Chapter 16: The Analysis and Valuation of Bonds

Part II: Term Structure theories and Bond Volatility

· Term Structure theories

- The term structure is the relationship between interest rates and maturity, at a given point in time. It is also called the yield curve. Yield curves are plots of rates of return against maturities for bonds that are otherwise alike.

- Such curves can be constructed from current observations. For example, one could take all outstanding corporate bonds from a group in which the bonds are almost identical in all respects except their maturities, then generate the current yield curve by plotting each bond's YTM against its maturity.

- The major difficulty in generating such curves is obtaining a large enough sample of bonds that are almost identical in terms of risk, liquidity, coupon yields, and the like. To address this problem, a widely used approach is to generate a spot yield curve from spot rates using Treasury securities, as we have done before.

- Regardless of the procedure used to plot such curves, a yield curve can take many shapes. Some alternatives are shown in Figure 16.5 of your text. Which “should” be the “correct” shape of the yield curve? Should it be rising, falling, flat or humped? The answer is that, as in asset pricing, we have theories, but no exact answers. There are three competing theories of term structure. Let us briefly talk about each of them below:

1) Expectations Hypothesis

- The expectations hypothesis says that forward rates are expectations of future spot rates. It is easiest to see this with an example:

Consider the following spot rates at time t.

Maturity	Spot rate	Fwd rate
1	6.00%
2	6.25%	6.50%
3	6.50%	7.00%
4	7.00%	8.51%

I have also calculated the implied forward rate from this spot rate curve.

- In terms of notation, s_1t =6%, s_2t = 6.25%, s_3t = 6.50% and s_4t=7.00% i.e. at time t., the yield curve is upward sloping. The question is why?

- The implied forward rates are _t+1r₁=6.50%, _t+2r₁=7.00%, _t+3r₁=8.51%. As we discussed earlier, each of these forward rates can be calculated by using the formulas:

- Now, the expectations hypothesis says that investors expect the one-year spot rate at time t+1, i.e. s_1t+1 to be exactly equal to the one-year forward rate at time t. i.e. _t+1r₁. In other words, E(s_1t+1)= _t+1r₁=6.50%.

Similarly, E(s_1t+2)= _t+2r₁=7.00%, and E(s_1t+3)= _t+3r₁=8.51%. This can be seen from the following diagram:

- This means the yield curve is sloping upwards at time t, only because investors expect short-term rates to rise in the future.

- On the other hand, if short-term interest rates were expected to fall, then we would see a negatively sloped or downward sloping yield curve. The following diagram shows just this case:

- The expectations hypothesis is widely used because of its simplicity, and intuitive appeal. Various tests by finance researchers have found it to be fairly adequate as an explanation of term structure.

2) Liquidity Preference Hypothesis

- This theory says that investors will expect a greater yield on longer term bonds compared to shorter-term bonds, simply because a bond’s risk or volatility increases with the maturity.

- This theory implies that the yield curve should be always upward sloping, whether short-term rates expected to rise or fall. Any other type of yield curve (for example, a downward sloping curve) is to be viewed as a temporary aberration.

- Let’s see this with the help of the previous example:

		Expected	Liquidity	Forward
Maturity	Spot rate	spot rate	Premium	Rate
1	6.00%
2	6.25%	5.95%	0.55%	6.50%
3	6.50%	5.90%	1.10%	7.00%
4	7.00%	5.80%	2.71%	8.51%

- Here, we have a rising yield curve, which means that implied one-year forward rates at t+1, t+2, and t+3 are increasing (as before). But investors expect one-year short rates in the future to fall, as can be seen from the second column!

