It seems that all anyone can talk about now is the potential of a Fed taper and rising interest rates. “Duration” is the word on everybody’s lips — and perhaps for good reason: it’s a pretty easy method to start exploring the potential damage caused on our portfolio by a hike in rates.

But duration is highly simplified. Firstly, it is a first-order estimate — eventually, convexity becomes important as well. Also important is how the fixed-income portfolio is structured. In other words, are we holding our bonds to maturity — and therefore our loss is never realized — or are we holding a constant maturity portfolio and selling at a loss to purchase the current liquid run of instruments?

The latter consideration is just as important as duration for investors holding any of the popular fixed-income ETFs, because the majority are pegged to a target maturity.

To explore the effects of rising rate environments on these instruments, we created code that would purchase a bond of a given maturity, collect the coupons monthly and re-invest them in bonds of the same run, and then sell the bonds at the end of the year to purchase the new on-the-run series.

Obviously, this is a crude approximation of how the ETFs work. The ETFs hold a variety of different maturities and yields at a given time and may hold onto bonds for longer than a year. In our model, we are treating the ETF *as if *it were a bond, using historical dividend yield (21-day EWMA of the sum of dividends occurring in the last 252 days) as a proxy for the current bond yield-to-maturity and turning over our entire portfolio at the end of every year.

But I think the approximation is close enough to yield some interesting results. On top of generating the actual portfolio (to check our calibration), in the images below I generate a portfolio from *reversed* yields: i.e. what would our portfolio have done if the previous decades yields for the security had occurred in reverse order. For most securities, this means rising rates.

It looks like the model is a fairly decent approximation for most of the portfolios — except for TIP, which pays some pretty wacky dividends. But all in all, we learn some interesting things when our yields occur in reverse order. IEF and TLT take a good knock on the nose, initially, as rates rise off of long-term lows — but the biggest drawdown is 25% — and both recover by the end of the period. Well … IEF at least. Perhaps the reverse-generated SHY has some of the most interesting returns: it seems that coupon rates become high enough in the latter period to swamp out losses from rising rates.

Which raises an interesting question: how do the dynamics of re-invested coupons, current rates, and increasing rates play out in a fixed-maturity portfolio? The below images explore this relationship. Each graph has a starting yield level and increases yields by X basis-points a year (equally spaced monthly, which explains the saw-tooth time-series). I assume the maturity is equal to that of TLT (approximately 27).

The results show an interesting dynamic: the percentage of rate increase relative to yield eventually creates an inflection-point whereby the increase has made the yield so high, the coupon size makes up for the reduction in bond valuation.

But what is this relationship?

Let’s consider the math of a more simplified problem. If you don’t like math, stick with us — there is an actionable take-away at the end.

Let’s say we purchase a bond today at par, wait a year and sell it, and collect the interim coupons without re-investing. Our profit or loss is a function of how rates have changed over that period. Specifically, the price change of the bond we are selling can be estimated as:

$latex \Delta P\approx f'(r)\Delta r+\frac{1}{2}f”(r)(\Delta r)^{2} $

Where f(r) gives the bond’s value for rate level r. By definition, -f'(r)/P is our duration, D, and f”(r)/P is convexity, C. So we can re-write our change in price as:

$latex \frac{\Delta P}{P}\approx-D\Delta r+\frac{1}{2}C(\Delta r)^{2} $

For small changes in r, the second part of the term goes to zero, so we can further reduce our equation to get a rough price change estimate as simply:

$latex \frac{\Delta P}{P}\approx-D\Delta r$

So far, this is nothing new — we’ve only just shown why people are so duration sensitive in the face of rising rates: percent price change is a linear function of change in rates. The higher our duration, the more the value of the bond decreases during a rate hike.

But what does this tell us about our over-all profit and loss within a *constant maturity portfolio*? Consider that in the portfolio, every year we will buy a bond of a certain maturity, hold it for a year, collecting any coupons, and sell it at the end of the year. For the trade to be profitable, we want the coupon we receive plus the amount we sell the bond for to be greater than what we bought the bond for in the first place.

Assuming par is $1, we can write this condition as:

$latex \$1*r_{0}+\$1*\Delta P>0$

If we plug in our definition for change of price from above, we see a relationship between the change in rates and duration emerge that gives us a positive return:

$latex \frac{r_{0}}{r_{1}-r_{0}}>D$

Re-arranging, we get:

$latex (\frac{r_{1}}{r_{0}}-1)<\frac{1}{D} $

In other words, the percent change in rates has to be less than 1/D for the coupon to make up for the loss in bond value due to the change in rates. So if a bond has a duration of 2, a relative increase in rates of less than 50% will still allow us to exit our position with a profit at the end of the year.

In a highly simplified, constant-maturity fixed-income portfolio, if the increase of rates per year, as a percentage of current yield, is less than 1/Duration, the portfolio can still be profitable.

Again — highly simplified because we are not re-investing coupons — but perhaps still an actionable, simple rule. Consider TLT, which currently has a yield of approximately 3.59%. With a duration of 16.30, we would need rates to rise at less than 22bp (3.59% / 16.30) a year to remain profitable. Again, while our rule is for a very simplified scenario, it holds fairly well in the simulation from above, which re-invests the coupons and has rates increasing monthly:

It can be seen that the environment where rates rise at 30 bp / Year has its inflection point shortly after the 1/16.30 (~6.10%) level.

Now consider HYG: ignoring credit-spread related issues, with a yield of 5.18% and a duration of 4.23, interest rates would need to rise nearly 122bp in a year before coupons could no longer cover up the losses.

So as we consider a future with rising interest rates, if we are using constant-maturity ETFs, we shouldn’t simply consider duration as the only factor in our decision, but yield relative to duration.

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