# Archives

## Valuing Risky Bonds

In this context, a risky bond is one for which there is a nonzero probability of default.  Conceptually, valuing such a bond is child’s play: you calculate the (present) value of the bond assuming no default (which the curriculum abbreviates VND: Value, No Default), then subtract the (present) value of the expected credit losses (i.e., […]

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## 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

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. ## Yield Curve Strategies – Static 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} We’ll use this breakdown later in this article. Earning a Return with a Static Yield Curve First, a couple of calculations, then some theory. Suppose that today’s yield curve looks like this:  Maturity, Years Par Rate Spot Rate Forward Rate 1 2.220% 2.220% 2.220% 2 2.521% 2.525% 2.831% 3 2.799% 2.810% 3.382% 4 3.057% 3.078% 3.887% 5 3.296% 3.331% 4.348% 6 3.518% 3.570% 4.774% 7 3.722% 3.794% 5.150% 8 3.912% 4.008% 5.516% 9 4.087% 4.209% 5.836% 10 4.249% 4.401% 6.139% (OK, technically it’s three yield curves.) You buy a 10-year bond that pays a 4% coupon annually. The yield on that bond is 4.249%, as the table above attests. You hold that bond for one year, then sell it. If the yield curve remains static for that year, then the yield on the bond when you sell it will be 4.087%: it will be a 9-year bond at that point. The price at which you buy the bond is: $FV = 1,000$ $PMT = 40$ $i = 4.249\%$ $n = 10$ $Solve\ for\ PV = -980.05$ The price at which you sell the bond is: $FV = 1,000$ $PMT = 40$ $i = 4.087\%$ $n = 9$ $Solve\ for\ PV = -993.56$ Your holding period return is: $HPR = \frac{993.56 – 980.05 + 40}{980.05} = 5.459\%$ Broken down according to our expected return formula, above: $Coupon\ return = \frac{40}{980.05} = 4.081\%$ $Rolldown\ return = \frac{993.56 – 980.05}{980.05} = 1.378\%$ $E\left(\%∆Price\ from\ investor’s\ view\ of\ benchmark\ yields\right) = 0\%$ $E\left(\%∆Price\ from\ investor’s\ view\ of\ yield\ spreads\right) = 0\%$ $E\left(\%∆Price\ from\ investor’s\ view\ of\ currency\ value\ changes\right) = 0\%$ $HPR = 4.081\% + 1.378\% + 0\% + 0\% + 0\% = 5.459\%$ Suppose instead that, noting that your holding period is one year, you buy a 1-year bond paying an annual coupon of 3%. The price when you buy the bond is 1,007.63 (you should verify this on your calculator), and 1,000 at maturity. Your holding period return is: $HPR = \frac{1,000.00 – 1,007.63 + 30}{1,007.63} = 2.220\%$ which is exactly the 1-year YTM. This illustrates one strategy that you can employ when you think that the yield curve will be static for your expected holding period; it’s called riding the yield curve. Static Yield Curve Strategies Amongst the strategies that you can employ when you think that the yield curve will remain static are: • Buy-and-hold • Rolling down (or riding) the yield curve • Carry trade • Long position in bond futures • Receive fixed, pay floating swap • Selling convexity Buy-and-Hold While this is listed as a static yield curve strategy, as far as I can tell it’s more a properly a whatever-happens-to-the-yield-curve (or a who-cares-what-happens-to-the-yield-curve?) strategy. You buy some bonds and hold them to maturity. What happens to the yield curve doesn’t matter (unless you’re Silicon Valley Bank, but that’s a topic for an entirely different article), because the bond will be worth par at the end no matter what. (OK: technically, what happens to the yield curve does matter a little if the bond pays coupons and you plan to reinvest those coupons. But that’s not likely to have a huge impact on your overall return. Furthermore, if that’s our concern, we’d prefer a buy-and-hold strategy when rates are rising, so that our reinvestment income increases.) The only reason I can imagine that this is included under the rubric of static yield curve strategies is for comparison with other strategies in that category. I believe that it’s unlikely that you’d see an exam question in which the correct answer to the question of which strategy to use with a static yield curve will be buy-and-hold. Bear it in mind, of course, but don’t dwell on it. Example Suppose that you have a 2-year holding period. You buy a 2-year par bond (with a coupon rate of 2.521%, paid annually) and hold it for 2 years. If the yield curve remains static, then in one year when you get a coupon payment of 25.21, you’ll reinvest it for one year at 2.220% (the prevailing 1-year rate). Your holding period return is: $HPR = \frac{1,000.00\ – 1,000.00 + 25.21\left(1.0222\right) + 25.21}{1,000.00} = 5.0980\%$ The effective annual return is: $EAR = \left(1 + HPR\right)^{1/n}\ – 1 = 1.050980^{1/2}\ – 1 = 2.5173\%$ Note that your effective annual return is less than your original YTM of 2.521%. Why? Because the reinvestment rate on the coupon you received after 1 year was less than your original YTM. Rolling Down (Riding) the Yield Curve When you expect the yield curve to remain static, then the duration of your portfolio matters very little (as the yield change will be only what exists in the current yield curve, which will likely be small, so the price change from the pull-to-par (shortening the maturity) will likely outweigh the price change from duration). To take advantage of that fact, if the yield curve is normal (i.e., upward sloping), you can purchase bonds whose maturity is longer than your expected holding period, with the intention of selling those bonds at the end of the holding period. The upward sloping yield curve ensures that you will be selling the bonds at a lower yield than the yield at which you buy them, and the longer maturity on the bonds will likely be accompanied by a higher coupon rate, both of which should enhance your return. This strategy requires that you have some scope to increase the duration of your portfolio, possibly significantly, so it may not be practical in a portfolio that has tight restrictions on duration (e.g., that the portfolio duration may not differ from the benchmark duration by more than, say, 0.25 years). Example Suppose that you have a 2-year holding period. You buy a 10-year par bond (with a coupon rate of 4.249%) and hold it for 2 years. If the yield curve remains static, then in one year when you get a coupon payment of 42.49, which you’ll reinvest for one year at 2.220%. At the end of two years, assuming a static (stable) yield curve, you’ll sell the (now 8-year) bond at a yield of 3.912%: the price will be calculated as: $FV = 1,000$ $PMT = 42.49$ $i = 3.912\%$ $n = 8$ $Solve\ for\ PV = -1,022.77$ Your holding period return is: $HPR = \frac{1,022.77 – 1,000.00 + 42.49\left(1.0222\right) + 42.49}{1,000.00} = 10.8695\%$ The effective annual return is: $EAR = \left(1 + HPR\right)^{1/n}\ – 1 = 1.108695^{1/2}\ – 1 = 5.2946\%$ Notice the substantial increase in total return of this strategy compared to buying a 2-year bond: 5.2946% vs. 2.5173%. The increase comes from a higher coupon as well as the price increase going from a yield of 4.249% to a yield of 3.912%. This change in yield is the origin of the the terms rolling down the yield curve and riding the yield curve. This approach would be notably less beneficial when the yield curve is flat or inverted. Nobody wants to roll up the yield curve. Carry Trade Carry trade involves much the same aspects as riding the yield curve: you purchase a bond with a longer maturity than your expected holding period, sell it at the end of your holding period, and earn the ensuing coupon payments plus the price change (which you hope is appreciation). What distinguishes carry trade from simply riding the yield curve is the way the investment is financed. When you ride the yield curve, it is understood that you purchase the bonds involved by using funds (i.e., cash) that are in the portfolio. For a carry trade, the purchase is financed essentially by borrowing the funds for the holding period. Often this borrowing is done using a repurchase agreement or repo: a contract in which you sell a security and and agree to repurchase it at a given (higher) price in the future, most commonly, the next day (an overnight repo). Although the legal structure of a repo is that of a sale and repurchase, the economic substance of the agreement is that it is a collateralized loan: the difference between the repurchase price and the original sale price is the interest on the loan. Carry trade is profitable if the coupon and price change on the long-term bond are greater than the cost of the financing. Note that if the expected holding period is longer than one day, the overnight repo could simply be rolled over every day for the entire holding period. Although an overnight repo is a common form of financing the purchase of a long-term bond for a carry trade, it is by no means the only way that a carry trade can be financed. I’ll show you an example in which the financing will involve simply selling (issuing) a bond whose maturity is your expected holding period. It will be a lot easier to analyze than rolling over an overnight repo every day for two years. Example A 2-year carry trade could be achieved by selling a 2-year bond and using the proceeds to buy a 10-year bond. The holding period cost, as we saw in the buy-and-hold example, above, would be 5.0980%, while the holding period (gross) return, as we saw in the rolling-down-the-yield-curve example, above, would be 10.8695%, giving a net holding period return of: $HPR = 10.8695\% – 5.0980\% = 5.7716\%$ The effective annual net return is: $EAR = \left(1 + HPR\right)^{1/n}\ – 1 = 1.057716^{1/2}\ – 1 = 2.8453\%$ Long Position in Bond Futures Taking a long position in futures (or forward) contracts is much the same as purchasing the underlying asset. Indeed, for assets that don’t generate cash flows (and don’t have storage costs or convenience yields), a long futures position is equivalent to purchasing the asset. For bond futures, it’s a bit different from purchasing actual bonds because the futures contracts don’t pay coupons, whereas the actual bonds do. Therefore, whereas a rolling-down-the-yield-curve strategy (above) will earn both the coupon payments and any price appreciation on the bonds, a long position in a bond futures contract will gain only the price appreciation. Against that, purchasing the bond requires an initial investment, whereas taking the long position in a futures contract does not. (It does, however, require that you post a margin, which is often in the form of short-term risk-free (e.g., government) securities. Therefore, the return you earn on the margin account will be a short-term risk-free rate, so there is an opportunity cost to consider.) Taking the long position in bond futures will increase the duration of your fixed income portfolio. However, if your view is that the yield curve will remain static, that increased duration will not be an issue as far as your return is concerned. It may be an issue if your portfolio has a duration constraint (as was mentioned in the rolling-down-the-yield-curve strategy, above). Example Suppose that you take the long position in a 2-year government bond futures contract. The (theoretical) underlying is a 10-year, option-free, 100,000-par bond that pays an annual coupon of 6%. The future price is 108,703.10. (I got this by discounting each of the payments on the bond at their appropriate spot rates, totting them up, subtracting the present values of the first two coupons (which we won’t receive as we don’t own the bond), then increasing that present value by the 2-year spot rate for two years. You needn’t worry about the details; on the exam they’ll give you the price.) Two years from today, the futures contract matures and the mark-to-market price of the underlying (now an 8-year bond) is 114,397.90. The holding period return is: $HPR = \frac{114,397.90}{108,703.10}\ – 1 = 5.2389\%$ The effective annual return is: $EAR = \left(1 + HPR\right)^{1/n}\ – 1 = 1.052389^{1/2}\ – 1 = 2.5860\%$ Receive Fixed, Pay Floating Swap You’ll recall from Level II that swaps are priced (i.e., the fixed rate is calculated) to prevent arbitrage: the underlying assumption, it turns out, is that interest rates will evolve according to the (1-period) forward curve. In this situation, that assumption means that while today the 1-period rate is 2.220%, the expected 1-year rate starting one year from today is 2.831%, the expected 1-year rate starting two years from today is 3.382%, and so on. (Look at the forward curve in the table at the top of this article.) When the yield curve slopes upward, the fixed rate will be higher than the (initial) floating rate. (Indeed, it should be the par rate for the tenor (maturity) of the swap). Furthermore, if the yield curve remains static, then rates won’t evolve according to the forward curve, and the fixed-rate receiver will make out like a bandit. Example Suppose that you enter into a 2-year, annual pay, plain vanilla interest rate swap as the fixed rate receiver / floating rate payer. Based on the yield curve at the top of this article, the fixed rate should be 2.521%, and the first year’s floating rate will be 2.220%. At the end of the first year, your net payment will be a receipt of 0.301% (= 2.521% − 2.220%) times the notional value of the swap. Should the yield curve remain static, then the net payment at the end of the second year will likewise be a receipt of 0.301% times the notional. Selling Convexity Convexity is useful when yields change: the more convex your portfolio, the greater the price increase when yields fall, and the