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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?
The rate of return anticipated on a bond if it is held until the maturity date. YTM is considered a long-term bond yield expressed as an annual rate. The calculation of YTM takes into account the current market price, par value, coupon interest rate and time to maturity. It is also assumed that all coupons are reinvested at the same rate. Sometimes this is simply referred to as "yield" for short.

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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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%.

remember that the YTM is an anual yeild. the geometric average which you calculated as the CAGR is one soloution.

A algebraic average would be another, however your calculation of algebaic average is not quite right. Assuming the stright line apreciation you presented. the yield in the second year would not be 9.99 but rather 38.72/426.22=9.08% ... add these up and devide by 15.8172 and well the geometric average and geometric average are the same if there is no variance in the rates.

I believe the answer to your question of why the bonds are listed as they are is one of simple mathematics rather than the vagries of bonds and the special relationship bond investors have with inflation.

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"Obviously, the bond offers a CAGR of something close to 6.09% and a YTM of something close to 9.99%"

A bit of semantics: It's a CAGR of 6.09%, YTM of 6.09%, and an annualized HPR of 9.99%.

YTM is well defined - you can't redefine that term (at least not without expecting a lot of disagreement and confusion). YTM is calculated with PV=F/(1+r)^T (or a summation of those terms for a bond with coupons)

Therefore, for a zero, there is only one PV (present value) required (rather than a summation across all the payments), so YTM=CAGR

As you've pointed out, the errors between 6.09 to 6.17 you are encountering are related to the 360 day year assumption (and based on your use of 15.8172, likely a small error since you need to account for the 3 days between purchase and settlement)

What you are calculating is a an annualized Holding Period Return (HPR). HPR is calculated as:

(End Price - Beginning Price + Dividends) / Beginning Price

Dividing the HPR by the holding period produces the annualized HPR, and matches your 9.99%

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Thanks for the feedback, but my numbers are correct.

As I think about yields, there is no second year for a zero, nor for a couponed bond, either. My purchase price is the only price that matters to me. All subsequent income-streams depend on that price. All actual coupons received are not added back to my original investment to enhance its yield. The coupons received are aggregated, and they become new investments. Thus, each initial investment I make can be benchmarked against any other.

The same --for me-- with zeros. The implied coupon payments get aggregrated, but they can't be redeployed until par is received. What I am always interested in knowing is what can I accomplish with the cash that I have to spend. Buying a couponed bond creates one kind of game. Buying a zero creates another. But I need to know when which game is the better choice in terms of the dollars eventually realized from the initial investment only.

Yes, how brokers choose to report yields is for them to determine. But investors need to interpret those numbers as they might make the best sense to them. The impact of inflation is a whole 'nother problem. Some people choose to ignore it or understate it. That's their choice. I prefer to think about my bond returns as CAGR, so I can make direct comparisons with inflation. Others will make different choices for themselves.
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You're right. HPR is exactly what I want to calculate, not YTM.

You're also right about "trade-date" versus "settlement-date". That's a difference I ignore in my calculations. If my typical holding-period were mere months instead of many years, I'd have to be more careful. But a one-time error of three days, versus a holding period of fifteen plus years, won't much matter.

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Junkman, thanks a lot for your continued invaluable posts for for readily sharing your knowledge and apparent passion for fixed income. I am curious to learn your general thoughts on inflation - not what your thoughts are on its extend or duration in the future but rather why we you, we, everyone universally, assume inflation on a secular basis (say over most 10-20-30 year periods?

I am an econ major, previous real estate investor, have worked on the Street...fairly financially literate...but this basic question perplexes me. I understand inflation as having to be "too much money chasing too few goods" and I understand that the government prints money and there is an ever increasing amount of money in circulation but why does that invariably lead to inflation? Why do we always assume inflation? The pull of population growth? Population and output of goods and services not matching demand quickly enough? Inflation confuses me a bit, a least in that it is an assumption most investors take as a given
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This probably doesn't add much to the discussion but it isn't just too much money, but the money has to be spent. People often refer to the velocity of money. Currently there seems to be excess money being put into the economy but IMO it doesn't seem to be circulating very much. People are saving more to try to make up for their losses and banks are trying to prop up their financial position.

M V = P Y

where M is the money stock, V is the velocity of circulation, P is the overall price level, and Y is the level of output, to the equation for the inflation rate:

Inflation (in % per year) = Money Growth (in % per year) + Velocity Growth (in % per year) - Output Growth (in % per year)

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