mouse123

A non-callable bond with 4 years remaining maturity has an annual coupon of 12% and a $1,000 par value. The current price of the bond is $1,063.40. Given a change in yield of 50 basis points, which of the following is closest to the effective duration of the bond?

A) 2.94.

B) 3.27.

C) 3.11.

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First, find the current yield to maturity of the bond as:

FV = $1,000; PMT = $120; N = 4; PV = –$1,063.40; CPT → I/Y = 10%

Then compute the price of the bond if rates rise by 50 basis points to 10.5% as:

FV = $1,000; PMT = $120; N = 4; I/Y = 10.5%; CPT → PV = –$1,047.04

Then compute the price of the bond if rates fall by 50 basis points to 9.5% as:

FV = $1,000; PMT = $120; N = 4; I/Y = 9.5%; CPT → PV = –$1,080.11

The formula for effective duration is:

(V-–V+) / (2V0Δy)

Therefore, effective duration is:

($1,080.11 – $1,047.04) / (2 × $1,063.40 × 0.005) = 3.11

mouse123

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) 8.24.

B) 9.63.

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.

mouse123

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.

mouse123

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) 6.8.

C) 6.5.

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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:

mouse123

A noncallable bond with seven years remaining to maturity is trading at 108.1% of a par value of $1,000 and has an 8.5% coupon. If interest rates rise 50 basis points, the bond’s price will fall to 105.3% and if rates fall 50 basis points, the bond’s price will rise to 111.0%. Which of the following is closest to the effective duration of the bond?

A) 6.12.

B) 5.54.

C) 5.27.

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The formula for effective duration is: (V- – V+) / (2V0Δy). Therefore, effective duration is: ($1.110 – $1.053) / (2 × $1.081 × 0.005) = 5.27.

mouse123

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 = 94.12The 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.

mouse123

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.

mouse123

The price of a bond is equal to $101.76 if the term structure of interest rates is flat at 5%. The following bond prices are given for up and down shifts of the term structure of interest rates. Using the following information what is the effective duration of the bond?

Bond price: $98.46 if term structure of interest rates is flat at 6%
Bond price: $105.56 if term structure of interest rates is flat at 4%

A) 1.56.

B) 3.49.

C) 1.74.

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The effective duration is computed as follows:

Effective duration =	105.56 − 98.46	= 3.49
	2 × 101.76 × 0.01

mouse123

Assume that the current price of a bond is 102.50. If interest rates increase by 0.5% the value of the bond decreases to 100 and if interest rates decrease by 0.5% the price of the bond increases to 105.5. What is the effective duration of the bond?

A) 5.37.

B) 5.50.

C) 5.48.

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The duration is computed as follows:

Duration =	105.50 − 100	= 5.37
	2 × 102.50 × 0.005

mouse123

Negative convexity for a callable bond is most likely to be important when the:

A) price of the bond approaches the call price.

B) bond is first issued.

C) market interest rate rises above the bond's coupon rate.

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Negative convexity illustrates how the relationship between the price of a bond and market yields changes as the bond price rises and approaches the call price. The convex curve that we generally see for non-callable bonds bends backward to become concave (i.e., exhibit negative convexity) as the bond approaches the call price.