Reading 53: Term Structure and Volatility of Interest Rates- center, change, butterfly, 2010, class

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Session 14: Fixed Income: Valuation Concepts
Reading 53: Term Structure and Volatility of Interest Rates

LOS a: Illustrate and explain parallel and nonparallel shifts in the yield curve, a yield curve twist, and a change in the curvature of the yield curve (i.e., a butterfly shift).

 

 

 

Suppose that there is a nonparallel downward shift in the yield curve. Which of the following best explains this phenomenon?

A) The absolute yield increase is different for some maturities.

B) The yield decrease is the same for all maturities.

C) The absolute yield decrease is different for some maturities.

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A nonparallel downward yield curve shift indicates an unequal yield decrease across all maturities, i.e., some maturity yields declined more than others.

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Key Rate Durations
Issue	Value ($1,000's)	weight	3 mo	2 yr	5 yr	10 yr	15 yr	20 yr	25 yr	30 yr	Effective Duration
Bond 1	100	0.10	0.03	0.14	0.49	1.35	1.71	1.59	1.47	4.62	11.4
Bond 2	200	0.20	0.02	0.13	1.47	0.00	0.00	0.00	0.00	0.00	1.62
Bond 3	150	0.15	0.03	0.14	0.51	1.40	1.78	1.64	2.34	2.83	10.67
Bond 4	250	0.25	0.06	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.06
Bond 5	300	0.30	0.00	0.88	0.00	0.00	1.83	0.00	0.00	0.00	2.71
Total Portfolio		1.00	0.0265	0.325	0.4195	0.345	0.987	0.405	0.498	0.8865	3.8925

Change in Portfolio Value

Change from 3-month key rate increase:	(20 bp)(0.0265)	= 0.0053% decrease
Change from 5-year key rate increase:	(90 bp)(0.4195)	= 0.3776% decrease
Change from 30-year key rate decrease:	(150 bp)(0.8865)	= 1.3298% increase
Net change		0.9469% increase

This means that the portfolio value after the yield curve shift is:

1,000,000(1 + 0.009469) = $1,009,469.00

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If the entire yield curve undergoes a parallel shift such that the rate at all key maturities increases by 50 basis points, what will the value of the portfolio be?

A) $980,537.50.

B) $1,019,462.50.

C) $961,075.00.

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Key Rate Durations
Issue	Value ($1,000's)	weight	3 mo	2 yr	5 yr	10 yr	15 yr	20 yr	25 yr	30 yr	Effective Duration
Bond 1	100	0.10	0.03	0.14	0.49	1.35	1.71	1.59	1.47	4.62	11.4
Bond 2	200	0.20	0.02	0.13	1.47	0.00	0.00	0.00	0.00	0.00	1.62
Bond 3	150	0.15	0.03	0.14	0.51	1.40	1.78	1.64	2.34	2.83	10.67
Bond 4	250	0.25	0.06	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.06
Bond 5	300	0.30	0.00	0.88	0.00	0.00	1.83	0.00	0.00	0.00	2.71
Total Portfolio		1.00	0.0265	0.325	0.4195	0.345	0.987	0.405	0.498	0.8865	3.8925

Since the yield curve underwent a parallel shift, the impact on portfolio value can be computed directly using the portfolio's effective duration. There are two methods that can be used to calculate effective duration in this situation. Both methods use the market weight of the individual bonds in the portfolio. As shown in the third column of the table above, the market weight of each bond equals: Bond value/Portfolio value, where the portfolio value is $1,000,000.

Method 1) Effective duration of the portfolio is the sum of the weighted averages of the key rate durations for each issue.

The 3-month key rate durations for the portfolio can be calculated as follows:

(0.10)(0.03) + (0.20)(0.02) + (0.15)(0.03) + (0.25)(0.06) + (0.30)(0) = 0.0265

This method can be used to generate the rest of the key rate duration shown in the bottom row of the table above and summed to yield an effective duration = 3.8925.

