DIFFERENT DISCOUNT RATE FOR CASH OUTFLOWS AND INFLOWS?

Readers,

I just finished up reading Appendix 14-A Present Value of Risky Outflows of the textbook Capital Budgeting and Long-Term Financing Decisions by Neil Seitz and Mitch Ellison (Third Edition, Harcourt Brace College Publishers, 1999, page 470 – 473). That Appendix’s example makes a reference to Laurence D. Booth’s article: Correct Procedures for the Evaluation of Risky Cash Outflows, Journal of Financial and Quantitative Analysis 27 (June 1982), page 287-300. Prof. Laurence D. Booth is a name that will be familiar to many finance students (https://www.rotman.utoronto.ca/FacultyAndResearch/Faculty/FacultyBios/Booth)

I am a bit intrigued to read further that article since that Appendix said that the present value of a series of cash outflows is the same as the present value of a series of cash inflows with identical characteristics. This is interesting, since generally speaking, when we do the present value, we just jump to the NET cash flows, but this Appendix shows that underlying this general practice, it is assumed that both cash outflows and cash inflows have the same characteristics, though the book doesn’t really elaborate more about what it means with “having the same characteristics”.

The summary of that Appendix said:

Note that a positive beta on a cash outflow means that the outflow tends to move with revenues, and the variability of the outflow, therefore, decreases risk. A risk-averse decision maker would prefer variable outflows that decrease risk over a known cost with the same expected amount. The higher discount rate applied to risky outflows means a smaller present value of the outflow, which is consistent with the risky outflow being preferred over a riskless outflow.

Likewise, a cash outflow with a negative beta increases risk and, therefore, is less desirable than a riskless outflow. The low discount rate and resulting high net present value reflect the undesirability of the increased risk.

I will be using the example taken from Neil and Mitch’s demonstration of the validity and application of the basic principle of valuation for risky outflows is that the present value of a series of cash outflows is the same as that of the present value of a series of cash inflows with identical characteristics. The intuitive behind this, is that someone’s cash outflow will be another person’s cash inflow. In an equilibrium capital market, the present value of a series of outflows is the present value of those flows to the person receiving them. The above article by Prof. Laurence D. Booth provides a CAPM-based proof and a state-preference-based proof as well as a more general example.

Example 1

Assumptions:

  • One-period capital investment (Note: this assumption rings familiar since one of the assumptions underlying the CAPM is that all investors make decisions for a single-period horizon and can revise their portfolios at the end of that horizon, albeit CAPM itself is silent about the length of the holding period. Generally speaking, it is treated as a short period. This assumption might be challenged when we are talking about capital investment, since such investment (i) will yield return over a number of years, and (ii) not always be marketable compared to stock portfolios, and (iii) even it is marketable, will the project owner be able to capture the market value of that projects?)
  • There is only two possible outcomes, State I and State II, each of which with a probability of 50%.
  • The current market value of the market portfolio is $100.
  • The cash flows for the capital investment is identical to the ending value of the market portfolio [Note: in this example, both market portfolio and cash inflows will have the same value of $100 and $130 under State I and State II.]
  • The risk free rate is 10%.

The table below shows the cash flows in each State for the investment and the market portfolio:

With the expected value of market portfolio at the end of the period is $115, then the expected return on the market portfolio is $115/$100 (current market value) – 1 = 15%.

Because the cash INflows from the capital investment are identical or the same to the ending market portfolio value, we could say that the equilibrium present value of the risky INflow stream will be equal to the value of the market portfolio, that is $100, its current market value.

Concerning NET cash flows, since both will have the same value either in State I and State II, that is $20, then it is risk free, and we could discount the NET cash stream using the risk free rate of 10%.

The present value (PV) of a risk-free of NET cash flows = $20/(1+10%) = $18.18.

With PV-out = equilibrium present value of cash OUTflows, then,

PV_out = PV_in (= current market value) – PV_net

PV_out = $100 – $18.18 = $81.82.

Once we have PV_out and the expected value of cash OUTflows, we could get the implied risk-adjusted discount rate (k_out) by discounting it:

PV_out = expected value_out/(1+k_out)

k_out = expected value_out/PV_out – 1 = $95/$81.82 – 1 = 16.11%

As we get 16.11%, we could now confirm that this will be the same risk-adjusted discount rate that would be applied to a series of cash INflows with the same characteristics. We could use CAPM to prove it.

