I was recently taken to task for a claim that the yield on BCE.PR.Y was 8.18% based on a dividend of $1.05715 and an end value of $25.00 – my correspondent stated – quite rightly – that:
the most recent monthly dividend, declared Oct 28, 2008, was $0.8333 or $1.00 per year. Also Prime has dropped to 3.5% from 4% earlier this month, (according to the BOC website), indicating a further cut in the dividend in the near future. Even at the rates and prices you quote I make the yield out to be 7.3%.
My defense is as follows:
They system estimates the average future rate of prime by looking at the past. If we stay at 3.5% prime for long enough, the estimated future rate will drop to this level, but for now it’s higher.
Additionally, the system estimates the end-value (a “limitMaturity” is treated as thirty years, remember) by determining the price at which the instrument is fairly valued; determining fairness by comparison with other floating-rate dependent issues. This was the result of some experimentation and proved to be a better predictor than assuming a constant price (as is done with fixed-rate perpetuals).
Basically, the assumption is that an Investment Grade issue will not pay 100% of prime forever. There will be shocks, of course, and every now and then such an issue will be downgraded and quite properly pay 100% of prime; but over the long term such a rate is not sustainable.
I will admit that this analytical framework was formulated when deviations were relatively small; an investment grade issue paying (25.00 / 14.25) = 175% of prime is not something that happens often enough to permit testing!
All the above is not very satisfactory, I know: but there are a lot of moving parts in the analysis of these ratchet rate issues and the framework was determined empirically. In some cases, to my chagrin, the results are not even internally consistent (e.g., I might be estimating a ratchet yield of less than 100% of prime with end values well below par).
All I can say is that the empirically derived system, while having theoretical holes in it, does have a statistical ability to rank the various issues with significantly better-than-random accuracy, which is all I ever wanted it to do.
Now lets do the cash flow analysis! I have uploaded the full HIMIPref™ output; the last part is:
2038-12-16 MATURITY 25.00 0.080242 2.01
Total Cash Flows 56.6052
Total Present Value 13.5028
Discounting Rate 8.5887 % (Annual rate compounded semi-annually)
So for starters, we see that the the discounted present value of the $25.00 maturity is only $2.01. It’s not a particularly important variable.
But compare four bonds priced at par, each one paying $12 p.a., but with differing frequencies (annual, semi-annual, quarterly, monthly). Each one is described by fixed income convention as having a yield-to-maturity of 12%. Which would you rather have? Obviously, the monthly payer, since then you get your money faster … and this is borne out when we look at the annualized internal rate of return for the four bonds: 12.00%, 12.36%, 12.55% and 12.69%, respectively. The limiting case is an infinite number of infinitesimally small payments totalling $12 and has an IRR of exp(0.12) – 1 = 12.75%.
We note from the HIMIPref™ report that the 30-year discounting factor is 0.080242 so
1 / (1 + I)^30 = 0.080242
(1 + I)^30 = 1 / 0.080242 = 12.4623
I = 8.7727%
To convert this annual value to semi-annual, bond-equivalent yield, we note:
1+I = (1+i)*(1+i)
(1+i) = 1.042942
i = 4.2942
and therefore, the bond-equivalent yield is 2*i = 8.5884%, which is slightly different from the quoted figure, but we’ll attribute that to rounding.
But how to calculate this ourselves? The “ratchet yield” is 4.1997% of par, which implies total payments of $1.049925. These are made monthly, so monthly payments are $0.087494, which has been shown as a rounded value of $0.09 in the HIMIPref™ report.
The normal quick-n-dirty calculation is:
i = [Annual Income + Annual Capital Gain]/[Average of Beginning and Ending Price]
Annual Income = oh hell, let’s just call it $1.05, shall we?
Annual Capital Gain = Total Capital Gain / Term = (25.00 – 13.50) / 30 = $0.38333
Average of Beginning and Ending Price = (25.00 + 13.50) / 2 = 19.25
resulting in a quick-n-dirty estimate of (1.05 + 0.3833) / 19.25 = 7.45%.
It’s a lousy estimate. Terrible. Why?
Mainly because the beginning and ending prices are so different. The calculation assumes that the capital gain is realized in a linear fashion … but in fact, if it accrues at a constant rate, nearly twice as much is accruing at the end of the period as at the beginning. Conversely, the $1.05 income is much more interesting at the beginning of the period (current yield = 1.05 / 13.50 = 7.78%) than at the end (current yield = 1.05 / 25 = 4.20%.
When the capital gain through the period is massive, simple methods become simplistic. Such is life! Fortunately, yield calculators and Excel Spreadsheets will be readily available to most people.
Listen, take it from an old bond guy: if anybody ever tells you yield is simple, don’t listen to him!