- This is only possible if the forward rates are not equal to expected future short rates, but are greater than them. In symbols, we have:

_t+1r₁(=6.50%) > E(s_1t+1) (=5.95%)

_t+2r₁ (=7.00%) > E(s_1t+2)(= 5.90%)

_t+3r₁ (=8.51%) > E(s_1t+3) (=5.80%)

The difference in each case, is called the liquidity premium, i.e. the premium that is demanded by investors in order to hold long-term bonds as opposed to short-term bonds. Let us see all this in a graph:

- Historically, yield curves have been upward sloping most of the time. Also actual long-term spot rates have been consistently above those predicted by the expectations hypothesis.

- This means that in practice, some combination of the liquidity premium hypothesis and the expectations hypothesis might be true.

3) Market Segmentation Hypothesis

- This theory says that different investors have different needs and some concentrate on short-term bonds, while others concentrate on long-term bonds.

- Hence, this theory contends that short-term and long-term bonds have yields depending on the supply and demand within each maturity segment.

· Bond Price Volatility

- Bond price volatility is measured in terms of percentage changes in bond prices with respect to the changes in the variables relating to the bond. These variables could be:

a) par value

b) coupon rate

c) number of years to maturity

d) prevailing market interest rate

· Six facts about bond price volatility

Fact 1: Bond prices move inversely to bond yields (interest rates)

Fact 2: Price movements from equal absolute increases or decreases in yield are not symmetrical. A decrease in yield raises bond prices by more than an increase in yield of the same amount lowers prices.

Fact 3: For a given change in yields, longer-maturity bonds have larger price changes.

Fact 4: These price changes are at a diminishing rate as term to maturity increases.

Fact 5: Higher coupon issues show smaller percentage changes for a given change in yield; thus, bond price volatility is inversely related to coupon rate.

Fact 6: The absolute level of bond yields affects bond maturity in the following way: at a higher bond yield level, the price volatility is higher. The same percentage change in yield means a bigger change at a higher bond yield level than at a lower bond yield level. For example, at a yield level of 20%, a 10% increase in yield means a change of 2%. At a 5% yield level, the same 10% increase means a change of 0.5%. Thus, at higher yield levels, a given change in yield results in a higher change in bond price, which means the bond volatility is higher.

Notes regarding these facts

- As we can see from the above six facts, bond price volatility depends upon coupon rate, term to maturity, yield level and the magnitude and direction of yield change. Now, we cannot choose the yield level of the direction or magnitude of yield change of bond in our portfolio, as these are all determined by market forces. However, we could pick the coupon rate or term to maturity of our bonds as we see fit. Thus, these are the two factors that we can control.

- We know that bond price changes are higher for a given change in bond prices for a longer maturity and a lower coupon rate. Thus, if we expect interest rates to fall, we want a portfolio of bonds that has the maximum price increase possible. Thus, we would invest in (very) low coupon bonds with long maturities.

- By the same logic, if we expect interest rates to increase, we want to protect ourselves to the extent possible from the imminent fall in bond prices. Thus, we want a bond portfolio that has as low a decrease in bond prices as possible.

- This line of reasoning suggests that we would like to have a measure of volatility that embodies both term to maturity and coupon rate. We would like this measure of volatility to increase with term to maturity and decrease with the coupon rate. Fortunately, such a measure does exist. It is called duration, and we study it next.

· Duration

- There are many types of “durations”. We shall study two of these, Macaulay Duration and Modified Duration.

Macaulay Duration

Macaulay Duration is defined as: , where:

C_t = Cash flow in time t

t = time period in which cash flow C_t occurs

i = yield to maturity on the bond

First, let us understand the mechanics of calculation. Then, we can understand the properties of Macaulay Duration. Consider a 3-year bond with annual coupons and a coupon rate of 15%. The par value of the bond is $1,000, and the YTM of the bond is 10% .

This bond’s cash flows are as follows:

This means that: C₁ = 150, C₂ = 150, C₃ = 1150. Now, the duration formula can be written as follows:

The denominator is simply the market price of this bond i.e. P_m. Hence, we can also write this equation as:

Now, the calculation proceeds as follows and results in a Macaulay Duration for this bond of 2.65 years.

