Two related ideas that seem to permeate discussions of CEF pricing on Seeking Alpha are (NYSE:I) that the buyer of a fund selling at a discount is getting the assets cheaply, and (ii) that a return of capital from a fund that sells at a discount creates alpha by distributing cash at the discounted rate. Both of these ideas are wrong. This article deals with the alleged alpha to be found in cash distributions from funds selling at a discount.

Consider a hypothetical CEF that pays a managed distribution of $.10 that goes ex-dividend at the end of each calendar quarter. (For convenience sake, I'll use "paying" to mean "going ex-dividend.") On December 31, 2017, after paying its Q4 distribution, the fund's NAV is $10.00, and it is selling at $9.00. Some would say that the CEF "sells at" a 10% discount, but I believe such a characterization is misleading. Let's look forward 90 days or so to see why.

On March 30, we are on the eve of the first distribution of the year. Here are some possible scenarios.

1. *The fund has earned ten cents in ordinary income, but the underlying securities are worth exactly what they were worth on December 31.*

The NAV of the fund is $10.10. What is its market price? If we say that the fund "sells at" a 10% discount, and nothing has happened to change the market's assessment of the quality of management or prospects for the underlying assets, it should be selling at a 10% discount on March 30, *i.e.*, for $9.09.

On March 31, the fund pays a .10 distribution. NAV is now $10, and, if the fund "sells at" a 10% discount, the market price of the shares becomes $9.00. But that can't be right. *Everybody knows* that the value of a share of stock falls on the ex-dividend date by the amount of the distribution. So the shares of our fund *must* fall by .10 on the morning of March 31. So, what's the discount? It cannot be 10% on both the day before a distribution and the day of the distribution.

Having posited a 10% discount on December 31, we should assume the same discount on March 31. So, the price of the fund on March 31, *i.e.*, after the Q1 distribution, is $9.00. Assuming no market-moving events overnight, the price of the fund on March 30, *i.e.*, right before the Q1 distribution, must have been $9.10 in order for the price to fall by ten cents the next morning. So, on March 30, the fund's NAV was $10.10, and its market price was 9.10, rather than 9.09. The discount had shrunk from 10% to 9.901%. Not very much, but important to understand in that it makes the claim that the shares "sell at" a 10% discount imprecise and, as it turns out, very misleading.

2. *The fund has no income, and the underlying securities are worth exactly what they were worth on December 31.*

The NAV of the fund is $10. What is its market price? After the distribution, the NAV will be $9.90. If the fund is to have the same 10% discount on March 31 as on December 31, the price of the shares on March 31 will be $8.91. But, as we noted in looking at Scenario 1, the market value of the fund on March 30 must be $.10 higher than its value on March 31. So, on March 30, the fund's market price must be $9.01 when the NAV was 10.00. Thus, the discount immediately before the distribution was not 10% but 9.9%(*i.e.*, .99/10.00).

3. *The fund has .05 in income, and the underlying securities are worth $10.50.*

This scenario shows how a more or less random selection of performance parameters produces the same result as the polar examples. On the day before the distribution, the fund's NAV is $10.55. After the distribution, the market price of the fund will be 90% of $10.45, or 9.405. Therefore, the market value on March 30 must have been 10 cents higher, *i.e.*, $9.505. The discount on March 30 is thus (10.55-9.505)/10.55, or 9.905%.

No matter how we tweak the numbers, two facts remain: the discount contracts as the ex-date approaches, and (ii) the discount expands when the ex-date arrives. The contraction and expansion of the discount explain why the distribution creates no alpha. Specifically, the shares are *not* selling at a 10% discount on the day before the distribution, but are selling at that discount the day after. Thus, any alleged "alpha" contained in the distribution is immediately absorbed by the increase in the discount, and the market value of what you hold (shares and distributed cash) on the day the distribution occurs is exactly the same as the market value of the shares you held on March 30.

These scenarios also show that it makes no difference whether the distribution is a "return of capital" (Scenario 2) or funded entirely by income (Scenario 1). The math is the same: for any given discount to apply on the ex-date, the dividend must be smaller on the day before the ex-date. No matter what.

We can reach a similar conclusion by looking at the discounted cash flows. The price of any security (*i.e.*, the cash outlay by a cash buyer, however denominated) is the discounted present value of future cash flows. The price can be illustrated as a series of equal cash flows subjected to a single discount rate, but the conceptual math is a bit more complex. A different discount rate must be applied to each future cash flow, because each succeeding cash flow is rendered less certain by the increasing number of things that can intervene to defeat it or decrease its value in dollars. Alternatively,we can adjust the cash flow itself to reflect the changing uncertainty of payment and then discount each cash flow by the risk-free rate (something like the Treasury rate) applicable to the time period remaining until the distribution. I'm not saying that anyone actually does this math - I don't - but it is important to understand in the context of the discount on a CEF around the time of a managed distribution.

If the price of the security is the sum of the present values of all future cash flows, we can deconstruct that price into the sum of the present value of the next cash flow, plus (ii) the sum of the present values of all subsequent cash flows. If the next cash flow is literally tomorrow, we can understand how that one cash flow is not discounted at all in the pricing process. It is like accrued interest, which is certain to be paid and so is explicitly excluded from the quoted price of a bond, but not from the amount the buyer actually pays to acquire it. And, like accrued interest, an imminent distribution from a CEF cannot be bought at a discount.

Because the nearest cash flow is discounted less than the far ones, the *aggregate* discount necessarily changes as the nearest cash flow approaches, so that the upcoming distribution is discounted at zero the day before the payment. The orderly cycling of the discount rate is masked by other events affecting the pricing of the CEF, but the price on the ex-dividend date will drop by 100% of the distribution when paid, even if the shares are selling at a discount to NAV. All of that discount is attributable to future cash flows. The discount (in dollars, not percentages) is the same after the ex-date as before it. But the numerator (market price) and denominator (NYSE:NAV) of the discount fraction have fallen by the same amount - the amount of the distribution - and so the perceived discount is larger after the distribution than before it.