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 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.

Freshman calculus: Formula A is the first two terms of the Taylor series approximation for Formula B.

P.S. Thank you for this blog. It’s a pleasure to read every day.

Thank you, Norbert, for both the compliment and the calculus refresher!

It occurs to me that the annualization calculation for the conventional yield for both preferreds and bonds can be considered as a normalization procedure giving the yield in the current pay period (quarterly or semiannual), calculated on an annualized basis.

[…] 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. […]

[…] Those who have suffered through endlessly repetitive disputes about the difference between “Annualized Internal Rate of Return” and “Yield to Maturity” in the posts Modified Duration and Yield from On-Line Calculator will be highly relieved to find an authoritative source for the conventions! […]