# Archives

## Yield Curve Strategies – Dynamic Yield Curve

Making Money with Bonds

In the article on yield curve strategies in general, I mentioned the two broad ways to make money with bonds:

1. Coupons (more generally, to incorporate synthetic strategies using, for example, swaps: interest payments)
2. Price changes

The curriculum breaks down the expected return on a bond in this manner:

\begin{align}E\left(R\right) &≈ Coupon\ income\\
\\
&\pm Rolldown\ return\\
\\
&\pm E\left(∆Price\ from\ investor’s\ view\ of\ benchmark\ yields\right)\\
\\
&\pm E\left(∆Price\ from\ investor’s\ view\ of\ yield\ spreads\right)\\
\\
&\pm E\left(∆Price\ from\ investor’s\ view\ of\ currency\ value\ changes\right)
\end{align}

In this article, we’ll largely ignore this breakdown; we’ll look at the price change on the bonds, not the yield income.  That’s the way of the curriculum.

What Is a Dynamic Yield Curve?

In general, a dynamic yield curve is one that we expect to change during our intended holding period.  Although the manner in which a yield curve change has myriad variations, we’ll break them down into a few simple building blocks.  Most changes will be (roughly) a combination of one of more of these building blocks, so the appropriate strategy will be (roughly) that same combination of the strategies for each of the separate blocks.  (At the end of the article, I’ll discuss what we do when we expect a wild change in the yield curve: one that cannot easily be constructed out of the blocks.)  The building blocks we will consider are a(n):

• Upward parallel shift (increase in level)
• Downward parallel shift (decrease in level)
• Steepening (increase in slope)
• Flattening (decrease in slope)
• Increase in curvature
• Decrease in curvature

Our Starting Yield Curve

Suppose that today’s yield curve looks like this:

 Maturity, Years YTM Maturity, Years YTM 1 2.220% 16 4.999% 2 2.521% 17 5.093% 3 2.799% 18 5.181% 4 3.057% 19 5.262% 5 3.296% 20 5.337% 6 3.518% 21 5.407% 7 3.722% 22 5.472% 8 3.912% 23 5.531% 9 4.087% 24 5.587% 10 4.249% 25 5.637% 11 4.399% 26 5.683% 12 4.539% 27 5.726% 13 4.667% 28 5.766% 14 4.787% 29 5.804% 15 4.897% 30 5.841%

Graphically:

Available Bonds

For simplicity, we’ll assume that we have available par bonds at all maturities from 1 year to 30 years.  As we’ll be talking quite a bit about duration and convexity, let’s take a look at the modified duration and convexity for each of these bonds:

 Maturity, Years Duration, Years Convexity, Years2 Maturity, Years Duration, Years Convexity, Years2 1 0.98 1.91 16 10.84 154.28 2 1.93 5.61 17 11.19 166.63 3 2.84 10.95 18 11.52 178.78 4 3.71 17.74 19 11.83 190.72 5 4.54 25.81 20 12.11 202.39 6 5.33 34.98 21 12.37 213.80 7 6.06 45.06 22 12.61 224.91 8 6.76 55.91 23 12.84 235.72 9 7.41 67.36 24 13.04 246.23 10 8.01 79.27 25 13.23 256.41 11 8.57 91.51 26 13.41 266.28 12 9.10 103.98 27 13.58 275.83 13 9.58 116.57 28 13.73 285.07 14 10.03 129.19 29 13.87 294.00 15 10.45 141.78 30 14.00 302.61