(1)	(2)	(3)	(4)	(5)
Year	Cash flow	PV at YTM	PV as a	Product
			% of price	(1) x (4)
1	150	136.36	0.1213	0.1213
2	150	123.97	0.1103	0.2205
3	1150	864.01	0.7685	2.3054
	Price =	1124.34	1.0000	2.6472

Properties of Macaulay Duration

We shall look at some important properties of Macaulay Duration, and see how it is useful.

1. The duration of a zero-coupon bond will always equal its maturity

Let us see this for the 3-year example bond above:

The duration is

If this bond were a zero-coupon bond, we have C₁=C₂=0, which means that the formula becomes:

In general, for an n-year zero-coupon bond, we have a duration of n years.

2. The duration of a coupon bond will always be less than its term to maturity

This should be again obvious from the following equation for our 3-year example bond:

Note that P_m=PV(C₁)+PV(C₂)+PV(C₃)

Thus, the duration of this bond is a weighted average of 1,2 and 3 using some weights that have to add up to 1. In this case, the weights are 0.1213, 0.1103, and 0.7685. Thus, duration for this bond is (0.1213´1)+(0.1103´2)+(0.7685´3), which has to be less than 3.

The extreme case is when the weights on the first two years are zero, and the weight on the third year is 1. This is the zero-coupon bond we discussed above, which has a duration of (0.0´1)+(0.0´2)+(1.0´3)= 3 years.

3. There is an inverse relationship between coupon rate and duration.

I have plotted below a graph of duration versus coupon rate, for our 3 year example bond below:

We can observe the inverse relationship between coupon rate and duration here. At 0% coupon (a zero-coupon bond), the duration equals the term to maturity of the bond. Intuitively, as the coupon rate increases, the present value of the coupon payments relative to the price of the bond increase, which means the “weights” on the earlier years increase relative to the weights on the last year. This decreases duration.

4. There is generally a positive relationship between duration and term to maturity

The positive relationship between duration and term to maturity can be see from the above graph. Intuitively, the longer the term to maturity, there is a positive weight on the years farther out, especially due to the face value of the debt (due in the last year) that is a large chunk of the bond’s value. Note from the diagram above that for a zero-coupon bond, the duration rises linearly with term to maturity, because for a zero-coupon bond, the duration is equal to the maturity. For coupons greater than zero, the duration increases with maturity at a decreasing rate.

5. There is an inverse relationship between yield to maturity and duration

Let us plot a graph between YTM and duration plugging in different YTM values for our 3-year example bond.

Modified Duration

- This is an adjusted measure of duration, that can be used to approximate the interest rate sensitivity of a bond’s price.

- Modified Duration is defined as: for a semi-annual coupon bond.

- This measure is useful because it can be used to approximate changes in bond price for small changes in yield.

- Let us study the mechanics and application of Modified Duration.

- Mechanics

Modified Duration is actually, D_mod =

For those who like calculus, here’s why modified duration is given by the equation given earlier. For those who don’t care about calculus, don’t worry! The derivative is not the important point.

We know that , which can be expanded to:

We can take the first derivative of this expression for P_m (a.k.a differentiate this expression) with respect to i.

which gives:

which means:

The bottomline is that, since we know how to calculate Macaulay Duration, we can calculate Modified Duration. For instance, our example 3-year bond had a duration of 2.6472 years, and an i=10%. Since this is a strange bond with yearly coupons, I have to divide Macaulay duration by (1+i), not (1+i/2). So, modified duration of this bond is given by:

- Application of modified duration

We have learned how to calculate duration. How is it used? It turns out that modified duration is very useful because it gives us an estimate of how much the bond price will change for a given interest rate change. The equation for this is the following:

Here, DP = change in bond price in response to change in yield i

Di = change in yield (interest rate), in percentage points

D_mod = modified duration of the bond, in years

P = beginning price of the bond

Let us understand this by means of an example. Consider a 5 year corporate bond with a coupon rate of 10%, and a par value of $1,000. The current yield to maturity (YTM) of this bond is 12%. We expect interest rates to decline by 90 basis points (or, 0.9%) over the next six months. We know that this will lead to an increase in the price of the bond. The question is: by how much will the bond’s price increase? Let’s do this example in two ways.

