Your critique of YTM is very good, but you must remember that it is a convention use for convenience in a relatively homogeneous world, where all bonds are issued at par, will mature at par and will pay a coupon in line with market rates at time of issue.

The basic problem with the math is that both YTM and IRR presume a flat yield curve that is subject solely to parallel shifts. This implicit assumption makes both of the methodologies highly suspect and – while they both provide good rules of thumb – make them useless for serious quantitative work.

IRR has its own special problems; when determining portfolio returns, for instance, you cannot determine IRR for the full period by linking two sub-periods. Additionally, in certain situations, IRR will return multiple answers. Assume, for instance, you have the following cash flows in your portfolio:

12/31/07 : Initiate with $1,000.00

12/31/08 : Take out $3,300.00

12/31/09 : Put in $3,627.5

12/31/10 : Portfolio valued at $1,328.25

Very strange cash flows, obviously! But now calculate the IRR … annualized returns of 5%, 10% and 15% are each completely accurate!

This is because IRR is just the roots of a polynomial expression. To get the cash flows required to get such a silly result, I expanded (x – 1.05)*(x-1.1)*(x-1.15)

Anyway, my point remains the same: the most telling critique of YTM is the reliance on assumptions about the yield curve that don’t make any sense. Since you know that all the assumptions are a little fishy, you must also realize that the answers are going to be a little fishy.

*However we can assume a standard shift in IRR for the entire portfolio (and even for balance sheet) and can do impact analysis (stress testing) by applying IRR-based portfolio Modi-duration and convexity on a shift in IRR.*

I’m not convinced. I will agree that if you have a population of financial instruments that is heterogeneous in terms of interim cash flows, IRR will probably do a better job than YTM. I am dubious as to the actual incremental value of this approach when doing something like Value-at-Risk for an institution with major assets and liabilities all along the curve. For such an analysis, I would greatly prefer to price everything at the base curve and see what happens to prices when you throw things at it.

For this you need to calculate a zero-coupon curve (so that each time, t, has a specific yield, Y(t), and therefore a specific discounting factor, D(t), instead of saying Y(t) = constant as you do with both YTM and IRR), do a principal component analysis of historical curves, and then throw things at your base curve. This takes an awful lot of computation – but nowadays, computing time is cheap.

To illustrate my point … 1994. The yield curve went from upward sloping in a relatively straight line, to extremely humped (the 2-10 curve was a LOT steeper than that 10-30 curve). If you had been hedging long 10s with a short 5/30 butterfly, you would have been killed. And a lot of people were.

]]>I have studied your response and would like accept slip on my part by considering IRR as equivalent to YTM. I have independently verified the pricing effect on account of IRR-based modified duration and YTM-based modified duration. The results are indeed matching and now I am convinced that if YTM is used for arriving Modified Duration from Macaulay Duration, the use of divisor is necessary.

However a careful analysis of YTM and IRR and their related modified duration has raised several questions in my mind. I have narrated my thoughts below:

YTM is sensitive to the frequency of coupon while IRR is always annualized and will not be impacted by frequency of coupon. For example, let us consider three bonds having same IRR but different frequency of coupon, say, 1, 2 and 4 per annum respectively. So if IRR of all three bonds is 10%, the YTM would be 10%, 9.762% and 9.6455% respectively. Due to this it is not possible to calculate YTM based Modi-Duration for a portfolio (consist of different bonds having different frequency of coupons). However it is seemingly easy to derive IRR-based M-duration for the portfolio (by applying value of each bond as weight to their respective Modi-duration). This will facilitate to do impact analysis (stress testing) on a portfolio by assuming a standard shift in IRR and applying the same on portfolio Modi-duration (IRR based). However we will not have such liberty of doing stress testing on a portfolio by assuming a standard shift in YTM, because x% change in YTM for 2-periods-per-annum bond is altogether different from x% change in YTM for 4-periods-per-annum bond. Further if portfolio consist of any zero coupon bond, it would make it more complex to do YTM based stress testing and it is virtually impossible to calculate YTM for zero coupon bonds (unless help of some assumption is taken)

Limited usage of YTM:

As I mentioned in our previous trail of discussion, it is not possible to calculate YTM for a bond having uneven coupon frequency. Further YTM curve is virtually useless to calculate convexity. We have to convert YTM into IRR to calculate convexity of a bond and such IRR-based convexity would not help for analyzing impact on account of shift in YTM curve.