Bullet, Ladder, and Barbell Portfolios

Throughout this article we’ll be talking about three portfolios in particular, each with an initial value of $10 million: a bullet portfolio (all of bonds in the portfolio having maturities very close to each other), a ladder portfolio (bonds at several maturities spread along the yield curve), and a barbell portfolio (a concentration of short-maturity bonds plus a concentration of long-maturity bonds, with no bonds with maturities in between those extremes). The benchmark (modified) duration is 10.5 years, and each starting portfolio will have that same duration. The specific portfolios are:  Weights Maturity, Years Bullet Ladder Barbell 1 5.90% 26.88% 5 9.60% 10 13.21% 14 25.84% 15 33.52% 15.75% 16 40.65% 20 17.47% 25 18.64% 30 19.44% 73.12% Total 100.00% 100.00% 100.00% Duration, Years 10.50 10.50 10.50 Convexity, Years2 143.61 177.36 221.78 (For the bullet and ladder portfolios, I wanted to keep the weights as close to equal as possible; the criterion I used was to minimize the standard deviation of the weights, while keeping the duration equal to that of the benchmark. With different criteria, other sets of weights are possible.) Changes in Level A change in the level of the yield curve is a nothing more than a parallel shift, upward or downward: Although I have never seen nor heard the adjectives “bull” and “bear” applied to changes in level (they are used with changes in slope), there is no reason that they couldn’t be: a bull shift would be a downward parallel shift (so called because it will result in bond prices increasing), and a bear shift would be an upward parallel shift (accompanied by bond prices decreasing). Thinking in this manner will help you when we get to the slope changes, and I think that there’s an advantage to using similar language for similar situations; it strengthens the memory. Adjust Duration When we anticipate an upward parallel shift (increase in level) in the yield curve, the time-honored strategy is to decrease the duration of our portfolio. There are a number of ways that the duration of our portfolio can be decreased, including: • Replace some or all of our bonds with bonds having shorter maturities • Purchase put options on bonds • Sell call options on bonds • Enter into the short position in a bond futures or forward contract • Enter into a plain vanilla interest rate swap as the fixed-rate payer / floating-rate receiver When we anticipate a downward parallel shift (decrease in level), the indicated strategy is to increase the portfolio’s duration. Of course, there are an equal number of ways that the duration of our portfolio can be increased, including: • Replace some or all of our bonds with bonds having longer maturities • Purchase call options on bonds • Sell put options on bonds • Enter into the long position in a bond futures or forward contract • Enter into a plain vanilla interest rate swap as the fixed-rate receiver / floating-rate payer All of these approaches are straightforward, and are described in detail in the article on yield curve strategies in general (see the link at the beginning of this article), so I won’t go into them again here. Increase Convexity Suppose, however, that the duration on the portfolio is constrained; e.g., the modified duration of the portfolio must be within 0.25 years of the modified duration of the benchmark. Lengthening or shortening the duration by 3 months won’t make much of a difference when there is a parallel shift, so we need to look beyond duration. Generally, the convexity of the portfolio is not constrained (the people who create investment policy statements (IPSs) generally aren’t particularly sophisticated, truth be told), so when we expect a parallel shift in the yield curve, we can benefit from an increase in the portfolio’s convexity: higher convexity gives a greater price increase when yields fall, and a smaller price decrease when yields rise. One way to increase the convexity of our portfolio is to increase the dispersion of the cash flows. Note that amongst the bullet, ladder, and barbell portfolios, the bullet has the lowest convexity, the ladder has middling convexity, and the barbell has the highest convexity. Let’s take a look at the values of each of the bonds in our existing portfolios under the existing yield curve, a 50 bps upward shift, and a 75 bps downward shift:  Bond Prices Maturity, Years Existing + 50 bps − 75 bps 1$1,000.00 $995.13$1,007.39 5 $1,000.00$977.61 $1,034.80 10$1,000.00 $960.92$1,062.38 14 $1,000.00$951.40 $1,079.03 15$1,000.00 $949.47$1,082.53 16 $1,000.00$947.69 $1,085.80 20$1,000.00 $941.89$1,096.81 25 $1,000.00$936.92 $1,106.89 30$1,000.00 $933.63$1,114.09

(You can verify these prices with your calculator, or you can take my word for it.  It’s probably not a bad idea to verify one or two just to make sure that you’re happy with the table.)