smaller the price decrease when yields rise. And, like most things that are useful, convexity costs money; a more convex portfolio is generally more expensive than a less convex, but otherwise comparable (e.g., identical duration), portfolio. Convexity is not particularly useful, however, when yields don’t change. When ∆y = 0, it doesn’t matter what the portfolio’s duration and convexity are: the price isn’t going to change (apart from the pull to par as time passes). Therefore, a sound strategy when you expect the yield curve to be static is to sell convexity. There are various ways to do this. You can: • Shorten the maturity of the bonds in your portfolio. Shorter maturity bonds generally have lower convexity than longer maturity bonds. However, with an upward sloping yield curve, shorter maturity bonds will also have lower yields than longer maturity bonds, so you may lose more than you gain by adopting this approach. • Replace straight bonds with callable bonds, or prepayable bonds (such as MBSs) which have negative convexity at lower yields, and less convexity than straight bonds even at moderate yields. • Sell call options and/or put options on bonds. Call options and (out-of-the-money) put options have positive convexity, so selling them will reduce the convexity of the portfolio. Call options increase the duration of the portfolio while put options decrease the duration. By selling them in an appropriate mix, you can reduce the portfolio’s convexity while keeping its duration unchanged, or lengthening the duration, or shortening the duration. Example (Note: you won’t be doing any of these calculations on the exam; you’ll simply have to know that to sell convexity you sell some calls and sell some puts. I’m including them simply to give you an understanding of how selling convexity works; i.e., how you figure out how many calls and puts to sell. If you don’t care about the calculations, then 1) shame on you, and 2) skip this section.) You own a portfolio with a market price of AUD 23,498,000, a modified duration of 8.32 years, and a (modified) convexity of 80.83 years2. You believe that the yield curve will remain unchanged for at least the next 6 months, so you decide to sell off half of your (money) convexity. Your investment mandate requires you to match the benchmark duration of 8.20 years within ±0.25 years, so you decide to leave the duration of your portfolio unchanged. You have available these options on 20-year government bonds:  Type Strike Price Duration, Years Convexity, Years2 Call AUD 1,275.00 AUD 81.50 108.8 8,327.8 Put AUD 1,275.00 AUD 70.38 −102.4 5,943.4 How many options should you buy/sell to achieve your goal? (Assume options can be bought/sold only in lots of 100.) How much will it cost? Both calls and puts have positive convexity, so we’re going to be selling them. First, we need to determine the ratio of puts to calls that we will sell. We want to keep the duration of the portfolio unchanged, so the net money duration of the options must equal zero: $\#\ calls × call\ value × call\ duration + \#\ puts × put\ value × put\ duration = 0$ $\#\ calls × AUD\ 81.50 × 108.8\ years + \#\ puts × AUD\ 70.38 × -102.4\ years = 0$ $\#\ calls × 8,867.20 = \#\ puts × 7,206.91$ $\#\ puts = \frac{\#\ calls × 8,867.20}{7,206.91} = \#\ calls\left(\frac{8,867.20}{7,206.91}\right) = 1.2304\left(\#\ calls\right)$ Therefore, for each call we sell, we need to sell 1.2304 puts to leave the duration unchanged. We want to sell off half of the money convexity. so the calculation is: $\#\ calls × call\ value × call\ convexity + \#\ puts × put\ value × put\ convexity\\ = \frac12 × portfolio\ value × portfolio\ convexity$ $\#\ calls × AUD\ 81.50 × 8,327.8\ years^2 + \#\ puts × AUD\ 70.38 × 5,943.4\ years^2\\ = \frac{AUD\ 23,498,000 × 80.83\ years^2}{2}$ $\#\ calls\left(678,715.7\right) + \left(1.2304 × \#\ calls × 418,296.5\right) = 949,671,670$ $\#\ calls\left[678,715.7 + \left(1.2304 × 418,296.5\right)\right] = 949,671,670$ $\#\ calls\left(1,193,377\right) = 949,671,670$ $\#\ calls = \frac{949,671,670}{1,193,377} = 795.8$ $\#\ puts = 1.2304 × 795.8 = 979.1$ Rounding to the nearest 100 options, we should sell 800 calls and 1,000 puts. Because of the rounding, we’ll change the money duration slightly and be off a little on the money convexity: \begin{align}∆money\ duration &= -800 × AUD\ 81.50 × 108.8\ years\\ \\ &\ \ \ \ \ -\ 1,000 × AUD\ 70.38 × \left(-102.4\ years\right)\\ \\ &= -113,152\ AUD-years \end{align} \begin{align}∆money\ convexity &= -800 × AUD\ 81.50 × 8,327.8\ years^2\\ \\ &\ \ \ \ \ -\ 1,000 × AUD\ 70.38 × 5,943.4\ years^2\\ \\ &= -961,269,052\ AUD-years^2 \end{align} This decreases the money duration by only 0.06% (0.0048 years, which is negligible), and reduces the money convexity by 50.61% (which is quite close to the 50% reduction we sought). ## Dividend Discount Models A time-honored method to determine the value of an investment is to discount to the present all of the investment’s (expected) future cash flows, and tot up those present values. It’s a method we use commonly when valuing bonds, and when valuing projects in which a company is considering investing (e.g., whether or not to purchase a machine that makes tennis balls): calculating the net present value (NPV), or, in a similar vein, its internal rate of return (IRR). I describe these ideas in detail in this article. Extending that idea to an investment in a company’s stock, when a company pays dividends regularly – and is expected to continue to pay dividends ad infinitum – a reasonable way to determine the value of the company’s stock is to use a dividend discount model (DDM): discount each of the expected dividends to the present (at your required rate of return) and tot up those present values; the resulting sum is the value of the stock (at least, its value to you). Example Euler Pharmaceuticals pays an annual dividend of CHF 2.00 per share. Historically, they have raised their annual dividend by CHF 0.25 per share every 10 years; they have been paying the current dividend for the last 5 years. Their expected dividends are:  Year Dividend 1 CHF 2.00 2 CHF 2.00 3 CHF 2.00 4 CHF 2.00 5 CHF 2.00 6 CHF 2.25 7 CHF 2.25 8 CHF 2.25 9 CHF 2.25 10 CHF 2.25 11 CHF 2.25 12 CHF 2.25 13 CHF 2.25 14 CHF 2.25 15 CHF 2.25 . . . . . . 96 CHF 4.50 97 CHF 4.50 98 CHF 4.50 99 CHF 4.50 100 CHF 4.50 Graphically: The value of the stock depends on the discount rate (required rate of return) used to determine the present value of the dividends. The value versus the required rate of return is:  Required Return V0 0.0%935.00 0.5% $529.11 1.0%$325.78 1.5% $217.43 2.0%$155.78 2.5% $118.33 3.0%$94.14 3.5% $77.61 4.0%$65.77 4.5% $56.94 5.0%$50.14 5.5% $44.75 6.0%$40.38 6.5% $36.78 7.0%$33.75 7.5% $31.18 8.0%$28.96 8.5% $27.04 9.0%$25.35 9.5% $23.86 10.0%$22.53

