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V: Bonds 15: Duration

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Chapter 15: Duration © Oltheten & Waspi 2012 Duration Concept Calculation Duration and Price Volatility

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Chapter 15: Duration © Oltheten & Waspi 2012 Fundamental Risk Reinvestment Risk: The risk that coupons, paid out of the bond, cannot be reinvested at the same YTM. Price Risk: The risk that the price of the bond will fall Note that this is a risk only if we sell the bond before it matures. There is no price risk if we hold the bond to maturity.

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Chapter 15: Duration © Oltheten & Waspi 2012 Duration Duration: Weighted by Net Present Value average term to maturity. Duration can be calculated on any cash flow structure.

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Chapter 15: Duration A Tale of Two Bonds $1000 5% Annual Coupon $50 Price: $918.00 $1000 9% Annual Coupon $90 Price: $1,082.00 © Oltheten & Waspi 2012

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Chapter 15: Duration A Tale of Two Bonds © Oltheten & Waspi 2012 Income Yield: 5.45% Income Yield: 8.32% Capital Gain 1.73% Capital Gain -1.56% Yield to Maturity: 7.00% 5 year capital gain = 8.93% Annual capital gain = 1.73% 5 year capital gain = - 7.58% Annual capital gain = - 1.56%

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Chapter 15: Duration A Tale of Two Bonds $50 $1050$50$90 $1090$90 How much of my investment faces a reinvestment risk every year? © Oltheten & Waspi 2012

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Chapter 15: Duration Calculating Duration I 5 year 5% Annual Coupon Bond at 7% TCash Flow NPVNPV/P 1 year$50$46.735.09% 2 years$50$43.674.76% 3 years$50$40.814.45% 4 years$50$38.144.16% 5 years$1050$748.6481.55% Total NPV =$918.00100.00% © Oltheten & Waspi 2012

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Chapter 15: Duration © Oltheten & Waspi 2012 A Tale of Two Bonds How much of each bond must be reinvested after 1,2,3,4 and 5 years? © Oltheten & Waspi 2012

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Chapter 15: Duration Calculation 5 year 5% Annual Coupon Bond at 7% TCash Flow NPVNPV/PDuration T*NPV/P Convexity D*(T+1) 1 year$50$46.735.09%.050903.101806 2 years$50$43.674.76%.095146.285439 3 years$50$40.814.45%.133383.533530 4 years$50$38.144.16%.166209.831044 5 years$1050$748.6481.55%4.07755324.465317 Total NPV =$918.00100.00%4.523 yrs26.217 yrs 2 © Oltheten & Waspi 2012

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Chapter 15: Duration A Tale of Two Bonds © Oltheten & Waspi 2012

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Chapter 15: Duration © Oltheten & Waspi 2012 Duration & Price Risk Volatility: Change in the price of the bond due to a change in market yield.

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Chapter 15: Duration Duration & Volatility 5% annual bond: 4.523 yrs * 1% = 4.227% 1.07 Modified Duration is 4.227 years If Yd 1% then P 4.227% 9% annual bond: 4.272 yrs * 1% = 3.993% 1.07 Modified Duration is 3.993 years If Yd 1% then P 3.993% © Oltheten & Waspi 2012

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Chapter 15: Duration © Oltheten & Waspi 2012 Price Yield Curve

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Chapter 15: Duration © Oltheten & Waspi 2012 Price Yield Curve 5 year 5% annual coupon 7% yield

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Chapter 15: Duration © Oltheten & Waspi 2012 Price Yield Curve 20 year 6% semi-annual coupon 8% yield

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Chapter 15: Duration © Oltheten & Waspi 2012 Calculating Duration II Calculate the duration and convexity of a semi-annual bond $10,000 6% coupon December 31, 2017 Settles March 2, 2014 102.000

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Chapter 15: Duration Calculating Duration II Base Price: 62/180 days Accrued Interest: Invoice Price: YTM: 5.41186% $10,200.00 $103.33 $10,303.33 © Oltheten & Waspi 2012

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Chapter 15: Duration Calculating Duration II © Oltheten & Waspi 2012

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Chapter 15: Duration © Oltheten & Waspi 2012 Exercise Calculate the duration and convexity of a semi-annual bond $1000 6% coupon 2.5 years to maturity Priced to yield 8%

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Chapter 15: Duration Semi-Annual Bonds 1 1/2 year 6% Semi-Annual Coupon Bond at 8% TCash Flow NPVNPV/PDuration T*NPV/P Convexity D*(T+1) 1 2 3 4 5 © Oltheten & Waspi 2012

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Chapter 15: Duration © Oltheten & Waspi 2012 Volatility Duration: First derivative of the Price Yield Curve.D = dP/dY Slope of the Yield Curve Convexity: Second derivative of the Price Yield Curve.C = dP2/d2Y Curvature of the Yield Curve

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Chapter 15: Duration © Oltheten & Waspi 2012 Volatility Taylor Expansion: Modified Duration Modified Convexity Yield at which duration was calculated

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Chapter 15: Duration Volatility $1000 6% semi-annual coupon 2 ½ years to maturity Priced to Yield 8% Duration: 2.355 yrs Modified D: 2.355 =2.264 (1.04) Convexity:6.922 yrs 2 Modified C:6.922 = 6.400 (1.04) 2 © Oltheten & Waspi 2012

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Chapter 15: Duration Δ Yield +200 basis points Duration onlyConvexity Correction Total -2.355(+.02)+16.922(+.02) 2 = (1.04)2(1.04) 2 © Oltheten & Waspi 2012

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Chapter 15: Duration Δ Yield -200 basis points Duration onlyConvexity Correction Total -2.355(-.02)+16.922(-.02) 2 = (1.04)2(1.04) 2 © Oltheten & Waspi 2012

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Chapter 15: Duration © Oltheten & Waspi 2012 Price Yield Curve Convexity corrections are always positive Price effect is asymmetric

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Chapter 15: Duration © Oltheten & Waspi 2012 Volatility Yields increase by 2% - 4.5288% + 0.128% = - 4.4009% Yields decrease by 2% + 4.5288% + 0.128% = + 4.46568% Convexity corrections are always positive Price effect is asymmetric

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Chapter 15: Duration © Oltheten & Waspi 2012 Spreadsheet Exercise 15-1 15-2

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Bonds IV © Oltheten & Waspi 2012

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