The values of the bullet, ladder, and barbell portfolios under the existing yield curve, a 50 bps upward shift, and a 75 bps downward shift are:

 Portfolio Existing + 50 bps − 75 bps Bullet $10,000,000$9,492,464 $10,829,549 Ladder$10,000,000 $9,496,342$10,840,237 Barbell $10,000,000$9,501,611 $10,854,041 The duration approximation for the 50 bps upward shift is: $∆Price ≈ -\10,000,000 × 10.5 × 0.5\% = -\525,000$ leaving a portfolio value of: $\10,000,000 – \525,000 = \9,475,000$ Because of the convexity, all of the portfolios under the 50 bps upward shift have a higher value than that, with the bullet (least convex) having the lowest value and the barbell (most convex) the highest, as expected. The duration approximation for the 75 bps downward shift is: $∆Price ≈ -\10,000,000 × 10.5 × -0.75\% = \787,500$ giving a portfolio value of: $\10,000,000 + \787,500 = \10,787,500$ Because of the convexity, all of the portfolios under the 75 bps downward shift have a higher value than that, with the bullet having the lowest value and the barbell the highest; again, as expected. What does this mean? If we have a bullet portfolio and anticipate a parallel shift, one way we can increase our convexity and improve the performance is to switch to a ladder or barbell portfolio with the same duration, and if we have a ladder portfolio we can switch to a barbell portfolio. One downside to this approach is that buying and selling bonds can incur significant transaction costs, as well as the possibility of having to recognize taxable gains, so it may not be the most efficient method to increase the convexity of our portfolio. Another downside is that convexity is not free: it’s akin to buying insurance and, as with all other forms of insurance, it costs money. In other words, the cost of the ladder portfolio should be higher than the cost of the bullet portfolio, and the cost of the barbell portfolio should be higher than the cost of the ladder portfolio. A third downside is that the amount of convexity we can achieve is limited: the barbell portfolio will have the maximum possible convexity using only straight bonds. If we want to increase convexity further, we have to look to other methods. Putable Bonds One possibility is to replace some or all of the straight bonds in our portfolio with putable bonds, which generally have greater convexity than otherwise comparable straight bonds. Replacing straight bonds with putable bonds has the same disadvantages as changing from a bullet structure to a ladder or barbell structure, or from a ladder to a barbell: buying and selling bonds is costly, taxable gains might be incurred, buying convexity can be expensive, and there’s a limit to the additional convexity in putable bonds. Another possible disadvantage is that putable bonds are generally less liquid than straight bonds, so you might not be able to buy them even if you really, really want to. Options on Bonds We saw in the article on static yield curve strategies that we can sell convexity whilst maintaining the duration of our portfolio by selling an appropriate mix of call options on bonds and put options on bonds. Call options have positive duration and positive convexity, put options have negative duration, and out-of-the-money put options have positive convexity. Therefore, if we want to buy convexity as protection against parallel shifts in the yield curve, we can buy an appropriate mix of call options on bonds and put options on bonds. Buying options generally has lower transaction costs than buying and selling bonds, buying options does not trigger the recognition of taxable gains, and the amount of convexity we can purchase is virtually unlimited, constrained only by our budget for insurance. An advantage of using options is that if we have some flexibility on the duration of our portfolio (say, within 0.25 years of the benchmark duration), we can choose the mix of calls and puts to add the convexity we want while adjusting the duration to the limit of that flexibility: increasing it when we expect a downward shift and decreasing it when we expect an upward shift. Changes in Slope For a normal (i.e., upward sloping) yield curve, the usual terminology for changes in slope make sense: what we call steepening makes the yield curve more steeply sloped upward (i.e., farther from horizontal, closer to vertical, long end up, short end down), and what we call flattening makes the yield curve less steeply sloped upward (i.e., closer to horizontal, farther from vertical, long end down, short end up, flatter). The usual language falls apart, however, when we have an inverted (i.e., downward sloping) yield curve: a steepening (long end up, short end down) actually brings the curve closer to horizontal, while a flattening (long end down, short end up) makes the curve less horizontal (more downward). Thus, we have the language of slope: what we call steepening is an increase in the slope, and what we call flattening is a decrease in the slope. Personally, I prefer to think of steepening as a rotation