that since cash flows are earned throughout the year, it is reasonable to assume average. That is the cash flows are received at the mid of each valuation period. I will not go into workings again, as it was all explained in the first post, but for comparisons with normal discounting, let me provide you with the screenshots of how the discounting FCF will look like.

1. Analysis is done on January 1:

2. Analysis is done on April 30:

Discounting-cases-examplesDownload

From the above analysis, it should be little clear how both mid-year convention as well as stub period will have impact on cash flows of explicit period.

Lets turn our attention to the Terminal Value.

Earlier, I mentioned that Terminal Value discounting only gets affected by the stubs and not the adoption of mid-year convention. Let us now try to understand, why this is the case with terminal value.

As you will already know, two of the most common ways of getting to the terminal value figure are – Multiple Value Method and Gordon Growth Method

1. Multiple Value Method:

With this approach, we assume an exit multiple and use the corresponding metrics (EBIT / EBITDA) as of the last year of the explicit forecast period to get our terminal value. Now, there are few things to note here.

• Since we use the full year metric, say EBITDA of Year 5 in our example above, the terminal value so obtained is value at the end of the Year 5. To elaborate, it does not matter what convention of discounting for free cash flow we are assuming (normal vs mid-year), since the implied assumption is that the company is sold basis full year EBITDA, this is the value they will receive “at the end” of the explicit period / full year
• The assumption that EBITDA will be “assumed” to have received at the middle of year 5 in case of mid-year convention does not change the fact that it is a full year EBITDA basis which the company is sold at the end of the period

As such, it should be clear as to why the discounting convention does not play a role in the terminal value.

In our example above, if we are using the Multiple Value Method and assume that the company gets sold at 6.5x EBITDA Multiple and assuming EBITDA for Year 5 is say $235 million:

Terminal Value = 235 x 6.5 = $1,527.5 million

To get the PV of this Terminal Value, use the same discount factor as used in the final year if we are using the simple discounting for the other cash flows of the explicit period. So we will refer back to the “Simple Discounting” cases discussed above (since mid-year convention case is not relevant for terminal value) and use discount period as 5 for the January 1 case and 4.671 for the April 30 stub case. (Refer to the tables above).

However, in case the previous free cash flows of the explicit period are assumed to be discounted using the mid-year convention, then we need to add 0.5 to the final year discount period to account for the fact that it is a full year EBITDA usage. Hence, in the above example we will take (4.5+0.5)= 5 for the Jan 1 case and (4.171+0.5)=4.671 for Apr 30 case. Notice, this is exactly what the discount rate we are using in the “simple discounting” example above. This is exactly what we should be aware of!

2. Gordon Growth Method:

Under this approach, since we are not assuming an abrupt sale of business at the end of explicit forecast period – as is the case with multiple method – we can safely assume that no matter what the convention used (simple or mid-year), we will continue getting the free cash flows to perpetuity in the year end or mid as the case maybe.

Since the terminal value assumption matches with the assumption so taken for the explicit forecast period, we can use the same discounting period as used for the final year of forecast. So here, for simple discounting it will be 5 for Jan 1 case and 4.671 for the Apr 30 case. And for the mid-year assumption, it would be 4.5 for Jan 1 case and 4.171 for the Apr 30 case.

To get the Terminal Value using this approach, lets continue with above example where final year EBITDA was $235 million. If the WACC was 9.88%, growth rate was 2.5% and last year FCF of forecasted period was $109.98 million, the TV using the Gordon Growth Method is:

TV = (109.98 x (1 + 2.5%)) / (9.88% - 2.50%) = $1,527.5

It is not a surprise that the TV derived from both the methods is exactly the same. This happened as I used the implied growth rate from the Multiple method and applied it here. Though in real life scenarios you will be using either one of the methods, but it is good to understand the relation between the two.

• Formula for getting the implied growth rate from the multiple method is:
  • ( Terminal Value * Discount Rate – Final Year FCF ) / (Terminal Value + Final Year FCF)
• Formula for getting the implied EBITDA multiple is pretty straight forward:
  • Terminal Value from Gordon Method / Final Year EBITDA

• Please keep in mind that in case mid-year convention is used for the explicit period and you wish to compare the terminal value from both the methods (Multiple and Gordon Growth), you would need to multiply the TV from Gordon Growth by (1 + Discount Rate)^0.5, to bring it to the end of the explicit period. This is required since in Multiple Method it is always assumed that the TV is received at the “end” of the year. Note, such comparisons are quite rare in real life scenarios.