For a series of cash OUTflows, under two possible end-of-period State I and State II, the returns on each State will be as follows:

State I = $80/$81.82 – 1 = -0.0222 (Minus sign), or -2.22%

State II = $110/$81.82 – 1 = +0.3444 (Positive sign), or 34.44%

The market portfolio return for each State will be as follows:

State I = $100/$100 – 1 = 0%

State II = $130/$100 – 1 = 30%

As the beta is the slope coefficient for the relationship between returns for some assets and return for the market portfolio, then the beta for the cash OUTflows is:

beta_out,m = [34.44% – (-2.22%)]/[30% – 0%] = 1.222 (Positive sign)

We now could calculate the required return using this beta_out,m and CAPM standard formulation, as follows:

K_out = Risk free + Beta_out,m (market portfolio return – risk free rate)

K_out = 10% + 1.222 (15% – 10%) = 16.11%

So we could see here that the equilibrium risk-adjusted discount rate is the same, regardless the cash flows is COSTs (for cash OUTflows) or Revenue stream (for cash INflows).

Under Example 1 here, the readers are requested to note that :

A positive beta on a cash OUTflow (in this case, +1.222), means that the OUTflows tend to move IN LINE with

  • the Revenues, in which from State I to State II, revenue is up from $100 to $130, and the cash OUTflows is also moving up from $80 to $110;
  • the variability of the cash OUTflows.

This IN-LINE movement, therefore, DECREASES the risk of the project.

A risk-averse decision maker, when faced with two options: (A) Variable cash OUTflows that decrease risks, or (B) A KNOWN cost with the same expected amount, might prefer Option A.

How do we know that Option A is preferable over Option B?

This is crystal clear if we compare the discount rate of Option A and Option B.

Option A have the discount rate of 16.11%, while Option B have the risk-free discount rate of 10%.

With higher discount rate being applied to the risky cash OUTflows, resulting to a lower PV of the cash OUTflows. The lower the cost (or cash OUTflows), this option will be preferred over a risk-less cash OUTflow.

Example 2

Assumptions:

  • We will carry all assumptions from Example 1, but we just change now the direction of the cash OUTflows which will move in the opposite direction from the general market returns, which implicitly this will INCREASE the risk. So when the market returns are up from State 1 to State II, along with the cash INflows going up as well, yet, the cash OUTflows are going down.

Table below displays the cash flows in each State:

Here we need to solve jointly :

  • The present value of the NET cash flows (= PV_net);
  • The beta of the net cash flows (beta_net,m)
  • The required rate of return for the NET cash flows (k_net)

The calculation is as follows:

PV_net = Expected Value of NET cash flows / (1 + k_net) = $20/(1+k_net)

K_net = Risk free rate + beta_net,m (market return – risk free rate) = 5% + beta_net,m (15% – 10%)

Beta_net,m = [($50 – PV_net)/PV_net – (-$10 – PV_net)/PV_net] divided by (30% – 0%) = $200/PV_net

By substituting k_net and beta_net,m above to PV_net, then

PV_net = $20/(1 + (10% + (200/PV_net) x (15% – 10%))

PV_net = $9.09

Once we have the PV_net of $9.09, PV of cash OUTflows will be:

PV_out = PV_in – PV_net = $100 (its current market value) – $9.09 = $90.91.

The implied risk-adjusted discount rate of the cash OUTflows =

PV_out = expected amount / (1 + k_out)

k_out = expected amount/PV_out – 1 = $95/$90.91 – 1 = 4.5%

The beta of this cash OUTflows will be :

State I : ($110/$90.91) – 1 = +20.9988%

State II : ($80/$90.91) – 1 = – 12.0009%

Beta_out,m = (-12.0009% – 20.9988%)/(30% – 0%) = – 1.1 (Minus sign)

The CAPM required rate of return = risk free + beta_out,m (market return – risk free)

K_out = 10% + (-1.1) x (15% – 10%) = 4.5%

Again, we have here the risk-adjusted present value for the risky cash OUTflows that is the same present value for risky cash INflows with identical characteristics.