counterclockwise (or anticlockwise, or widdershins), and flattening as a rotation clockwise (or deasil). I encourage you to use whatever language works best for you. When we talk about a change in slope (separate from a change in level), we generally agree to leave one point on the yield curve fixed, and move the other points around it. (This is another reason I like the visual of a rotation.) Although we could choose any point (i.e., the yield at any given maturity) to remain fixed, tradition has come to leaving one of two points fixed: either leaving the short end of the curve fixed or else leaving the long end of the curve fixed. As we will see, the names we give to the specific types of steepening or flattening tell us which point is fixed. Hang on. In general, when there is a change in the slope of the yield curve, modified duration and convexity are of little use in telling us how the portfolio will behave (because those measures assume a parallel shift – a change in level only – not a change in slope); we need more sophistication in our analysis of our portfolio. More on that in a moment. First, let’s make sure that we understand changes in slope in general. Steepening When long-term yields rise, or short-term yields fall, or both, we say that the yield curve steepens: the slope of the yield curve increases. Graphically, it looks something like this: If long-term yields increase while short-term yields remain unchanged, we refer to the change as a bear steepening. If long-term yields remain unchanged while short-term yields decrease, we refer to the change as a bull steepening. Both are illustrated here: Note that the general steepening illustrated above can be thought of as a bull steepening combined with a positive (bear) parallel shift: or as a bear steepening combined with a negative (bull) parallel shift: Without a duration constraint on our portfolio, we would want to shorten its duration when a steepening is expected: sell long-term (hence, long-duration) bonds, buy short-term (hence, short-duration) bonds. If we have a duration constraint (so we’re pretty much restricted to the bullet, ladder, and barbell portfolios, above), we want primarily to limit our exposure to the 30-year bonds, so we would prefer the ladder to the barbell, and the bullet to the ladder. For the graphs, above, the yield changes run: • From 0 bps at 0 years to +100 bps at 30 years for the bear steepening • From −50 bps at 0 years to +50 bps at 30 years for the general steepening • From −100 bps at 0 years to 0 bps at 30 years for the bull steepening The bond prices are:  Bond Prices Maturity, Years Existing Bull Steepening General Steepening Bear Steepening 1$1,000.00 $1,009.55$1,004.59 $999.67 5$1,000.00 $1,038.76$1,015.28 $992.47 10$1,000.00 $1,055.21$1,013.46 $973.73 14$1,000.00 $1,055.40$1,003.35 $954.55 15$1,000.00 $1,054.08$1,000.00 $949.47 16$1,000.00 $1,052.30$996.40 $944.33 20$1,000.00 $1,041.52$980.09 $923.57 25$1,000.00 $1,022.42$957.28 $898.09 30$1,000.00 $1,000.00$933.63 $873.92 The values of the bullet, ladder, and barbell portfolios under the existing yield curve, and the three steepenings are:  Portfolio Existing Bull Steepening General Steepening Bear Steepening Bullet$10,000,000 $10,536,962$9,994,012 $9,486,940 Ladder$10,000,000 $10,315,241$9,791,734 $9,309,770 Barbell$10,000,000 $10,025,664$9,527,025 $9,077,271 These values verify what we expected: the bullet performs the best in each of the three scenarios, the ladder is in the middle, and the barbell performs the worst in each scenario. Synthetic Adjustments Another approach is to adjust your exposure to various maturities synthetically. Here, you want to increase your exposure at the short end of the yield curve and decrease your exposure at the long end. You can increase your exposure at the short end by: • Taking the long position in futures or forwards on short-term bonds • Buying call options on short-term bonds • Selling put options on short-term bonds You can decrease your exposure at the long end by: • Taking the short position in futures or forwards on long-term bonds • Buying put options on long-term bonds • Selling call options on long-term bonds If you have duration constraints, you’ll have to adjust the exposures so that the net result doesn’t violate those constraints. And note that while futures, forwards, and options on long-term bonds are fairly common, futures, forwards, and options on short-term bonds may be much less common, so these strategies may be difficult or impossible to implement in practice. Flattening When short-term yields rise, or long-term yields fall, or both, we say that the yield curve flattens: the slope of the yield curve decreases. Graphically, it looks something like this: If short-term yields increase while long-term yields remain unchanged, we refer to the change as a bear flattening. If short-term yields remain unchanged while long-term yields decrease, we refer to the change as a bull flattening. Both are illustrated here: Note that the general flattening illustrated above can