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# What Does “Discount Factor” Really Mean?

While I have been dabbling with stock investment for three years now, I have not actually given any serious thought to the question until very recently. As taught in many investment textbooks, an investor should apply some sort of a "discount factor" to the expected cash flow generated by a business, to derive a "fair value". Ben Graham suggested using 2X long-term Treasury yield. Others prefer assigning P/E values to specific sectors. Personally, I have also been using such methods indiscriminately, throwing in a 15% discount for business that I am uncertain about, or an 8% for business that I am really sure. But what exactly does this "discount factor" mean?

United States vs. Banana Republic

First of all, I think we should go back to the simple time value of money equation:

(1) For a perpetuity with fixed payment of "\$A" per year (payment starting at Year 1) and no growth, the net present value "S", at prevailing interest rate "r", is:

Let's say we now have two Governments issuing bonds: United States and Banana Republic. Suppose that US is considered absolutely safe with no risk of default while BR is less trust worthy. If both are issuing a perpetuity with fixed payment of "\$A" per year, how will an investor react? Well, if these two bonds are the only alternatives to holding cash, then he will be willing to pay a higher price for US's bond than BR's bond.

Formally, the buyer is willing to pay "S" for the US bond, but less than "S" for the BR bond. In other words, the implied interest rate is exactly "r" for US bond, but larger than "r" for BR bond. The BR bond is sold with a "discount factor" with respect to the US bond, to account for the less trust-worthiness.

Banana Republic's Point of View

How will the picture look like from US and BR governments' points of view? Well, in both cases, both US and BR governments will have the same payment schedule, i.e. pay "\$A" per year in perpetuity, so we simply have to calculate the present value. What "interest rate" should we use? Well, assume that the cash received by both US and BR governments can only be used to invest in the safest asset, which in this case is the US bond. Therefore, the discounted cash flow is exactly the same for both US and BR governments - it is equal to "S".

For the US government, it is like receiving "S" in its left hand from the investor, buying a US bond from its right hand, and then passing "\$A" payment every year from the right hand directly to the investor (as to how the right hand generates the yearly return remains a mystery). For the BR government, it receives less than "S" from the investor, pays less than "S" to buy the US bond, and gets paid less than "\$A" every year from the US bond, but will have to pay exactly "\$A" back to the investor - so BR government had better figure out a way to make up for the difference.

If that is all there is about the world, then it doesn't make sense for BR government to issue the bond - it gets no benefit itself, and yet still has to make up for the difference for the investor. However, if now BR government somehow finds an Orange County that issues a bond with interest rate "2r" (i.e. for a payment of "S" it will get annual payment of "\$2A"), then it may make sense for BR to issue its own bond, get less than "S" from investor, invest in Orange County bond, pay "\$A" to the investor, and then keep the rest for itself.

How much is it worth?

Let's focus on BR government's perspective. The question I post is this: how should BR government evaluate its own bond (a liability)? Well, using the US bond interest rate to discount the perpetual annual payment of "\$A", it should be valued exactly at "S" (as liability). Should BR government "discount" the bond simply because the investor is willing to pay less than "S"? Or should BR government place a "premium" on it simply because it can by the higher-yield Orange County bond?

I think the answers to both questions, in my opinion, are "No" (even though certain accounting rules may say otherwise). A liability is just a liability: it is equal to the total future cash payment discounted to the present value using a risk-free interest rate. It does not matter whether the investor is willing to pay "S", "0.5S" or "0.1S" - from BR government's point of view, it simply has to keep paying "\$A" per year to the investor. The "discounting" on the investor's side has no bearing whatsoever on the size of liability assumed by BR government. It does not matter either that BR government can buy the Orange County bond with "2r" interest rate. Whether BR government can get interest rates of "10r", "100r" or "0" has no bearing whatsoever on the size of liability it assumed; these other investment opportunities only serve to encourage (or discourage) BR government from issuing the bond. But once the bond is issued, the liability does not change regardless of whether BR government is successful or not in its chosen investment opportunity.

Investor's Profit

What, then, does the "discount factor" mean? From the investor's point of view, the BR bond is worth less than "S" due to discounting on his side. From BR government's point of view, the BR bond is valued exactly at "S" as a liability. So what does the difference between the two represent? Well, I may call it "investor's profit". This is the profit the investor gains (or demands) by buying the BR bond.

Since the BR bond is riskier than the US bond, the investor demands a higher profit from the BR bond than from the US bond (the "profit" from the US bond is simply 0). This is justified on various ground: in the current case, BR bond has greater risk, i.e. the annual "\$A" payment may not always come as promised. In other cases, it may be due to higher rate of inflation, longer maturity time, higher rate of return offered by bonds of similar qualities, etc. If this is the consensus view of the investing public on BR, and if BR government has no means to raise fund other than from the investing public, and if BR government is in urgent need to raise funds, then it has to accept "discount factor" demanded by the investor and provide him with "investor's profit".

What does it mean exactly?

Let's say the investor applies a discount factor that assumes 3 missing annual payments. Does it mean that 3 annual "\$A" payment WILL NOT come as promised? If the investor knows FOR SURE that exactly 3 annual payment will be missed, and yet his discount factor assumes 3 missing annual payments, then in reality he is not getting any "investor's profit" at all! He might as well spend the money buying US bond instead.