Although convexity has some resemblance with the term complexity, it is surprisingly very easy to calculate convexity of a portfolio (the same can be calculated by taking value of a bond as a weight factor for its respective convexity…provided we are considering IRR and not YTM…seems life is much more easy with IRR then YTM)

Some surprising result of YTM

Let us consider following two bonds

Bond – 1 Bond – 2

Value $100 $100

Maturity Value $120 $120

Period 3 Years 3 Years

Annual Coupon Frequency 4 2

Coupon per period $0.01 $0.01

IRR of bond-1 is 6.304425% while bond-2 is 6.28499%. The result is obvious (as IRR of bond-1 is greater than IRR of bond-2). However let us look at YTM. YTM of bond-1 is 6.160633% and bond-2 is 6.189226%. Here YTM of Bond-1 is less than YTM of bond-2 Inspite of bond-q is giving all the cash flows of bond two plus some more additional cash flows. The reason for the said result is that YTM is giving too much importance to frequency of coupon (irrespective of how insignificance value of coupon is)

My last defense (for IRR):

Are we justifying our self by doing an impact analysis (stress testing) on a portfolio (or a balance sheet) by applying standard shift in YTM for YTM-based Modi-duration of the portfolio? (That is what currently many banks and financial institution are doing). As we know, it is incorrect to assume a standard shift in YTM curve for the entire portfolio (or a balance sheet). By doing this we are invariably accepting the presence of arbitrage opportunity, which is unrealistic in today matured markets and hence it is incorrect to assume standard shift in YTM for the entire portfolio (or balance sheet). However we can assume a standard shift in IRR for the entire portfolio (and even for balance sheet) and can do impact analysis (stress testing) by applying IRR-based portfolio Modi-duration and convexity on a shift in IRR.

Conclusion: YTM based Modi-duration is useful only to do impact analysis for standard plain vanilla bond and that too at individual bond level (and not at portfolio level). IRR based Modi-duration and convexity can be used to do impact analysis of non-conventional bonds, zero coupon bonds, portfolio having combination of such bonds and even for entire balance sheet.

Inspite of IRR-based Modi-duration is adaptable under all situations, it is very surprising that the same is ignored unanimously by authors of all great books. Even Microsoft Excel formula on Modified duration has no provision for calculating Modi-duration based on IRR. (I will get back to you on limitations of Microsoft Excel M-duration formula very soon).

]]>You have given me lots of food for thought. I will carefully go through your response over the week-end and will revert back on Monday.

I am really impressed with your promptness in responding to queries.

Thank You

Triyog

According to triyogpandya’s calculations above (edited to refer to IRR, not YTM):

Therefore

At 12% IRR the bond value is $89.5704 on 1st Jan 2007.

Macaulay duration is 1.885151 years

IRR-Modified-Duration is 1.885151 / 1.12 = 1.68317If market expectation changes and IRR shifts to 11.75%, the bond pricing will increase to $89.9486 on 1st Jan 2007 (calculated taking present value of coupon and maturity value at IRR 11.75%)

By repeating the calculation of my last post, we find that an IRR = 11.75% implies Bond-Equivalent-Yield (BEY) of 11.42%

A change of 11.66% – 11.42% = 24bp in BEY has resulted in a price change of $89.9486 to $89.5704; the price change is 42.2bp.

This implies the modified duration is 42.2 / 24 = 1.758333. [My significant figures are rather erratic, aren’t they?]

We know the Macaulay duration : 1.885151 as pre triyogpandya. This, being the PV-weighted average time to cash receipt, shouldn’t change. At least, not much! I’ll have to think about that a little more!

Therefore, the multiplier to convert Macaulay to Modified(YTM) is 1.758333 / 1.885151 = 0.932728

BUT:

The figure (1 / (1+ BEY)) = (1 / (1 + 0.1175)) = 1 / 1.1175 = 0.894855

WHILE:

The figure (1 / (1+BEY/2) = (1 / (1 + 0.1175/2)) = 1/1.05875 = 0.944510

is much closer. I believe that if the calculations were to be re-run with a smaller change in yield to eliminate the effects of convexity (readers will recall I changed yield by 1bp in my initial response above), the necessity for division of BEY by 2 would be much more clear.

]]>Say, for example, triyogpandya’s rather unusual bond was trading in the Canadian universe. As he has claimed and I have verified, the internal rate of return of this bond is 12.00%.

[This is an inconveniently high number for current purposes. To resolve this difficulty, assume that 2-year corporates are trading to yield 10% to 14%]

OK, so say we have a NORMAL corporate bond, which is quoted to yield 11.75% according to the convention. We cannot say it is trading 25bp through triyogpandya’s bond, because our bond – like every other bond in the Canadian universe – has its yield calculated according to the Canadian convention, while triyogpandya’s bond has been quoted with IRR.

No problem! We can convert!

Recall:

Internal Rate of Return: IRR = (1+R)^2 – 1

Yield to Maturity: YTM = 2R

Given IRR = 0.12, we continue

(1+R)^2 – 1 = 0.12

(1+R)^2 = 1.12

(1+R) = 1.0583

R = 0.0583

YTM = 11.66%

triyogpandya’s bond does not yield 25bp more than our alternative; it yields 9bp less, WHEN YOU COMPARE APPLES TO APPLES (as best as we can). This is, shall we say, important information!

We should still be careful! When talking about triyogpandya’s bond, we should be precise and talk about an 11.66% bond-equivalent-yield, without getting sloppy and calling it a 11.66% YTM. This will remind us that triyogpandya’s bond is rather special, has unusual cash-flows and will have a somewhat different duration & modified duration from a conventional bond of the same tenor.

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