Reading 66: Introduction to the Measurement of Interest Rate 2010, effective, interpret, decrease, increase

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LOS d Compute 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) 6.0.

C) 5.0.

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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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What happens to bond durations when coupon rates increase and maturities increase?

       As coupon rates increase, duration:           As maturities increase, duration:

A) increases  increases

B) decreases   increases

C) decreases   decreases

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As coupon rates increase the duration on the bond will decrease because investors are recieving more cash flow sooner. As maturity increases, duration will increase because the payments are spread out over a longer peiod of time.

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A non-callable bond with 10 years remaining maturity has an annual coupon of 5.5% and a $1,000 par value. The current yield to maturity on the bond is 4.7%. Which of the following is closest to the estimated price change of the bond using duration if rates rise by 75 basis points?

A) -$47.34.

B) -$5.68.

C) -$61.10.

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First, compute the current price of the bond as: FV = 1,000; PMT = 55; N = 10; I/Y = 4.7; CPT → PV = –1,062.68. Then compute the price of the bond if rates rise by 75 basis points to 5.45% as: FV = 1,000; PMT = 55; N = 10; I/Y = 5.45; CPT → PV = –1,003.78. Then compute the price of the bond if rates fall by 75 basis points to 3.95% as: FV = 1,000; PMT = 55; N = 10; I/Y = 3.95; CPT → PV = –1,126.03.

The formula for effective duration is: (V_-–V₊) / (2V₀Δy). Therefore, effective duration is: ($1,126.03 – $1,003.78) / (2 × $1,062.68 × 0.0075) = 7.67.

The formula for the percentage price change is then: –(duration)(Δy). Therefore, the estimated percentage price change using duration is: –(7.67)(0.75%) = –5.75%. The estimated price change is then: (–0.0575)($1,062.68) = –$61.10

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An $850 bond has a modified duration of 8. If interest rates fall 50 basis points, the bond's price will:

A) increase by $34.00.

B) increase by $4.00.

C) increase by 22.5%.

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ΔP = (-)(P)(MD)(Δi)

ΔP = (-)(8)(850)(-0.005) = +$34

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A bond has the following characteristics:

• Modified duration of 18 years
• Maturity of 30 years
• Effective duration of 16.9 years
• Current yield to maturity is 6.5%

If the market interest rate decreases by 0.75%, what will be the percentage change in the bond's price?

A) +12.675%.

B) 0.750%.

C) -12.675%.

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Approximate percentage price change of a bond = (-)(effective duration)(Δy)

= (-16.9)(-0.75%) = +12.675%

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Par value bond XYZ has a modified duration of 5. Which of the following statements regarding the bond is TRUE? If the market yield:

A) increases by 1% the bond's price will increase by $50.

B) increases by 1% the bond's price will decrease by $50.

C) increases by 1% the bond's price will decrease by $60.

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Approximate percentage price change of a bond = (-)(Duration)(Δy)

(-5)(1%) = -5%

($1000)(-0.05) = –$50

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Given a bond with a modified duration of 1.93, if required yields increase by 50 basis points, the expected percentage price change would be:

A) -1.025%.

B) -0.965%.

C) 1.000%.

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Approximate percentage price change of a bond = (-)(duration)(Δ y)

(-1.93)(0.5%) = -0.965%

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A non-callable bond has an effective duration of 7.26. Which of the following is the closest to the approximate price change of the bond with a 25 basis point increase in rates using duration?

A) -0.018%.

B) 1.820%.

C) -1.820%.

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The formula for the percentage price change is: –(duration)(Δy). Therefore, the estimated percentage price change using duration is: –(7.26)(0.25%) = –1.82%.

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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 approximate percentage price change of the bond using effective duration and assuming interest rates decrease by 0.5%?

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) 0.174%.

B) 1.74%.

C) 0.0087%.

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The effective duration is computed as follows:

Using the effective duration, the approximate percentage price change of the bond is computed as follows:

Percent price change = -3.49 × (-0.005) × 100 = 1.74%

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A bond with a semi-annually coupon rate of 3% sells for $850. It has a modified duration of 10 and is priced at a yield to maturity (YTM) of 8.5%. If the YTM increases to 9.5%, the predicted change in price, using the duration concept decreases by:

A) $85.00.

B) $77.56.

C) $79.92.

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Approximate percentage price change of a bond = (-)(duration)(Δy)

Δy = 9.5% ? 8.5% = 1%

(-10)(1%) = -10%

($850)(-0.1) = -$85