
Subject: YTM vs. CAGR Date: 12/2/2009 2:26 PM
Recommendations: 2
How to interpret reported bond returns has been on my mind for a while, especially the difference between the YTMs for zeros versus those for couponed bonds. So I queried Investopedia to see what they might have to say.

What Does Yield To Maturity - YTM Mean?

What Does Compound Annual Growth Rate - CAGR Mean?
The year-over-year growth rate of an investment over a specified period of time. The compound annual growth rate is calculated by taking the nth root of the total percentage growth rate, where n is the number of years in the period being considered. CAGR isn't the actual return in reality. It's an imaginary number that describes the rate at which an investment would have grown if it grew at a steady rate. You can think of CAGR as a way to smooth out the returns.

For now, set aside the fact that brokers might report YTMs using a 360-day year (instead of a more reasonable “Actual/actual”, which would include leap days). Also, set aside the possibility that coupons are being reinvested at the coupon rate (which is a near impossibility if the coupon-rate is higher than the prevailing, benchmark interest-rate.)

In my experience, if a couponed bond is priced near par, the YTM a broker will report, and the YTM I will calculate using Excel, will be very close. But when the bond is priced at a steep premium or discount, then my results differ significantly from theirs. I trust my own numbers, not theirs, so the fact of a difference is simply an annoyance, and I make the following mental adjustments: If I’m buying at discount, my yield will be higher than they report. If I’m buying at a premium, my yield will be lower than they report.

So far, so good. But here’s the problem and the reason for this post. If the bond is a zero, then brokers report a YTM that is actually a CAGR. Don’t trust my word on this. Run the numbers yourself. But here is a concrete example.

This morning, a lot of 25 of Toyota’s 0’s of 25 came onto the market (for a customer having sold it a couple days previously at 36 something). The offer was 38.750 with a reported YTM of 6.09%. For now, set aside the fact that a commission would have to be paid and prorated over the mandatory minimum purchase of 10 bonds. Just focus on the purchase price and the reported YTM. Today is 12/02/2009. The bond is due 09/26/2025, or a holding period of 15.8172 years, which becomes the root to be extracted when par is divided by the purchase price. On my trusty TI-34 calculator, the result of dividing par (aka, $1,000) by the purchase price of $387.50 is 2.580645161. When the nth root of that is extracted, where “n” is equal to the holding period of 15.8172 years, the resulting number is 1.061769916. Drop the “1” before the decimal point, and you end up with CAGR and a number very close to the broker’s reported yield of 6.09%.

Now, let’s consider an alternative method of figuring yield. Let’s assume we’re just a humble investor who wants to know what his implied annual return will be on his purchase without compounded the interest on the implied interest. In other words, if I put up X dollars, and end up with Y dollars after Z number of years, how much implied money is being added to my account? The numbers will be the same as before. The purchase price is $387.50. Par is $1,000. The holding period is the same 15.8172 years. But this time, let’s subtract the purchase price from par and then divide the result by the holding period. That result is $38.72, or the implied annual gain on our purchase price. Now, divide that gain by the purchase price. That result is 0.09992258, which can be rewritten as 9.99%.

So, which of the two previous results is “correct”? Does Toyota’s 0’s of ’25 offer a YTM of 6.09% or a YTM of 9.99%? Obviously, the bond offers a CAGR of something close to 6.09% and a YTM of something close to 9.99%. So, both answers are correct. But brokers report the CAGR for zeros as if it were the YTM. Why might CAGR be preferred, and not just for zeros, but for all bonds? Because the nemesis of bond investors, inflation, is stated as a CAGR. Therefore, to properly discount the impact of inflation, bond yields should also be stated in terms of CAGR, not YTMs.

PS Yeah, I did buy 10 of the offered 25 for my IRA account. Toyota is in a lot of trouble these days. But a YTM of nearly 10% seemed to be adequate compensation for the risks, just as a CAGR of 6% something seemed to be adequate. After I pay taxes on my gains and subtract inflation, my net probably won’t be a real rate of return, meaning, I will not have preserved purchasing-power. But the ding from inflation will be tolerable compared to suffering the low rates offered by CDs these days, and there’s a chance the bond will be called. So I might pick up a windfall profit from that. Under a worst case scenario of default, the capital risked is small, and the loss --after a Chapter 11 workout-price is deducted-- will be tolerable. Buying the bond was a risk I was willing to accept, so that I could move some unneeded, current purchasing-power forward to the future.
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