Graphically:

The problem with this (sort of) real-world example is that analyzing a dividend that increases every once in a while and otherwise remains constant is . . . um . . . difficult (unless you’re doing it in Excel, as I am).

Therefore, the curriculum makes some simplifying assumptions to make it easier to determine the value of a stock based on its expected dividends.  We acknowledge that these assumptions are unreasonable in the real world, but they make the models easy to understand.  Once you understand these simplified models thoroughly, it’s much easier to graduate to more complex (i.e., realistic) models and understand them.  So . . . tally ho!

(Short) Holding Period Models

We start with the most unrealistic model possible: you plan to buy a stock today, hold it for one year (receiving one dividend at the end of the year), then sell it.  You know your required rate of return, you know the dividend you’ll receive in one year, and you know the price at which you’ll be able to sell the stock in one year.  (That’s the unreasonable part: how can you possibly know the price at which you’ll sell the stock in one year without knowing its price today?)  You need to compute the price of the stock today.  Let’s look again at Euler Pharmaceuticals:

• Next year’s expected dividend: CHF 2.00
• Expected price in one year: CHF 31.52
• Required rate of return: 7.50%

The price you’d be willing to pay today is:

\begin{align}V_0\ &=\ \dfrac{D_1}{1 + r} + \dfrac{P_1}{1 + r}\\
\\
&=\ \dfrac{CHF\ 2.00}{1.075} + \dfrac{CHF\ 31.52}{1.075}\\
\\
&=\ CHF\ 1.86 + CHF\ 29.32\\
\\
&= CHF\ 31.18
\end{align}

Suppose, instead, that you plan to hold the stock for two years:

• Expected dividend in one year: CHF 2.00
• Expected dividend in two years: CHF 2.00
• Expected price in two years: CHF 31.88
• Required rate of return: 7.50%

The price you’d be willing to pay today is:

\begin{align}V_0\ &=\ \dfrac{D_1}{1 + r} + \dfrac{D_2}{\left(1 + r\right)^2} + \dfrac{P_2}{\left(1 + r\right)^2}\\
\\
&=\ \dfrac{CHF\ 2.00}{1.075} + \dfrac{CHF\ 2.00}{1.075^2} + \dfrac{CHF\ 31.88}{1.075^2}\\
\\
&=\ CHF\ 1.86 + CHF\ 1.73 + CHF\ 27.59\\
\\
&= CHF\ 31.18
\end{align}