be thought of as a bull flattening combined with a positive (bear) parallel shift: or as a bear flattening combined with a negative (bull) parallel shift: Without a duration constraint on our portfolio, we would want to lengthen its duration when a flattening is expected: sell short-term (hence, short-duration) bonds, buy long-term (hence, long-duration) bonds. If we have a duration constraint (so we’re pretty much restricted to the bullet, ladder, and barbell portfolios, above), we want primarily to maintain as much exposure to the 30-year bonds as possible, so we would prefer the barbell to the ladder, and the ladder to the bullet. For the graphs, above, the yield changes run: • From +100 bps at 0 years to 0 bps at 30 years for the bear flattening • From +50 bps at 0 years to −50 bps at 30 years for the general flattening • From 0 bps at 0 years to −100 bps at 30 years for the bull flattening The bond prices are:  Bond Prices Maturity, Years Existing Bull Flattening General Flattening Bear Flattening 1$1,000.00 $1,000.33$995.46 $990.63 5$1,000.00 $1,007.60$985.01 $963.04 10$1,000.00 $1,027.15$986.76 $948.31 14$1,000.00 $1,048.27$996.66 $948.27 15$1,000.00 $1,054.08$1,000.00 $949.47 16$1,000.00 $1,060.06$1,003.62 $951.06 20$1,000.00 $1,085.44$1,020.47 $960.73 25$1,000.00 $1,119.77$1,045.57 $978.29 30$1,000.00 $1,156.52$1,073.95 $1,000.00 The values of the bullet, ladder, and barbell portfolios under the existing yield curve, and the three flattenings are:  Portfolio Existing Bull Flattening General Flattening Bear Flattening Bullet$10,000,000 $10,550,067$10,006,094 $9,498,087 Ladder$10,000,000 $10,805,252$10,229,878 $9,702,084 Barbell$10,000,000 $11,145,325$10,528,484 $9,974,817 These values verify what we expected: the bullet performs the worst in each of the three scenarios, the ladder is in the middle, and the barbell performs the best in each scenario. Synthetic Adjustments As with steepening, when you anticipate the yield curve flattening you can make adjustments using derivatives. This time, you want to decrease your exposure at the short end of the yield curve and increase your exposure at the long end. You can decrease your exposure at the short end by: • Taking the short position in futures or forwards on short-term bonds • Buying put options on short-term bonds • Selling call options on short-term bonds You can increase your exposure at the long end by: • Taking the long position in futures or forwards on long-term bonds • Buying call options on long-term bonds • Selling put options on long-term bonds Once again, mind any duration constraints, and make sure that the net result doesn’t violate those constraints. And remember that derivatives on short-term bonds may be difficult to find. Changes in Curvature Changes in curvature of the yield curve look something like this: Note that the change in curvature generally assumes that the ends of the yield curve are fixed, so that all of the action, if you will, takes place in the middle of the yield curve. Using the bull/bear language, an increase in curvature would be a bear change, while a decrease in curvature wold be a bull change. When a change in curvature is combined with a smaller magnitude parallel shift in the opposite direction (i.e., an increase in curvature plus a downward parallel shift, or a decrease in curvature plus an upward parallel shift), the combination is called a butterfly. A positive butterfly looks like this: while a negative butterfly looks like this: (Note that the adjective – positive or negative – describes the parallel shift, or the direction in which the ends of the yield curve (the wings) move relative to the original yield curve.) (Note, too, that , broadly, butterflies are neither bull nor bear necessarily.) Increase in Curvature When we expect that the curvature of the yield curve will increase, we want to reduce the exposure to the middle of the curve: mid-term bonds (the body of the butterfly). Generally, this is accompanied by an increase in the exposure to the ends of the the yield curve (the wings), to keep the portfolio invested fully. Note that there is no particular reason to change the (overall) duration of the portfolio; we can maintain the same duration while adjusting the exposures at different maturities. Considering our three portfolios, the barbell should perform best (as it has no exposure to the middle of the yield curve), and the bullet should perform worst (as it has 100% exposure to the middle of the yield curve), while the ladder should perform somewhere in between. For the graphs, above, the yield changes run: • From 0 bps to +53.2 bps for the increased curvature • From −26.6 bps to +26.6 bps for the negative butterfly The bond prices are:  Bond Prices Maturity, Years Existing Increased Curvature Negative Butterfly 1$1,000.00 $999.05$1,001.66 5 $1,000.00$983.28 $995.20 10$1,000.00 $959.47$979.98 14 $1,000.00$949.05 $974.44 15$1,000.00 $948.00$974.41 16 $1,000.00$947.69 $975.06 20$1,000.00 $952.44$983.24 25 $1,000.00$970.60 $1,005.18 30$1,000.00 $1,000.00$1,038.34

