### Introduction

In a prior blog post, I wrote about a simple formula that can be utilized to understand the impact that rising rates may have upon a constant maturity fixed-income index. Emanuel Derman wrote a similar post about constant duration indices. While most people are aware of duration and how to use it to estimate price changes in a bond for given rate changes, ETF investors commonly find themselves in constant maturity portfolios, which are slightly different.

The key difference is that a constant maturity portfolio is going to roll the portfolio over on a frequent basis, selling older bonds and buying newer ones to keep a constant maturity profile. Rolling over in a rising rate environment will force the portfolio to realize losses – but it theoretically also moves the portfolio into bonds offering a higher yield, potentially a lower duration, and a higher expected return. So depending on what duration is and how quickly rates rise, it may be advantageous to roll the portfolio.

So if we expect rates to rise in the near future, are we better off just buying and holding or are we better off holding a constant maturity index and trying to stop, drop our bonds, and roll our way out of trouble?

### Assumptions & Methodology

To perform this test, we assume that rates will start with where they were on 2/26/2015 and will evolve over a 2-year cycle to be equivalent to where they once were on 12/31/2004. This is a more-or-less linear shift in the curve over a 2-year period back to a more “normal” environment. Whether this is truly how rates will unfold in the future, I have no idea. I chose a two year period because the last 3 real rate hikes happened over a 2-year period (see our weekly commentary here).

To construct our constant maturity index, we use a methodology proposed by Aswath Damodaran. Specifically, we assume that we initially purchase our bond at par. Each coupon we receive is then re-invested on the date it was received back into the bond (so, we assume fractional shares and infinite liquidity) based on the rate. Finally, on the roll-over date, we sell all of our shares and roll over into a new bond, assumed to be selling at par. We assume zero transaction costs and taxes.

We assume coupons are received every 6 months (starting the date the bond was purchased) and the portfoio is rolled over on a monthly basis.

Of course, *how *the rates go from point A to point B is critical, as it will impact the price at which coupons are re-invested and the dynamics of the bonds we roll into. To account for this, for each point on the yield curve, we will create 1000 sample evolutions using a process called a Brownian Bridge. This methodology creates a stochastic process that is guaranteed to start at point A and end at point B, but is more or less a random walk in between. Volatility from these random walks is estimated based on rate changes from 12/31/2004 to 12/31/2007, a “normal” rate environment. 100 sample evolutions of the 10-year rate are plotted below. We can see they all start and end at the same point, but take very different paths to get there.

For each of the random 1000 sample evolutions, we build a buy & hold index and a constant maturity index. This way, we can compare how the two strategies do when run over the same rate evolution, but also evaluate how each strategy performs over different rate evolutions.

### Initial Results

So how does this process unfold for simply buy & hold strategy? We plot the results below.We can see that while the mark-to-market prices drift between the start date and the point at which the bond matures, we can see that the prices all converge at maturity. However, since at maturity we purchase a new bond for the second year, *how *the rate has evolved at this point makes a significant difference. We can see that if rates are high at the half-way point, we can see that returns go up – if rates were low, then the second period returns are dampened.

The results of a constant maturity index, however are quite different.

The best performing index here returned 2.87% while the worst performing returned -1.33%. The difference between the rate evolutions that created this performance difference? At first blush, it looks pretty minor.

The subtle difference here appears to be in the fact that at several roll-over points, one rate index spiked while one sank. Spiking caused higher realized losses. So how the rate evolved was key in determining the ultimate performance of the constant maturity index.

So how often did the constant maturity index beat the buy & hold portfolio for the same rate evolution? 0% of the time. That’s right: not once did rolling result in better performance for the 1-year rate.

So how does this look as we walk out further on the yield curve? Let’s look at the 5-year.

While the mark-to-market performance over the 2-year period dramatically varies based on each unique rate evolution, as all the rates converge to the same final point, so do the prices. This makes sense: we started with the same bond and ended at the same rate so each portfolio should have the same value. The minor variations are due to the coupons we collected and re-invested at different prices.

Interestingly, the results for the constant maturity index look very similar, with slightly greater variance in results (look at the y-axis scale). The similarity in end results is due to the fact that rates end up in the same place. So while there is greater variance at the end because we’ve not only re-invested coupons but also rolled our bonds every month, ultimately the portfolio returns are dwarfed by where rates end up at the end of the period.

Despite the visual similarity, the constant maturity portfolios beat their respective buy & hold counter-parts only 1.3% of the time.

In fact, the story remains much the same across the board.

Rate | % of Buy & Hold that beat CM |
---|---|

1-Year | 0.00% |

2-Year | 0.20% |

3-Year | 0.30% |

5-Year | 1.30% |

7-Year | 3.00% |

10-Year | 8.40% |

20-Year | 77.20% |

But … there’s the 20-year. A glaring exception. What is going on here?