Note that on your calculator, you can solve these with the TVM buttons.  For example, the second one can be solved this way: (with your calculator in END mode):

n = 2
i = 7.5%
PMT = 2.00
FV = 31.88
Solve for PV = −31.18

Single-Stage (Gordon Growth) Model

When the holding period is long-term (essentially, infinite, or perpetual), a common simplifying assumption is that the dividend grows at a constant rate.  The model incorporating this assumption is known as the Gordon Growth model, and the formula for the value of the stock is simplicity itself:

\begin{align}V_0\ &=\ \dfrac{D_1}{1 + r} + \dfrac{D_2}{\left(1 + r\right)^2} + \dfrac{D_3}{\left(1 + r\right)^3} + \cdots\\
\\
&= \sum_{i=1}^\infty \dfrac{D_i}{\left(1 + r\right)^i}\\
\\
&=\ \dfrac{D_0\left(1 + g\right)}{1 + r} + \dfrac{D_0\left(1 + g\right)^2}{\left(1 + r\right)^2} + \dfrac{D_0\left(1 + g\right)^3}{\left(1 + r\right)^3} + \cdots\\
\\
&= \sum_{i=1}^\infty \dfrac{D_0\left(1 + g\right)^i}{\left(1 + r\right)^i}\\
\\
V_0 &=\ \dfrac{D_0\left(1 + g\right)}{r\ -\ g} =\ \dfrac{D_1}{r\ -\ g}
\end{align}

where:

• $r$: required rate of return
• $g$: dividend growth rate

Note that this model makes sense only if $r > g$.  Otherwise, the sum is infinite.

Let’s take a careful look at this formula, to see whether or not it makes sense.  We’ll compare the stocks of two companies, not particularly cleverly known as Company A and Company B.

D1

Suppose first that the stocks of Companies A and B are considered equally risky (so that they command the same required rate of return), and that their dividends will grow at the same rate unto perpetuity.  Company A just paid a dividend of GBP 1.00 while Company B just paid a dividend of GBP 1.25.  I hope that it’s clear that you would be willing to pay more for Company B’s stock than for Company A’s stock, and that’s exactly what we get from the Gordon Growth formula: when D0 (and, consequently, D1) is higher, all else equal, V0 is higher (because the numerator is bigger and the denominator is unchanged), and conversely.

r

Suppose now that Companies A and B just paid the same dividend, and that their dividends will grow at the same rate unto perpetuity, but that Company B’s stock is considered riskier than Company A’s stock.  I trust that it’s clear that your required rate of return for Company B’s stock will be higher than your required rate of return for Company A’s stock, and that, therefore, you would be willing to pay less for Company B’s stock than for Company A’s stock.  We see that that’s exactly what we get from the Gordon Growth formula: when r is higher, all else equal, V0 is lower (because the numerator is unchanged while the denominator is bigger), and conversely.

g

Suppose now that Companies A and B just paid the same dividend, and are considered equally risky, but that Company B’s dividend is expected to increase at a faster rate than Company A’s dividend.  I assume that it’s clear that you would be willing to pay more for Company B’s stock than for Company A’s stock.  We see that that’s exactly what we get from the Gordon Growth formula: when g is higher, all else equal, V0 is higher (because the numerator is unchanged while the denominator is smaller), and conversely.

In all cases, the Gordon Growth model passes muster: it behaves exactly as we should expect it to behave, and the analyses are simple and straightforward.  That’s the point of making these simplifying assumptions: it’s easy to understand how the model works, and it works the way it’s meant to work.  Once we understand that, presumably we can move on to more complex (and reasonable) models and be able to understand how they work as well.

Example

Suppose that Ramanujan Technologies just paid an annual dividend of INR 200.00 per share.  Its dividends are expected to grow at 1.5% per year, and you determine that 8.4% is an appropriate rate of return for such an investment.  The amount per share you should be willing to pay for Ramanujan Technologies stock is:

$V_0 = \dfrac{D_1}{r\ -\ g} = \dfrac{INR\ 200 \times 1.015}{0.084\ -\ 0.015} = INR\ 2,942.03$

Delayed Gordon Growth Model

Suppose that instead of the first dividend coming one year from today, we expect the first dividend five years from today; i.e., the company does not pay a dividend yet, but we expect that it will start paying one in five years (and that the dividend will grow at a constant rate thereafter).  Before we analyze the value of the company’s stock, let’s take another look at the Gordon Growth model:

$V_0 = \dfrac{D_1}{r\ -\ g}$

There’s nothing special about the subscript “0” on the value, or the subscript “1” on the dividend; what’s important is that the dividend comes one year after the date for the value.  So, for example,

$V_1 = \dfrac{D_2}{r\ -\ g}$

$V_2 = \dfrac{D_3}{r\ -\ g}$

and, in general,

$V_t = \dfrac{D_{t+1}}{r\ -\ g}$

Therefore, we can use the Gordon Growth model to determine the value of the stock one year before the first dividend is paid.  If the date on the value isn’t today, then we simply discount that value back to today.