Method 1: Find the bond’s price at 12% (current YTM), and at the expected future YTM of 11.10% (after interest rate decrease), and compare the prices directly. This can be done using our basic bond pricing equation, where we plug in the following values for the variables:

C_i=$100, P_p=$1,000, n=5

Price of the bond at 12% YTM (let’s call this price P₁):

Price of the bond at 11.10% YTM (let’s call this price P₂):

958.64

Therefore, DP = P₂-P₁ = 958.64-926.40 = $32.24. In percentage terms, that’s a change of: 3.48%

Method 2: Use the modified duration to figure out the percentage price change. As a first step, let’s figure out the Macaulay Duration of this bond, using the current YTM of 12%.

(1)	(2)	(3)	(4)	(5)
Period	Cash flow	PV at YTM	PV as a	Product
			% of price	(1) x (4)
1	50	47.170	5.09%	0.0509
2	50	44.500	4.80%	0.0961
3	50	41.981	4.53%	0.1359
4	50	39.605	4.28%	0.1710
5	50	37.363	4.03%	0.2017
6	50	35.248	3.80%	0.2283
7	50	33.253	3.59%	0.2513
8	50	31.371	3.39%	0.2709
9	50	29.595	3.19%	0.2875
10	1050	586.315	63.29%	6.3290
	Sum	926.40	100.00%	8.0225

Thus, the macaulay duration of this bond is 8.0225 half-years, which is equal to (8.0225/2) = 4.0113 years. The modified duration can be calculated as:

years

Now, we can apply our formula for percentage change in bond price directly

Here, Di = - 90 basis points = - 0.90 percentage points

3.41%, which is a very close approximation to the 3.48% value we obtained in Method 1.

- Note that the modified duration results only in an approximate value for the bond price volatility, or change in bond price.

- Also note that the smaller the yield change, the better the approximation. The higher the yield change, the worse the approximation. Let’s see this in our example.

What if the expected change in bond yield were much smaller, say 15 basis points, i.e. Di==0.15%,

Now, , as before, and

Therefore, DP = P₂-P₁ = 931.68-926.40 = $5.28. In percentage terms, that’s a change of: 0.5697%

Now, our modified duration formula says that:

0.5676%, which is very, very close.

Conversely, what if the expected change in bond yield were larger, say 150 basis points, i.e. Di==1.50%?

Now, , as before, and

Therefore, DP = P₂-P₁ = 980.93-926.40 = $54.53. In percentage terms, that’s a change of: 5.89%

On the other hand, our modified duration formula says that:

5.68%, which is not as close an approximation at all.

· Convexity

- By now, we know that duration can give us only an approximate estimate of bond price volatility. To improve on the accuracy of this approximation, especially with larger bond yield changes, we need to understand convexity.

- Convexity = , which means that it is the second derivative of bond price with respect to the yield, divided by the bond price.

- In terms of computation, for a semi-annual coupon bond, the formula is:

Let us calculate convexity for our example 3-year bond with annual coupons and a coupon rate of 15%. The par value of the bond is $1,000, and the YTM of the bond is 10%.

(1)	(2)	(3)	(4)	(5)
Year	Cash flow	PV at YTM	(t^2+t)	Product
				(3) x (4)
1	150	136.36	2	272.73
2	150	123.97	6	743.80
3	1150	864.01	12	10368.14
	Price =	1124.34		11384.67
	(1+i)^2=	1.3225

Now, Convexity =

- Intuitively speaking, convexity is a measure of how convex or how curved the price-yield curve is. Some price-yield curves are more curved or convex than others.