### Hold or Fold?

Let’s consider a simplified example: we’ll buy a 5-year bond and every 6 months make a decision about whether we want to continue to hold, re-investing our coupon, or sell our bond and re-invest the proceeds and the coupon into a new on-the-run bond.

So we buy a 5-year bond yielding 1.54% for $100. Six months later we receive our coupon and have a choice: based on what we believe will happen in the next six months, do we hold (i.e. re-invest our coupon in the now off-the-run bond), or roll (i.e. sell our off-the-run bond and use the proceeds, plus our coupon, to buy the new on-the-run bond).

Mathematically, the dollar return from holding over the next period is roughly equivalent to:

$latex (1 + \frac{c_0}{B_{0,1}})(-\mathbb{E}[\Delta r]D_{0,1}B_{0,1} + c_0)$

To translate math to English, we’re calculating the new number of shares we have of the first bond and multiplying it by the dollar change of the bond over the next period and the coupon we’ll collect over the next period. The choice to roll would give us dollar returns roughly equivalent to:

$latex (\frac{B_{0,1} + c_0}{B_{1,0}})(-\mathbb{E}[\Delta r]D_{1,0}B_{1,0} + c_1)$

The big difference here is that we’re getting shares of the new on-the-run bond, and therefore we’ll also collect the coupon of the new bond and use the duration of the new bond.

So we really have a few factors to consider in our decision. (1) If we sell now, how many shares can we buy of the new bond? (2) What do we expect rates to do? (3) What is the duration profile of both bonds? (4) What is the coupon difference between the bonds? These factors will all balance each other. In rolling, we may end up in a situation where duration is higher, but we realize such a loss that the number of shares we can purchase is now enough to offset our exposure.

### A Simplified Two Period Case

To simplify things, we’ll consider just a 2-period case. At T=0, we buy our 20-year bond, yielding 2.39%, for par at $100. At T=1, rates are now at 3.62%. Our bond is now worth $83.21, we’ve collected $2.39 in interest, and our duration is 14.69. If we choose to re-invest, we’ll have 1.03 shares.

A new bond is now on the market – a 20-year yielding 3.62%. It’s duration is 14.14. If we sell our first bond and use our collected interest, we can buy 0.86 shares.

We expect rates to climb by 1.23% over the next period. What should we do? We’ll, let’s calculate our expected dollar return.

In the first case, we get 1.03 * (-13.70 * 0.0123 * $83.85 + $2.39) = -13.02. In the second case, we get 0.86 * (-14.14 * 0.0123 * $100 + $3.62) = -11.84. Interestingly, rolling still leaves us with greater dollar exposure to an interest rate change (-$0.1259 vs -$0.1216 per basis point increase) – but in this case, the coupon is large enough to make up for it ($3.11 vs $2.46). So in this simplified 2-period process, we expect to lose less if we roll.

If we don’t know $latex \mathbb{E}[\Delta r]$, we can do some re-arranging to calculate a break-even point. It is better to hold if:

$latex \mathbb{E}[\Delta r]>\frac{s_1*c_1 – s_0*c_0}{s_1*D_{1,0}*B_{1,0} – s_0*D_{0,1}*B_{0,1}}$

Where $latex s_0$ and $latex s_1$ are the shares held of each bond. So it all comes down the excess coupon collected relative to the duration exposure difference. So, as before, we have ($3.11 – $2.46) / ($1216.04 – $1259.03) = -1.52%. Rolling and holding will lead to the same return if rates *fall* by 1.52%. Since we thought rates were going to climb, we were better off rolling.

Of course, the difference is quite minor, and we see that play out in the relative performance results of a constant maturity 20-year index versus the buy & hold counterpart.

### Conclusion

Much of this analysis was predicated on the fact that we were going to have about a 200bp level change in the yield curve that we expected to occur over a 2-year period.

The choice to roll will predominately be governed by the tradeoff between our exposure (i.e. shares we hold), the duration of our bonds, our expectation for rate changes, and the coupons we’ll collect.

In shorter maturity bonds, rolling increased dollar duration exposure faster than the coupon benefit received. Since we expected rates to rise, it was better to simply hold.

In 20-year bonds, however, the losses were so great, and the duration numbers so similar between rolling and not rolling, it was actually advantageous to roll because we were reducing our exposure.

So if your view is that rates are going to rise several hundred basis points in the next couple of years, you may be better off just holding your bonds, as with each coupon that peels off you’ll be reducing your duration to the next set of rate increases faster than realized losses can decrease exposure or increased coupons can make up for losses.

It may make sense to roll if you are holding long-dated bonds – then again, at the 20-year level, either way you go, you’re probably going to be holding a large paper loss at the end of the day.

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