Example

Suppose that Abel Industrial is expected to start paying a dividend five years from today.  The first (annual) dividend is anticipated to be EUR 2.50 per share, increasing 1% per year thereafter.  If your required rate of is 8.2%, how much would you be willing to pay for a share of Abel stock?

\begin{align}V_4\ &=\ \dfrac{D_5}{r\ -\ g}\\
\\
&= \dfrac{EUR\ 2.50}{8.2\%\ -\ 1.0\%}\\
\\
V_4 &= EUR\ 34.72\\
\\
V_0 &= \dfrac{V_4}{\left(1 + r\right)^4}\\
\\
&= \dfrac{EUR\ 34.72}{1.082^4}\\
\\
&=\ EUR\ 25.33
\end{align}

Multi-Stage Models

In a two-stage growth model, a company’s dividends are assumed to grow a one rate (presumably a high rate) for a finite period of time, then to grow at another rate (presumably a lower rate) forever thereafter.  In a three-stage growth model, the dividends are assumed to grow at one (high) rate for a finite period of time, then at a second (middling) rate for another finite period of time, then at a third (low) rate forever thereafter.  I’ll leave it to your imagination what a four-stage, or a five-stage, or a six-stage, or a more-than-six-stage model might be.

The approach for determining the value of the company’s stock under all of these models is the same:

• Determine all of the dividend amounts for all of the finite periods
• Discount those amounts to the present
• Use the Delayed Gordon Growth model for the final (infinite) period
• Tot up all of the present values
• Voilà!

Two-Stage Example

Jacobi Materials just paid an annual dividend of EUR 1.75 per share.  They expect the dividend to grow 10% per year for the next five years, then to grow at 2% per year forever thereafter.  If you require a return of 7.7% to invest in Jacobi, how much would you be willing to pay for a share of its stock today?

The dividends for the first six years will be:

 Year Dividend 1 EUR 1.9250 2 EUR 2.1175 3 EUR 2.3293 4 EUR 2.5622 5 EUR 2.8184 6 EUR 2.8748

The value is computed as:

\begin{align}V_0\ &=\ \left[\sum_{i=1}^5 \dfrac{D_i}{\left(1 + r\right)^i}\right] + \dfrac{D_6}{\left(r\ -\ g_{low}\right)\left(1 + r\right)^5}\\
\\
&= \dfrac{EUR\ 1.925}{1.077} + \dfrac{EUR\ 2.1175}{1.077^2} + \cdots + \dfrac{EUR\ 2.8184}{1.077^5} + \dfrac{EUR\ 2.8748}{\left(7.7\%\ -\ 2\%\right)1.077^5}\\
\\
&=\ EUR\ 44.13
\end{align}

(Note that you could add the discounted values of the first four dividends, then use the Delayed Gordon Growth formula on the fifth dividend and arrive at the same total.  Just an interesting fact.)

Three-Stage Example

Dirichlet Products just paid an annual dividend of EUR 2.25 per share.  They expect the dividend to grow 10% per year for the next three years, then to grow at 5% for two more years, then to grow at 2% per year forever thereafter.  If you require a return of 7.3% to invest in Dirichlet , how much would you be willing to pay for a share of its stock today?

The dividends for the first few years will be:

 Year Dividend 1 EUR 2.4750 2 EUR 2.7225 3 EUR 2.8586 4 EUR 3.0016 5 EUR 3.1516 6 EUR 3.2147

The value is computed as:

\begin{align}V_0\ &=\ \left[\sum_{i=1}^5 \dfrac{D_i}{\left(1 + r\right)^i}\right] + \dfrac{D_6}{\left(r\ -\ g_{low}\right)\left(1 + r\right)^5}\\
\\
&= \dfrac{EUR\ 2.475}{1.073} + \dfrac{EUR\ 2.7225}{1.073^2} + \cdots + \dfrac{EUR\ 3.1516}{1.073^5} + \dfrac{EUR\ 3.2147}{\left(7.3\%\ -\ 2\%\right)1.073^5}\\
\\
&=\ EUR\ 54.11
\end{align}

(Note that, again, you could add the discounted values of the first four dividends, then use the Delayed Gordon Growth formula on the fifth dividend and arrive at the same total.)

Combining Dividend Discount Models and Multiplier Models

Occasionally, an analyst will combine a dividend discount model with a multiplier model: the terminal value is computed using a multiplier rather than using the Gordon Growth model.  It’s not a big deal.

Example

Suppose that Dedekind Cutlery just paid a dividend of EUR 1.25 per share, which was 50% of its earnings per share (EPS).  Its EPS is expected to grow at 4% per year for the next five years (with its payout ratio remaining constant), whereupon its trailing P/E ratio is expected to be 15.4.  If you require a return of 8.1% to invest in Dedekind, how much would you be willing to pay for a share of its stock today?

The earnings and dividends for the next five years are expected to be:

 Year EPS Dividend 0 EUR 2.50 EUR 1.25 1 EUR 2.6000 EUR 1.3000 2 EUR 2.7040 EUR 1.3520 3 EUR 2.8122 EUR 1.4061 4 EUR 2.9246 EUR 1.4623 5 EUR 3.0416 EUR 1.5208

The share price five years from today is expected to be:

$V_5 = EUR\ 3.0146 \times 15.4 = EUR\ 46.84$

Today’s value is calculated as:

\begin{align}V_0\ &=\ \left[\sum_{i=1}^5 \dfrac{D_i}{\left(1 + r\right)^i}\right] + \dfrac{V_5}{\left(1 + r\right)^5}\\
\\
&= \dfrac{EUR\ 1.30}{1.081} + \dfrac{EUR\ 1.352}{1.081^2} + \cdots + \dfrac{EUR\ 1.5208}{1.081^5} + \dfrac{EUR\ 46.84}{1.081^5}\\
\\
&=\ EUR\ 37.31
\end{align}

When to Use a Dividend Discount Model

A valid question to ask is, “When should I use a dividend discount model to estimate the value of a stock?”