The readers are requested to note that under Example 2, which we have NEGATIVE Beta for the risky cash OUTflows, this has given rise to increase the risk. When the market is going down, from State II to State I, followed by the decline in the cash INflows, the cash OUTflows shows it has higher costs. This is of course, less desirable compared to a risk-less OUTflows.

When we compare the discount rate of 4.5% vs risk free rate of 10%, then with lower discount rate, this will result in higher net present value (= higher cost), reflecting the un-desirability of the increased risk.

One closing note on Example 1 and Example 2 is that if components of NET cash flows are discounted separately, the sum of the present values (for cash INflows and cash OUTflows) will equal the present value of the NET cash flows.

My understanding, that the discount rate for cash inflows and cash outflows, theoretically, could be different. Only if both shares the same risk factors, then we can use the same discount rate.

Under CAPM world, seems market risk, this single risk, is claimed to impact almost everything of the company’s cash flows, including INflows and OUTflows, and this has implicated that we should use one discount rate for both. When we are talking about cash outflows, this could occur horizontally or vertically. Horizontally, what I meant, is those cash OUTflows during the initial stages of the project, or in many cases, put as Io (Initial Investment) which could take years or months before the project will generate positive cashflows. The risk factors during this initial phases of the project could be said will be quite different from the risk factors impacting the later stages of the project life.

Additionally, cash OUTflows could occur vertically, that is during the period when the project has booked cash INflows, and the project will have costs of manufacturing, production, buying inventory, labor cost, overhead cost, marketing cost, project administration, etc. It is possible that the risk factors around this vertical cash OUTflows will be different from those leading to the cash INflows (which might more market oriented).

This will be logically bringing us to be able:

1. using different discount rate for cash INflows and vertical cash OUTflows

2. using different discount rate for horizontal cash OUTflows (early stage or seed phase of the project)

Respondent 1:

I agree to the general idea that the PV of cash inflow = cash outflow. However, there is an underlying assumption and that is same risk for inflows and outflows.

However, in reality capital outflow will have my risk rating and capital inflow has the payer’s risk rating (I would rather get an inflow from the US government than from a guy at the street). This should be reflected by the risk factors, though.

To make things short: I agree to the statement GIVEN same risk factors, however I do not agree that the risk factors ARE the same.

Karnen to Respondent 1:

In the practical leve, it is not that easy, to say that the risk factors impacting cash inflows and cash outflows are not the same, and this will require different discount rate.

In many cases, the analysts just assume away both cash flows (out and in) are having the same risk factors, and this is why, the NPV formula is applied to the NET cash flows, using one single discount rate.

Respondent 1 to Karnen:

Agree. That is my assumption as well.

Respondent 2:

Well, I would say that there are some generalizations that oversimplify the situation.

They say that an outflow has an equivalent inflow to other person. Yes, in general, but inflows and outflows usually are composed of outflows and inflows of MANY other people. 

For me, a cashflow (in or out) poses a given risk as perceived by me, the investor. This is the risk that is relevant: the ones I perceive. What other parties perceive about the risk of the contrary sign cashflow, is irrelevant to me.

Respondent 3:

A couple of marginal notes.

“a positive beta on a cash outflow means that the outflow tends to move with revenues”. This sounds counterintuitive. Generally, a positive beta means that an asset’s returns (let them be outflows) tend to move with the MARKET. Perhaps, the “beta” should be replaced with “correlation”, shouldn’t it?

“when we do the present value, we just jump to the NET cash flows, but this Appendix shows that underlying this general practice, it is assumed that both cash outflows and cash inflows have the same characteristics” Not sure that the latter statement is correct. We discount an expected net cash flow (a mix of inflows and outflows) at a rate corresponding to its risk, and that does not mean that we assume all components of the net cash flow have the same characteristics. Formally one obtains the same result when splitting the cash flow into parts and discounting them all at, say, the project’s (firm’s) cost of capital, but, in fact, according to corporate finance fundamentals each component should be discounted at it’s specific discount rate, and the weighted average if these rates aggregates into the discount rate  applied to the net cash flow.

Respondent 3:

Yes, that is an interesting issue that you mention and it’s something
I’ve worked out my own explanation for based on microeconomic theory.

Respondent 4:

I totally agree with this.

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