Reading 67: Introduction to the Measurement of Interest Rate 2011, effective, interpret, decrease, increase

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Session 16: Fixed Income: Analysis and Valuation
Reading 67: Introduction to the Measurement of Interest Rate Risk

LOS d: Calculate and interpret the effective duration of a bond, given information about how the bond's price will increase and decrease for given changes in interest rates.

 

 

A bond's yield to maturity decreases from 8% to 7% and its price increases by 6%, from $675.00 to $715.50. The bond's effective duration is closest to:

A) 7.0.

B) 5.0.

C) 6.0.

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Effective duration is the percentage change in price for a 1% change in yield, which is given as 6.

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An investor finds that for every 1% increase in interest rates, a bond’s price decreases by 4.21% compared to a 4.45% increase in value for every 1% decline in interest rates. If the bond is currently trading at par value, the bond’s duration is closest to:

A) 4.33.

B) 8.66.

C) 43.30.

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Duration is a measure of a bond’s sensitivity to changes in interest rates.

Duration = (V_- – V₊) / [2V₀(change in required yield)] where:

V_- = estimated price if yield decreases by a given amount
V₊ = estimated price if yield increases by a given amount
V₀ = initial observed bond price

Thus, duration = (104.45 – 95.79)/(2 × 100 × 0.01) = 4.33. Remember that the change in interest rates must be in decimal form.

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An international bond investor has gathered the following information on a 10-year, annual-pay U.S. corporate bond:

• Currently trading at par value
• Annual coupon of 10%
• Estimated price if rates increase 50 basis points is 96.99%
• Estimated price is rates decrease 50 basis points is 103.14%

The bond’s duration is closest to:

A) 3.14.

B) 6.15.

C) 6.58.

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Duration is a measure of a bond’s sensitivity to changes in interest rates.

Duration = (V_- ? V₊) / [2V₀(change in required yield)] where:

V_- = estimated price if yield decreases by a given amount
V₊ = estimated price if yield increases by a given amount
V₀ = initial observed bond price

Thus, duration = (103.14 ? 96.99) / (2 × 100 × 0.005) = 6.15. Remember that the change in interest rates must be in decimal form.

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When interest rates increase, the duration of a 30-year bond selling at a discount:

A) decreases.

B) increases.

C) does not change.

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The higher the yield on a bond the lower the price volatility (duration) will be. When interest rates increase the price of the bond will decrease and the yield will increase because the current yield = (annual cash coupon payment) / (bond price). As the bond price decreases the yield increases and the price volatility (duration) will decrease.

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A bond with a yield to maturity of 8.0% is priced at 96.00. If its yield increases to 8.3% its price will decrease to 94.06. If its yield decreases to 7.7% its price will increase to 98.47. The effective duration of the bond is closest to:

A) 2.75.

B) 7.66.

C) 4.34.

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The change in the yield is 30 basis points.

Duration = (98.47 ? 94.06) / (2 × 96.00 × 0.003) = 7.6563.

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A non-callable bond with 18 years remaining maturity has an annual coupon of 7% and a $1,000 par value. The current yield to maturity on the bond is 8%. Which of the following is closest to the effective duration of the bond?

A) 9.63.

B) 8.24.

C) 11.89.

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First, compute the current price of the bond as:

FV = $1,000; PMT = $70; N = 18; I/Y = 8%; CPT → PV = –$906.28

Next, change the yield by plus-or-minus the same amount. The amount of the change can be any value you like. Here we will use ±50 basis points.

Compute the price of the bond if rates rise by 50 basis points to 8.5% as:

FV = $1,000; PMT = $70; N = 18; I/Y = 8.5%; CPT → PV = –$864.17

Then compute the price of the bond if rates fall by 50 basis points to 7.5% as:

FV = $1,000; PMT = $70; N = 18; I/Y = 7.5%; CPT → PV = –$951.47

The formula for effective duration is:

(V_- – V₊) / (2V₀Δy)

Therefore, effective duration is:

($951.47 – $864.17) / (2 × $906.28 × 0.005) = 9.63.

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Calculate the effective duration for a 7-year bond with the following characteristics:

• Current price of $660
• A price of $639 when interest rates rise 50 basis points
• A price of $684 when interest rates fall 50 basis points

A) 3.1.

B) 6.5.

C) 6.8.

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The formula for calculating the effective duration of a bond is:

where:

• V_- = bond value if the yield decreases by ?y
• V₊ = bond value if the yield increases by ?y
• V₀ = initial bond price
• ?y = yield change used to get V_- and V₊, expressed in decimal form

The duration of this bond is calculated as:

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Consider an annual coupon bond with the following characteristics:

• Face value of $100
• Time to maturity of 12 years
• Coupon rate of 6.50%
• Issued at par
• Call price of 101.75 (assume the bond price will not exceed this price)

For a 75 basis point change in interest rates, the bond's duration is:

A) 5.09 years.

B) 8.79 years.

C) 8.17 years.

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Since the bond has an embedded option, we will use the formula for effective duration. (This formula is the same as the formula for modified duration, except that the “upper price bound” is replaced by the call price.) Thus, we only need to calculate the price if the yield increases 75 basis points, or 0.75%.

Price if yield increases 0.75%: FV = 100; I/Y = 6.50 + 0.75 = 7.25; N = 12; PMT = 6.5; CPT → PV = 94.12

The formula for effective duration is

Where:
V-	= call price/price ceiling
V+	= estimated price if yield increases by a given amount, Dy
V0	= initial observed bond price
Dy	= change in required yield, in decimal form

Here, effective duration = (101.75 – 94.12) / (2 × 100 × 0.0075) = 7.63 / 1.5 = 5.09 years.