PrefBlog

I mentioned this recent article briefly in the post DeCloet & National Policy 51-201.

Anyway … credit ratings have been in the news, big-time, for the past year. The problem, however, is not so much with the Credit Ratings Agencies themselves, but rather with the regulators and with buck-passing investment managers.

Look for the opinion link!

Many wonderful – or, at least, wonderful sounding – investment strategies have come to grief through poor trading … either lack of attention to detail or insufficent care in minimizing trading costs (and commission expense is simply where trading costs start!).

Don’t let your strategy be among the bad examples! Look for the research link!

Assiduous Readers will be well aware that there are many levels of bank debt, each with its own mix of risk and reward.

In the following article, I attempted to summarize the levels to provide a little perspective. Look for the research link!

Update, 2008-7-19: There is an error in Table 1. Innovative Tier 1 Capital should be shown with a subordination of 7.09% (the same as preferred shares) and a yield of 5.55%.

2007 was a rotten year for preferreds. In the January, 2008, edition of Canadian Moneysaver, I attempted to explain why.

Look for the research link!

Update, 2008-2-19: In the article I indicated my amusement at the mention of preferred shares in the BMO annual report:

It is noteworthy that BMO revealed a charge of “$160 million in respect of trading and structured-credit related positions and preferred shares” – surely one of the few times that preferred share trading has been mentioned as a significant element of a Canadian bank’s profitability!

BMO has just announced another writedown including:

Trading and structured credit-related positions, preferred shares, third party Canadian conduits and other mark to market losses, approximately $175 million pre-tax.

Those durn preferred shares, eh?

In practice, banks guarantee the credit quality of the Money Market Funds they sponsor. This guarantee should be reflected when computing their capital ratios.

Look for the opinion link!

Update, 2008-9-18: After Reserve Primary Fund broke the buck on September 16, there were some avowals of credit support from sponsors:

Bank of America Corp. and Federated Investors Inc. disclosed their most recent holdings and pledged to protect investors after the $3.45 trillion money-market fund industry was jolted by its first loss in 14 years.

As U.S. stocks fell 4.7 percent for the second time in three days, money-fund managers said yesterday they will commit capital to offset any investment losses they incur. They sought to calm investors after Reserve Primary, the nation’s oldest money fund, did what no competitor had done since 1994 — allow its net asset value to fall below $1 a share, or break the buck.

Why do I want extra yield for holding a perpetual that is priced near par? I try to explain the rationale in this article, published in the November 2007 edition of Canadian Moneysaver.

Look for the research link!

Preferred shares are often issued with terms that are virtually identical with another issue from the same issuer; in some other cases, the terms of two series may be different, but the series may be convertible into each other at specified times. Investors should look for this kind of pairing, because the market sometimes does not price these issues as similarly as one might expect.

Look for the research link!

Regular readers will know that I am currently enraged by the persistent difference in price between BAM.PR.M and BAM.PR.N!

Update, 2015-5-12: Note that I have also provided a link to the calculator.

Well, I talk about it often enough! This one leans a little too far to the “mathematical” to be considered mainstream fare, but I had to provide it in order that the keeners would have a rough idea of what I say*. Look for the research link!

*Assuming, of course, that the keeners care about understanding what I say!

Update and Bump, 2007-10-27: I have received a rather patronizing communication from a correspondent who claims that the formula given in the article for modified duration is incorrect.

He claims that the variable “f” should always be equal to one and attaches two pages of calculus (including patronizing commentary) that proves it.

Alas, my correspondent has forgotten to ask himself the questions that any investment manager should ask himself – particularly if holding himself out to be a quant:

• What assumptions are made, explicitly and implicitly, in the course of this calculation?
• Have I thoroughly checked these assumptions (insofar as it is possible to check assumptions!) ?
• What if I’m wrong?

In this particular case, my correspondent has slipped up by assuming that “yield-to-maturity” (the “y” in the equation) is equal to the internal rate of return.