The values of the bullet, ladder, and barbell portfolios under the existing yield curve, the increased curvature, and the negative butterfly shift are:

 Portfolio Existing Increased Curvature Negative Butterfly Bullet $10,000,000$9,481,453 $9,746,786 Ladder$10,000,000 $9,710,104$9,984,529 Barbell $10,000,000$9,997,452 $10,284,799 As expected: the bullet performs the worst in each of the three scenarios, the ladder is in the middle, and the barbell performs the best in each scenario. Note, too, that because the bullet and barbell portfolios each have a modified duration of 10.50 years, you can create a custom portfolio that is short the bullet and long the barbell, which would enhance the performance even more. For example, a portfolio that is short$2 million of the bullet and long $12 million of the barbell will still have a modified duration of 10.50 years, an initial value of$10 million, and a final value after the increase in curvature of:

$-0.2 × \9,481,453 + 1.2 × \9,997,452 = \10,100,651$

and a final value after the negative butterfly of:

$-0.2 × \9,746,786 + 1.2 × \10,284,799 = \10,392,402$

Decrease in Curvature

When we expect that the curvature of the yield curve will decrease, we want to increase the exposure to the middle of the curve: mid-term bonds (the body of the butterfly).  Generally, this is accompanied by a decrease in the exposure to the ends of the the yield curve (the wings), to keep the portfolio invested fully.  Once again, there is no particular reason to change the (overall) duration of the portfolio; we can maintain the same duration while adjusting the exposures at different maturities.  Considering our three portfolios, the barbell should perform worst (as it has no exposure to the middle of the yield curve), and the bullet should perform best (as it has 100% exposure to the middle of the yield curve), while the ladder should perform somewhere in between.

For the graphs, above, the yield changes run:

• From 0 bps to −53.2 bps for the decreased curvature
• From +26.6 bps to −26.6 bps for the positive butterfly

The bond prices are:

 Bond Prices Maturity, Years Existing Decreased Curvature Positive Butterfly 1 $1,000.00$1,000.95 $998.35 5$1,000.00 $1,017.07$1,004.83 10 $1,000.00$1,042.67 $1,020.52 14$1,000.00 $1,054.51$1,026.43 15 $1,000.00$1,055.65 $1,026.36 16$1,000.00 $1,056.06$1,025.68 20 $1,000.00$1,050.89 $1,017.16 25$1,000.00 $1,030.72$994.86 30 $1,000.00$1,000.00 $963.80 The values of the bullet, ladder, and barbell portfolios under the existing yield curve, the increased curvature, and the negative butterfly shift are:  Portfolio Existing Decreased Curvature Positive Butterfly Bullet$10,000,000 $10,555,203$10,261,017 Ladder $10,000,000$10,307,107 $10,022,318 Barbell$10,000,000 $10,002,553$9,730,882

As expected: the bullet performs the best in each of the three scenarios, the ladder is in the middle, and the barbell performs the worst in each scenario.

We can create a custom portfolio here as well: a portfolio that is long $12 million of the bullet and short$2 million of the barbell will still have a modified duration of 10.50 years, an initial value of \$10 million, and a final value after the decrease in curvature of:

$1.2 × \10,555,203 – 0.2 × \10,002,553 = \10,665,733$

and a final value after the negative butterfly of:

$1.2 × \10,261,017 – 0.2 × \9,730,882 = \10,367,044$

Summary

How do our standard portfolios – bullet, ladder, barbell – perform under various dynamic yield curve scenarios?

 Bullet Ladder Barbell Upward Shift Worst Middle Best Downward Shift Worst Middle Best Steepening Best Middle Worst Flattening Worst Middle Best Increased Curvature Worst Middle Best Decreased Curvature Best Middle Worst

What does this tell us?

If we know that there’s going to be a parallel shift in the yield curve, or a flattening of the yield curve, or an increase in the curvature, choose a barbell portfolio.  If we know that there’s going to be a steepening of the yield curve, or a decrease in the curvature, choose a bullet portfolio.  If we have no idea what’s going to happen to the yield curve, choose a ladder portfolio: never the best, but never the worst.