- In particular, low coupon rate bonds and long maturity bonds have high convexities, while high coupon rate bonds and short maturity bonds have low convexities.

- Let us see this in the following graph

The zero-coupon, 10 year bond has a convexity of 87.69

The 20% coupon, 3 year bond has a convexity of 7.73

· Combined effect of duration and convexity

- It turns out that convexity can improve upon the modified duration in estimating bond price changes due to change in bond yields.

- The effect of convexity on a bond’s volatility or price change is given by:

Price change due to convexity, DP = ½ ´ Price ´ Convexity ´ (Di)²

- Recall that, from before, the price change due to modified duration is given by:

Therefore, including convexity means that we can get a better estimate of price change for a given yield change as:

- Let’s do an example from your text: An 18-year bond with a 12% coupon rate and a 9% YTM, and a par value of $100, with yearly coupon payments.

- The price of this bond is given by our familiar equation:

- The duration and convexity calculations of this bond are as follows:

(1)	(2)	(3)	(4)	(5)	(6)	(7)
Year	Cash flow	PV at YTM	% of price	Product	(t^2+t)	Product
				(1) x (4)		(3) x (4)
1	12	11.01	8.72%	0.09	2.00	22.02
2	12	10.10	8.00%	0.16	6.00	60.60
3	12	9.27	7.34%	0.22	12.00	111.19
4	12	8.50	6.73%	0.27	20.00	170.02
5	12	7.80	6.18%	0.31	30.00	233.98
6	12	7.16	5.67%	0.34	42.00	300.52
7	12	6.56	5.20%	0.36	56.00	367.61
8	12	6.02	4.77%	0.38	72.00	433.61
9	12	5.53	4.38%	0.39	90.00	497.26
10	12	5.07	4.01%	0.40	110.00	557.58
11	12	4.65	3.68%	0.41	132.00	613.85
12	12	4.27	3.38%	0.41	156.00	665.56
13	12	3.91	3.10%	0.40	182.00	712.37
14	12	3.59	2.84%	0.40	210.00	754.10
15	12	3.29	2.61%	0.39	240.00	790.67
16	12	3.02	2.39%	0.38	272.00	822.10
17	12	2.77	2.20%	0.37	306.00	848.50
18	112	23.74	18.80%	3.38	342.00	8120.21
	Sum	126.27	Duration	9.07		16081.76
			(1+i)=	1.09	(1+i)^2=	1.1881
			Modified	8.32	Convexity	107.20
			Duration

Case 1: Decline in bond yield =100 basis points (bp) = 1%

I calculated that the bond price actually increases to $137.49, when the bond yield is 8% (Do this and verify for yourself!). That means the DP, or the increase in bond price with a 100 basis point decline in yield is $11.22. Let us see how our approximation formula works.

Here, Di =1.00% = -0.01, we have

Not bad at all, but duration does most of the work. The convexity part contributes, but not too much in this case.

Case 2: Decline in bond yield =300 basis points (bp) = 3%

Once again, I calculated that the bond price actually increases to $164.97, when the bond yield is 6% (Again, do this and verify for yourself!). That means the DP, or the increase in bond price with a 300 basis point decline in yield is $38.70. Let us see how our approximation formula works.

Here, Di =3.00% = -0.03, we have

This is an OK approximation. The more important point to see is that if we had ignored the convexity effect, we would be way off at a DP of $31.52. Surely, $37.61 is a much, much better approximation than $31.52.

This is merely a repeat of what I said earlier: Estimated price changes by using only modified duration work only for very small changes in bond yield. With bigger changes, convexity or the curvilinear nature of the price-yield curve must be taken into account. That is the moral of this lecture.

Final Word

Your text contains lots of other material that is advanced. I think the material we have covered is perfectly adequate in that we have learned the tools of the bond trade. We shall make a brief stop in Chapter 17 before moving forward.