It ain’t. Yield-to-Maturity is, by street convention, equal to N times the coupon-period return, where N is the number of coupons per year. This point comes up over and over again – on this blog, for instance, there is a very similar discussion on Research : Yield from On-Line Calculator. It’s a tricky point and, in this degenerate age where one can find internal rate of return on Excel more easily than by explicit trial-and-error approximations, one that is easily forgotten.

Ah, for the good old days! When men were men, when you had to understand a calculation before you could perform it, and no back-of-an-envelope was too humble to serve as a scratch pad!

I have uploaded an MS-Excel Spreadsheet in which two sets of calculations are made – the way in which my correspondent does calculations, and the way in which we do it on Planet Earth. I believe that my computer – and my Excel programme – are bug-free, so there will be no virus transmission; I believe that all reasonable precautions are made with my site and my hosting provider so that the spreadsheet will not be hacked after upload; and I believe that even if a macro-virus should reach my readers’ computers, their anti-virus software will prevent harm.

But what if I’m wrong? I have also uploaded PDF file that reflects the spreadsheet, for use by those who don’t want to take the chance.

In cells a1:d22, I have input the cash flows reflecting the November 1, 2007, purchase of a par bond that pays $5 every six months commencing May 1, 2008, and repays its principal on November 1, 2017. Since all calculations will be performed as of November 1, 2007, I feel perfectly justified in referring to this bond as a ten-year, 10% bond.

The crucial point of the analysis is found in cell B24 – the XIRR function provided by Microsoft. It indicates that the internal rate of this bond is not 10%, as one might expect, but 10.2425%. This is easy enough to understand – for each year, we’re getting half of our quoted 10% six months early and can earn interest on this – but it is often forgotten. And this bond will not be quoted – anywhere – as yielding 10.24%. As an exercise, I suggest that interested readers sell such a bond to their brokerage house “at a price to yield 10.24%” and then invoice them for $100. Let me know what they tell you.

In cells E1:E22, I have discounted each cash flow by time in years at the yearly rate of 10.25% indicated in cell E24. Cell E23 shows the sum of the present values thus calculated, 99.99945, so we may conclude that we understand how the XIRR functions works.

In cells F1:F22, I have discounted each cash flow by period at the period rate of return of 5% indicated in cell F24. Cell F23 shows the sum of the present values thus calculated, 99.99945258, which I’m willing to conclude is a rounding error (particularly since I was lazy, and determined the period number by doubling the time in years, which will be off due to day-count and leap-year approximations. It’s OK for a quant to be lazy, as long as he knows he’s being lazy!). Anyway, this calculation confirms we know how to calculate returns by period.

Cells H1:H26 calculate the Macaulay Duration of the bond according to the “time” cash flows; cells I1:I26 perform this calculation using the “period” cash flows; the results are as equal as we can expect anything in this wicked world to be.

The next four cells to be examined, H28:I29, are a little more complex. The Modified Duration is calculated by the disputed Equation 2 for each calculation method using the indicated value of “y” and setting “f” to either “1” (as my correspondent insists is always the case) or “2” (which is dependent upon the number of coupon periods per year, as explained in the article. We obtain the following results:

That’s quite the range of differences! It should be clear from the calculations that the “correct” answer is given by “f=1” for the “Time” method and by “f=2” for the “Period” method … but what if I’m wrong?

Cells E31:E53 calculate the present value of the bond according to the “time” method when the Internal rate of return is increased by 1 basis point. Cells F31:F53 calculate the present value when the “Yield to Maturity” (= twice period return) is increased by 1 basis point. The change in price effected by these changes is calculated as PVBP (Price Value of a Basis Point) in cells H53 and I53, respectively.

We find that the Effective Modified Duration (we will refer to the results of this calculation as “Effective” modified duration because the change of 1bp in yield, while small, is not infinitesimal) is 5.93 for the “Time” method, which is equal to the answer calculated in cell H28 using f=1; it is 6.23 for the “Period” method, equal to the answer calculated in cell I29 using f=2.

Thus, we find that Equation (2) of the article is absolutely correct – given the universally accepted definition of yield-to-maturity as being (periods per year) * (period-rate-of-return).