Obviously, a necessary condition for using a DDM is that the company pays a dividend; either it pays one now, or it is expected to start paying one in the foreseeable future, and that it will continue to pay dividends.

Assuming that that criterion is met, why use a DDM instead of, say, a free cash flow model?  The general answer is that dividends tend to be more stable than either free cash flow to equity (FCFE) or free cash flow to the firm (FCFF), so they’re more easily predicted (i.e., estimated).  This is certainly true for the way that most companies pay dividends: their dividend remains constant over a period of time, and they raise it only when they reasonably expect that they will be able to maintain the new (higher) dividend into the future.

However, if, instead of maintaining a constant dividend amount, a company chooses to pay a dividend amounting to a constant percentage of its net income, its FCFE, or its FCFF, then there’s no particular advantage to using a DDM over a FCFE model or a FCFF model.  And if they’re even less disciplined – their dividends being based on little more than whim and caprice (“Oh, what the heck?  Let’s pay a dividend!  Just for giggles!”) – then a DDM is even less likely to be an appropriate valuation model.

## Equity Valuation Models

The CFA curriculum discusses a variety of methods for estimating the value of a company’s equity.  The approaches, or models, fall broadly into these categories (covered in separate articles; follow the links):

• Present value models (also known as discounted cash flow models)
• Multiplier models (also known as market multiple models)
• Share price models
• Price-to-earnings (P/E) ratio
• Price-to-sales (P/S) ratio
• Enterprise value (EV) models
• EV/EBITDA ratio
• Asset-based models

If you haven’t read the article on option strategies in general, that’s a good place to start, then return here. In particular, if you haven’t read the warning about calculating profit that appears at the end of that article, you should go read it now; the way I’m calculating the profit here is correct, but […]

CFA® Level II Derivatives Membership, CFA® Level II Membership, CFA® Level III Membership, CFA® Level III Derivatives Membership

This will give you access to this and all other articles at that membership level.

## Short Strangle

If you haven’t read the article on option strategies in general, that’s a good place to start, then return here. In particular, if you haven’t read the warning about calculating profit that appears at the end of that article, you should go read it now; the way I’m calculating the profit here is correct, but […]

CFA® Level II Derivatives Membership, CFA® Level II Membership, CFA® Level III Membership, CFA® Level III Derivatives Membership

This will give you access to this and all other articles at that membership level.

If you haven’t read the article on option strategies in general, that’s a good place to start, then return here.  In particular, if you haven’t read the warning about calculating profit that appears at the end of that article, you should go read it now; the way I’m calculating the profit here is correct, but […]

CFA® Level II Derivatives Membership, CFA® Level II Membership, CFA® Level III Membership, CFA® Level III Derivatives Membership

This will give you access to this and all other articles at that membership level.

## Yield Curve Strategies – General

In 2019, CFA Institute revised all of their Level III Fixed Income readings, and, in particular, came up with a 10-page blue box example that was absurdly complex, which I simplified here.

That example’s gone.  Fortunately.  (In point of fact, I’m sorry that the idea is gone from the curriculum, because it was interesting and useful.  The problem was that the example was far too complicated for candidates, many of whom do not employ any fixed income in their day-to-day work, much less foreign fixed income.)

For 2022, CFA Institute replaced that Yield Curve Strategies reading with a new Yield Curve Strategies reading.  I’m going to address the new reading here, and in the companion articles linked here.  It’ll be interesting, but not as interesting as it could have been.

What are Yield Curve Strategies?

First, when we talk about yield curve strategies, we’re generally talking about investments in risk-free government bonds, either directly or synthetically.  This is distinguished from credit strategies in which we’re talking about investments in corporate bonds, for example, for which potential changes in credit spreads can play a significant rôle in our investment decision; the bonds we consider in yield curve strategies have essentially no credit spread component.  Thus, we may be talking about:

• Buying or selling government bonds (domestic or foreign)
• Taking the long or the short position in futures contracts on government bonds (domestic or foreign)
• Taking the long or the short position in forward contracts on government bonds (domestic or foreign) with reputable counterparties (i.e., those for whom the default risk is negligible)
• Buying or selling options on government bonds (domestic or foreign)
• Entering into plain vanilla interest rate swaps with reputable counterparties
• Entering into currency swaps with reputable counterparties

The objective in a yield curve strategy is, as you might expect for any investment strategy, to try to enhance the returns on our portfolio.  This may be accomplished by:

• Adjusting the overall duration of the portfolio
• Adjusting the key rate durations of the portfolio
• Adjusting the convexity of the portfolio
• Adjusting the cash flow of the portfolio

Which of these approaches we will use depends on our view of what will happen to the yield curve (or yield curves, if we’re considering investments denominated in more than one currency) over our investment horizon (and, of course, on the scope we’re allowed based on the investor’s investment policy statement (IPS)).  I’ll cover the possibilities, and help you to understand how to select an appropriate strategy given a particular view on the yield curve(s).

Making Money with Bonds

There are broadly two 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}

For yield curve strategies we won’t be concerned with the fourth term (changes in yield spreads), and we’ll be concerned with the last term only when we discuss bonds denominated in foreign currencies, at the end of this article.  Our main focus, therefore, will be on coupon income, rolldown return (i.e., the price change in a bond that results from the passage of time, assuming that the yield curve does not change), and price changes that result from changes in the yield curve.