Therefore, when applying a "discount factor", the investor must know with pretty high degree of confidence that "\$A" annual payment will come as promised. I suppose the thinking process is along the line:

(1) I know with pretty high degree of confidence that "\$A" annual payment will come as promised, which has a present value of exactly "S" using the risk-free interest rate.

(2) However, even if I am very sure about my estimation, it does not mean that I am willing to pay for full price at "S". I am only willing to pay less than "S" and demand "investor's profit", because I know from BR's recent performance that it has not been very consistent in bond payment (but now with a new Prime Minister, I think BR government will behave in a more honorable way).

(3) The "investor's profit" is also a buffer zone, or a "margin of safety". Even if you really do miss some payment, I am still sufficiently compensated and will not lose out too much.

You want to buy it from me?

Now, let's say I myself have bought the BR bond at "0.8S" in the morning (i.e. with "0.2S" investor's profit), and now you offer to buy my BR bond in the afternoon of the same day - how will I think? Well, first of all, I know with high degree of confidence that the total value that I am going to get from the BR bond is "S" (suppose for the moment that I live till eternity) and I am having a "0.2S" investor's profit. If you are paying me only "0.8S" to buy my bond, then I am effectively transferring my "0.2S" investor's profit to you, and I get back the "0.8S" I spent this morning, with no gain whatsoever. Unless there is now another bond offering investor's profit of "0.5S" (and I couldn't have bought this bond without selling the BR bond), why should I even be bothered? I may be much better off just waiting and getting out that "0.2S" myself.

But if you are offering me "0.9S", then I will get to keep my "0.1S" investor's profit and transfer only "0.1S" to you - it depends on how I weigh the two options: (1) wait till eternity for the perpetuity payment in full (and get "S"); or (2) take the "0.9S" cash now and give up the "0.1S" that I could have gotten by waiting. From your point of view, you will get an investor profit of "0.1S"(which comes from me), which you may or may not considered adequate (but if not, then you shouldn't enter into the transaction in the first place).

How much is it worth (again)?

Importantly, in both cases the underlying BR bond is still going to produce a present value of "S". The fact that I am willing to buy only at "0.8S", or that you are willing to pay for it at "0.9S", has no bearing on the true value of the BR bond (which is still "S" based on discounted cash flow).

Therefore, we need to distinguish between (.A) what is the present value of the future cash flow of an investment, and (.B) what other investors are willing to pay for it. The difference between the two is "investor's profit". To the extent that (.A) can be estimated with a high degree of certainty, then (.A) is not going to change much (assuming the risk-free interest rate does not change). On the other hand, (.B) varies depending on the investor. For "value investor", (.B) is always small - they always demand a large "investor's profit", or a large margin of safety. But for the less conservative investors, (.B) may get dangerously close to (.A) - and sometimes more than (.A). Maybe that should be called "investor's premium?"

Conclusion

Now back to the more practical issue: how should one evaluate any asset? I would suggest the following. Firstly, one needs to find the present value of future cash flow, discounted using the risk-free interest rate. The future cash flow should be estimated conservatively, in the sense that he has a high degree of confidence that this will actually come pretty close to actual performance -- this is the actual return he is going to get for his investment. Secondly, he should apply a discount factor to the calculated present value. For example, if one demands a return 2x that of risk-free interest rate, then the discount factor is 50%. This is the price that he is willing to pay, or in other words, this is the investor's profit that he is demanding.

To sum it up, the value of an asset, in terms of the total amount of return it will bring to its owner, is different from what an investor is willing to pay for the asset. The difference between the two is the profit for the investor (for engaging in this investment activity), which is reflected by the discount factor. The fact that many investors use a large discount factor (or are willing to pay only a low price) does not mean that the asset suddenly becomes less valuable. It only means that if the owner of the asset wants to (or is forced to) sell on the market, he will not be able to get a good price. If he is confident in the future performance of the asset, he should simply keep it to himself -- after all, he himself is an investor too, and he can enjoy his own "investor's profit" instead of passing that off to other people. The bottom line is: always pay less than what something is worth, but one should first have a good grasp on what something is actually worth.

P.S. The strange case of infinity

Now look at a slightly modified scenario with a growing perpetuity:

(2) For a growing perpetuity with payment of "\$A" at Year 1 and growing at rate of "g", the net present value "S", at prevailing interest rate "r", is:

Therefore, if "g" is equal to "r", then the denominator will be zero, and "S" will be infinite. What it means is this: if a financial instrument provides an annual payment "\$A" that grows at the same rate as risk-free interest, then the present value of this instrument is infinite.

Probably a lot of people will laugh it off as being ridiculous: how can something be worth infinite amount of money? Well, they are absolutely correct in this aspect: one should never pay for the full price, and even if "S" is a mathematical infinity, an investor should pay only a certain amount (perhaps discount it using some values).

But if I turn the table and ask: how much do I have to pay you now, so that you promise to pay me an annual payment of "\$A" growing at the same rate as risk-free interest rate, for perpetuity? Then I don't think it is so funny any more. You will be very bold if you accept any number less than infinity (you may try using my lum sum payment to invest in things that offer higher returns e.g. stocks -- but you have to keep doing it for ever), and you will be out of your mind to say okay if I have told you that the money you receive from me can only be invested in risk-free bond.

It is often useful to think from a different angle -- it may bring out something new.