So, we can summarize our characterization of this bond as:

Which method is right? Who knows? Who cares? What is truth? The street convention is simply that: a convention adopted in order to communicate.

Immediately after posting this update, I will reply to my correspondent and ask for permission to publish his correspondence.

Update, 2007-10-29 The idea “YTM = (coupon period return) * (coupons per year)” is usually expressed formulaicly as “coupon period return = 1 + (YTM / coupons per year)” or “coupon period discounting factor = 1 / [1 + (YTM / coupons per year)]”. See, for example BANK OF ENGLAND FORMULAE FOR CALCULATING GILT PRICES FROM YIELDS

Updated versions of this publication are available from the UK Debt Management Office.

Update, 2007-10-31 I have uploaded a note from the Bank of Canada on calculation conventions; there is also a Department of Finance Web-Page available with the same information.

Update, 2009-2-15: Note that the Modified Duration of a PerpetualDiscount is dependent solely upon its yield.

There are a number of misperceptions held among otherwise sophisticated investors regarding perpetual preferred shares. Now that the October edition of Canadian Moneysaver has been published, I can release this article, which attempts to address two of them, published in their September edition.

Look for the research link!

Update, 2007-10-15: An assiduous commenter asks how much the numbers would have changed without rebalancing … so I’ve done the calculation.

The difference is not as much as my correspondent suspected! Raw data (showing the returns for the period March 30-July 31) has been uploaded for reader inspection for both the PerpetualDiscount and PerpetualPremium indices.

The relatively small difference between the rebalanced and non-rebalanced indices illustrates the point that there is a very sharp point of inflection between “Premium” and “Discount” perpetuals; once that point is crossed, duration changes significantly and the price reaction to yield changes becomes much more like one group than the other, with very little “grey area” between the two camps.

Update, 2007-10-15, later: The immediately preceeding paragraph is nonsense. Sorry!

Update, 2009-1-29: Assiduous Reader PN writes in and says:

I have found your PrefBlog website to be an extremely useful source of information on preferred shares. I have recently delved back into the preferred share market after concentrating on common shares over the last 25 years.

I am continuing to debate the pros and cons of perpetual discounts vs. 5-year fixed resets. In this regard I found your “Perpetual Misconceptions” article in the September 2007 edition of the Canadian Moneysaver to be very useful. I liked Table 1 so much that I reproduced it as a Excel Spreadsheet so I could compute implied future yields for different x and y values (where the resultant return is x% and the current perpetual return is y%). In doing so I discovered a slight discrepancy in the calculation of the discount factors in your Table 1. For a 2% return you have
calculated the discount factor after year 1 as 1.00-.02= .9800 rather than 1/(1.02)=.9804 The small error is continually compounded for years 2 to 20. I was wondering why you choose not to use the generally accepted mathematical formula for discount rates?

I have attached a spreadsheet based on two sets of calculations: the first is based on the generally accepted mathematical formula and the second is based on your computations for Table 1. You can see the results are only very slightly different for the 2% and 5% rates you have chosen and would not affect any of the conclusions you have drawn in your article.

My question is a very minor point and I am sure you must have a good reason for your calculation of the discount factors. I am just wondering what was your reasoning was?

Well, PN, there’s a very simple answer to your question: I am an idiot.

I cannot, at this point, remember anything much about the preparation of this article; what may have happened is that a rough draft of the table made it into the final product without thorough checking; the checking being performed in a cursory fashion because, as you say, the errors are small and the conclusion robust. Let’s just pretend that we’re seeking a total return of 2.04% and that that figure is cited on the table title, shall we?

PN has won a complimentary issue of PrefLetter.