Your View of the Yield Curve

Broadly (i.e., really, really broadly), there are two possibilities for the yield curve in the future:

1. It can stay the same as it is today: a static yield curve
2. It can change: a dynamic yield curve

Your choice of strategy begins with your view of how the yield curve will look at the end of your holding period: the same as today, or different.  If you believe that the yield curve will remain static, you’ll choose from one set of strategies, but if you believe that the yield curve will change, you’ll choose from a different set of strategies.  I’ll cover both possibilities (and their corresponding strategy sets) in these articles:

The Yield Curve

For the articles on yield curve strategies, I’ll start with this yield curve:

 Maturity, Years Par Rate Spot Rate Forward Rate 1 2.220% 2.220% 2.220% 2 2.521% 2.525% 2.831% 3 2.799% 2.810% 3.382% 4 3.057% 3.078% 3.887% 5 3.296% 3.331% 4.348% 6 3.518% 3.570% 4.774% 7 3.722% 3.794% 5.150% 8 3.912% 4.008% 5.516% 9 4.087% 4.209% 5.836% 10 4.249% 4.401% 6.139%

(OK, technically it’s three yield curves.)

(For some of the graphs I’ll extend the par curve to 30 years, but for calculations we’ll mainly stick to 10 years maximum.)

Here’s how they look for 10 years:

and here’s how they look for 30 years:

The Strategies in a Nutshell

This is about as straightforward as it gets: you have a bond portfolio, so you buy some bonds.  Then, perhaps, you sell some of those bonds and buy other bonds.

By selling one bond and buying another, you can:

• Adjust the overall duration of the portfolio (sell a bond with one duration, buy a bond with a different duration)
• Adjust the key rate durations of the portfolio (for example, by switching from a bullet portfolio to a barbell portfolio with the same duration)
• Adjust the convexity of the portfolio (for example, a laddered portfolio will have more convexity than a bullet portfolio with the same duration)
• Adjust the cash flow of the portfolio (sell a bond with one coupon rate, buy a bond with a different coupon rate)

You can also adjust your currency exposure by selling a bond denominated in one currency and buying a bond denominated in another currency.

Taking the Long or the Short Position in Futures Contracts on Government Bonds

Although this is extremely similar to buying or selling (respectively) government bonds, there are some significant differences:

• Futures contracts require no upfront payment (although you will likely have to post a margin); therefore, they are leveraged positions (which can increase or decrease your total return significantly)
• Futures contracts do not pay coupons, so their duration is generally greater than the duration of the underlying bond, and their convexity is generally less than the convexity of the underlying bond

Apart from these, taking a long position in a forward contract is much the same as buying the underlying bond, and taking the short position is much the same as selling the underlying bond.

Taking the Long or the Short Position in Forward Contracts on Government Bonds

This is nearly identical to taking the long or short position in futures contracts.  The main difference is that these are custom contracts, so you can adjust such characteristics as the underlying bonds, the time to maturity, and the collateral (if any).  Apart from that, they work much as the futures contracts (above) work.

Buying or Selling Options on Government Bonds

The effects of options on government bonds are:

• Long call options increase duration and increase convexity
• Short call options decrease duration and decrease convexity
• Long put options decrease duration
• Long out-of-the-money puts increase convexity
• Long in-the-money puts may decrease convexity negligibly
• Short put options increase duration
• Short out-of-the-money puts decrease convexity
• Short in-the-money puts may increase convexity negligibly

Entering into Plain Vanilla Interest Rate Swaps

The fixed leg on a plain vanilla interest rate swap has an effective duration that is roughly 75% of the maturity of the swap, while the floating leg has an effective duration that is roughly half the time between swap payments.  Therefore, if you enter into a pay-fixed, receive-floating swap you will reduce the duration (and the convexity) of your portfolio, and if you enter into a pay-floating, receive-fixed swap you will increase the duration (and the convexity) of your portfolio.

Note that this will also have an effect on your cash flow, but apart from the payment at the first settlement date, you cannot say for certain whether it will increase or decrease your cash flow.

Entering into Currency Swaps

A fixed-for-fixed currency swap isn’t likely to change the duration or convexity of the portfolio much, nor will a floating-for-floating currency swap; however, a fixed-for-floating currency swap can change the duration much as a plain vanilla interest rate swap does.  It can also change your cash flows, similar to a plain vanilla interest rate swap.

Of course, in this case the two legs of the swap will pay different currencies, so they’ll be subject to different yield curves.  Additionally, a currency swap will be subject to changes in currency exchange rates, which adds another dimension to the effect they will have on a fixed income portfolio.

One common use of currency swaps is to combine them with bonds denominated in foreign currencies, effectively changing the foreign currency cash flows into domestic currency cash flows  (i.e., they can be a tool to hedge currency exchange rate risk).  Depending on the scope allowed by the IPS and the manager’s view on exchange rates, the manager may choose to overhedge or underhedge the foreign currency payments; i.e., use the swap as tool to engage in active currency management.

## Tax Bases

The tax base (it used to be tax basis, but somehow it changed to tax base; I think the reason is that the plural for each word is “bases” (pronounced <base-eez> and <base-ezz>, respectively), that someone mistook “bases” (plural of “basis”) to be “bases” (plural of “base”), and the error caught on) of an asset […]