Is Setting Depreciation = Capex at Terminal Year A Right Assumption in DCF?

It is a common belief that Capital Expenditure (Capex) and Depreciation must converge over time when looking at the projection window. In this article, we will try to explore the possible reasoning behind this approach and situations this might not necessarily be the right way to go forward.

It is important to note that, D&A and Capex might be higher or lower to the other anytime during the projection period. But what we are trying to understand is their relation to the other right at the terminal year in DCF analysis.

Firstly, let us understand what both the scenarios imply:

1. Capex > D&A: This reflects expansionary scenario, where the company is growing its assets base faster than it is depreciation them
2. Capex < D&A: This reflects the opposite of above scenario, where company’s asset base is declining as it is depreciating faster than it is growing and will eventually become zero

As a refresher, there are broadly 2 types of Capex – Maintenance or replacement and Expansionary. Maintenance Capex is required to maintain or replace the current asset base and expansionary capex is required to grow the company further such as new product line or new capacity for an existing product line. Thus, any outlays that expand the output capabilities of the firm will fall under expansionary Capex umbrella.

Note 1: When a company undertakes a Capex, the assets so generated is capitalized, meaning they are reflected on the balance sheet only and as the assets are used over their useful life, they get expensed in the form of D&A and then move through the income statement.

Note 2: It also essential to always remember that Capex always precedes depreciation. This is intuitive, as one cannot depreciate an asset they don’t own. Hence, it is important to adjust the depreciation to forecasted Capex, instead of doing the reverse.

While trying to ascertain the terminal value, it is important to think of it as – You are trying to project the company’s financial performance forever. Since a perpetual expansionary Capex is not practically viable, it might be prudent to assume that company’s asset base will remain at fairly constant level. This implies that company should invest in maintenance level Capex to maintain the current asset base. Thus, in the terminal year our focus is to determine the “normalized” D&A and Capex.

So, the question is – “Is the simplifying normalization assumption of setting depreciation equal to Capex in the terminal year right assumption?”

This assumption is typically appropriate in zero-growth/inflation scenarios. If growth is projected in the cash flow forecast, capex should typically exceed depreciation in the terminal period. The reason for the same is quite straightforward. Capex has a direct correlation to both growth and depreciation expense.

Increased levels of Capex should in turn lead to increased future growth. Likewise, increased Capex will raise future levels of depreciation expense. As long as a company earns a positive return on its capital investment, then capital Capex in excess of maintenance capital requirements should result in some level of future growth.

Please note, some models might assume that Capex drives revenue, but this is not necessarily true. Capex grows as revenue grows, but it doesn’t always drive revenue. Eg: Software companies are far less dependent on Capex to boost its revenue.

Let’s discuss this further to get a better understanding:

Projected capital expenditures should always reflect the expected long term growth rate assumed (LTG Rate), or conversely, a selected LTG rate should be supported by a certain level of Capex and an assumed rate of return on that investment (ROI). If growth expectations are increased or decreased, then either Capex need to be adjusted accordingly or new assumptions needs to be established regarding return on invested capital. Assuming the rate of return on invested capital is held constant, then any change to the LTG rate assumption should require the adjustments to the assumptions for both Capex and depreciation.

Note: Formula for calculating the sustainable long term growth rate for a company is,
g = b x ROE,
where, g = LTG Rate, b = Reinvestment Rate or plowback ratio and ROE = Return on Equity

This formula also illustrates the connected relationship between Capex and the LTG rate (i.e., that the two variables increase or decrease in tandem).

Nearly all company projections and discount rate data are presented in nominal terms. Thus, the LTG rates is based on nominal LTG rates that is including the impacts of both inflation and real returns.

Now consider a scenario where expected inflation of Country A is say 4%,

1. LTG Rate = 4%; inherent assumption of 0% real growth rate
2. LTG Rate = 3%; implies negative real growth rate
3. LTG Rate = 5%; implies 1% real growth rate

Thus, for normalized cash flow projections that include an assumption of a positive nominal LTG rate, equating Capex and depreciation expense may be flawed. When the estimated LTG rate is positive (i.e., any selected LTG rate greater than 0%, even if that growth rate results in negative or zero expected real growth) Capex may exceed depreciation expense.

Mid-Year Discounting With Stubs – Straight & Simple Explanation !!

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

Click on the link below to download the excel file for the workings of above table

Example-Mid-Year-ConventionDownload

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:

1. 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)
2. 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