Most readers will know about my article Yield Ahead, which I link in various places with:

In that article – and on this blog, right hand column, under “On-Line Resources” – I recommend Keith Betty’s On-Line Yield Calculator (Update, 2022-9-20: Link broken. Click HERE for a copy or try the version on googledocs) as a method whereby retail investors may quickly and easily calculate the yields to the various call dates for any given preferred issue, once they know its characteristics. I have published characteristics for the issues tracked by HIMIPref™ at PrefInfo.com:

Keith recently received a note from a market practitioner stating that “the formula in U105 should be =((1+((U104+1)^0.25-1))^4)-1 for the quarterly yield”, as opposed to the current formula, =4*((U104+1)^0.25-1) and asked what I thought.

To think about this, we first have to get rid of Excel Spreadsheet cell references. Let’s refer to the existing formula as (A) and the suggested formula as (B). We’ll also let the internal rate of return on the cash flows be “r”. Therefore:

(A) 4*((1+r)^0.25 – 1)

(B) ((1+((1+r)^0.25 -1))^4 – 1

It’s now easier to see that the expression (1+r)^0.25 appears in both expressions; this is the quarterly rate of return; we’ll set that equal to R, so:

(A) 4R

(B) (1+R)^4 – 1

Now we know what we’re talking about, which is always a pleasant state of affairs! Keith’s formula, (A), converts the quarterly value to annual without compounding (many, including myself, will refer to this as a simple yield), while the suggested formula, (B), converts the quarterly value to annual with compounding.

Once I had realized this, I basically shrugged my shoulders and drew my own conclusions, which I’ll explain at the end of this post. But I poked around in some reference material anyway and found, in Choudhry, Analysing and Interpreting the Yield Curve, Wiley Finance 2004, ISBN 9780470821251, page 22:

The market convention is sometimes simply to double the semi-annual yield to obtain the annualized yields, despite the fact that this produces an inaccurate result. It is only acceptable to do this for rough calculations. An annualized yield obtained by multiplying the semi-annual yield by two is known as a ‘bond equivalent yield’.

Well, I haven’t heard that definition of “bond equivalent yield” before, but if Choudhry has and wants to state it as a convention, I won’t complain.

Anyway … Keith’s point is that formula (A) ties in with material he gets from brokerage houses and that (i) he is therefore justified in referring to it as a market convention, and (ii) the spreadsheet will be more valuable to users if it is compatible with calculations shown by brokerages houses.

I agree with him; I also don’t think it matters very much.

All the math on the spreadsheet is accessible to the user; anybody sophisticated enough to ask the question can get the answer very easily and revise the spreadsheet for his own needs very easily. So as a practical matter, I don’t think it matters.

Additionally, it should be noted very carefully that the Internal Rate of Return (IRR) calculation assigns the same yield to every cash flow. This implies that (i) the yield curve is flat and (ii) all cashflows may be reinvested at this yield. Since it is known that these implications are both completely bogus, I take the view that worrying about the difference in presentation between formulae (A) and (B) is basically a waste of time.

And finally, another practical point: the purpose of the spreadsheet is to provide a tool for an apples/apples comparison between two sets of cash-flows. To the limits imposed by the calculation of IRR, that’s exactly what it does.

And, finally finally, I had a look at Bloomberg, on the grounds that nowadays the phrase “market convention” has an identical meaning to “the way Bloomberg does it”. What they refer to as “Actual” yield (as opposed to “S/A Street” yield) is pretty close to that given by formula (A), but not precisely. I looked at it for a while and, frankly, I don’t know how the $%##! they come up with their number.

Update: In looking at this again, I note that there are too many brackets in equation (B). It simplifies to:

= (1+((1+r)^0.25 – 1))^4 – 1

= (1+ (1+r)^0.25 – 1)^4 – 1

= ((1+r)^0.25)^4 – 1

= 1 + r – 1

= r

Update, 2007-10-28: This issue has reared its head again with respect to Modified Duration and is discussed in the post Research : Modified Duration.

It should also be noted that – in complete accordance with the convention that Keith has applied – a new issue preferred priced at $25.00 with an annual dividend of $1.20 paid quarterly is advertised as having a yield of 4.80%. If we were to use the IRR method suggested by Keith’s correspondent, we would be forced to advertise the yield as (1.012)^4 – 1 = 4.887% … and salesmen would be getting a lot of angry calls from clients.