Method 2) Effective duration of the portfolio is the weighted average of the effective durations for each issue. The effective duration of each issue is the sum of the individual rate durations for that issue. These values are shown in the right-hand column of the table above. Using this approach, the effective duration of the portfolio can be computed as:

(0.10)(11.4) + (0.20)(1.62) + (0.15)(10.67) + (0.25)(0.06) + (0.30)(2.71) = 3.8925

Using an effective duration of 3.8925, the value of the portfolio following a parallel 50 basis point shift in the yield curve is computed as follows:

Percentage change = (50 basis points)(3.8925) = 1.9463% decrease

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What is the value of Bond 4 if 3-month rates remain constant and all other rates increase by 135 basis points?

A) $250,000.00.

B) $229,750.00.

C) $243,375.00.

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Key Rate Durations
Issue	Value ($1,000's)	weight	3 mo	2 yr	5 yr	10 yr	15 yr	20 yr	25 yr	30 yr	Effective Duration
Bond 1	100	0.10	0.03	0.14	0.49	1.35	1.71	1.59	1.47	4.62	11.4
Bond 2	200	0.20	0.02	0.13	1.47	0.00	0.00	0.00	0.00	0.00	1.62
Bond 3	150	0.15	0.03	0.14	0.51	1.40	1.78	1.64	2.34	2.83	10.67
Bond 4	250	0.25	0.06	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.06
Bond 5	300	0.30	0.00	0.88	0.00	0.00	1.83	0.00	0.00	0.00	2.71
Total Portfolio		1.00	0.0265	0.325	0.4195	0.345	0.987	0.405	0.498	0.8865	3.8925

Since the 3-month rate did not change, and all other key rate durations for Bond 4 are zero, a 135 basis points change will have no effect on the value of the bond. Hence, Bond 4 remains valued at $250,000.00.

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Suppose that there is a parallel upward shift in the yield curve. Which of the following best explains this phenomenon? The yield:

A) decrease is the same for all maturities.

B) increase is the same for all maturities.

C) increase is proportional to the original level for all maturities.

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A parallel upward shift indicates an equal yield increase across all maturities.

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A yield curve is flat, and then it undergoes a non-parallel shift. After the shift, which of the following must be FALSE? The new yield curve is:

A) curvilinear.

B) flat.

C) a straight line.

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If a yield curve begins flat and then experiences a non-parallel shift, this means that some rates changed more than others. After the non-parallel shift the formerly flat yield curve can no longer be flat.

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Which of the following is TRUE if there is a positive butterfly shift in the yield curve?

A) The curvature of the yield curve increases.

B) The yield curve becomes less humped at intermediate maturities.

C) The curvature of the yield curve decreases.

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A butterfly shift occurs when yields increase (decrease), the yields in the short maturity and long maturity sectors increase more (less) than the yields in the intermediate maturity sector.

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Which of the following is TRUE if there is a twist in the yield curve?

A) The curvature of the yield curve increases.

B) The yield curve flattens or steepens.

C) The yield curve becomes humped at intermediate maturities.

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Twists refer to yield curve changes when the slope becomes either flatter or more steep. A flattening (steepening) of the yield curve means that the spread between short- and long-term rates has narrowed (widened).

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With respect to yield curve, a negative butterfly shift means that the yield curve has become:

A) more curved.

B) negatively sloped for all regions.

C) flat.

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By definition, a negative butterfly shift means the curve has become more curved or “humped.” Such a shift could lead to an increase in slope in some regions and a decrease in slope in other regions.

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Suppose the yield curve becomes steeper. Which of the following is a consequence of the steepening?

A) Long-term bonds become less sensitive to interest rate changes.

B) Long-term bonds become more sensitive to interest rate changes.

C) The yield spread between long and short-term securities increases.

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This is by definition. A steepening yield curve means that the slope of the yield curve increases. The slope is the difference (i.e. the term spread) between the yields of two maturities. Consequently, as the yield curve steepens this spread increases.