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

V: Bonds

15: Duration

Chapter 15: Duration © Oltheten & Waspi 2012

Duration

Concept Calculation Duration and Price Volatility


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.


Duration

Duration:Weighted by Net Present Value average term to maturity. Duration can be calculated

on any cash flow structure.


A Tale of Two Bonds

$10005% Annual Coupon

$50 $50 $50 $50 $50

Price: $918.00

$10009% Annual Coupon

$90 $90 $90 $90 $90

Price: $1,082.00


A Tale of Two Bonds

5

1t5t

1.07

1

$1000

1.07

1

$90$1082.00

5

1t5t

1.07

1

$1000

1.07

1

$50$918.00

Income Yield: 5.45%

Income Yield: 8.32%Capital Gain 1.73%

Capital Gain -1.56%

Yield to Maturity: 7.00%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%


A Tale of Two Bonds

$50 $50 $50 $1050$50

$90 $90 $90 $1090$90

How much of my investment faces a reinvestment risk every year?


Calculating Duration I

5 year 5% Annual Coupon Bond at 7%

T CashFlow

NPV NPV/P

1 year $50 $46.73 5.09%

2 years $50 $43.67 4.76%

3 years $50 $40.81 4.45%

4 years $50 $38.14 4.16%

5 years $1050 $748.64 81.55%

Total NPV =$918.00 100.00%


A Tale of Two Bonds

How much of each bond must be reinvested after 1,2,3,4 and 5 years?

© Oltheten & Waspi 2012

7.8% 7.3%

6.8%

6.3%71.8

%

9% Bond5.1%4.8% 4.4

%4.2%

81.6%

5% Bond


Calculation

5 year 5% Annual Coupon Bond at 7%

T CashFlow

NPV NPV/P DurationT*NPV/P

ConvexityD*(T+1)

1 year $50 $46.73 5.09% .050903 .101806

2 years $50 $43.67 4.76% .095146 .285439

3 years $50 $40.81 4.45% .133383 .533530

4 years $50 $38.14 4.16% .166209 .831044

5 years $1050 $748.64 81.55% 4.077553 24.465317

Total NPV =$918.00 100.00% 4.523 yrs 26.217 yrs2


A Tale of Two Bonds


Duration & Price Risk

Volatility:Change in the price of the bond due to a change in market yield.

Yd Δ*Yd1

Duration

Price

Price ΔVolatility


Duration & Volatility

5% annual bond: 4.523 yrs * 1% =

4.227%1.07

Modified Duration is 4.227 years If Yd1%

then P4.227% If Yd 1%

then P4.227%

9% annual bond: 4.272 yrs * 1% =

3.993%1.07

Modified Duration is 3.993 years If Yd1%

then P3.993% If Yd 1%

then P3.993%


Price Yield Curve

60

70

80

90

100

110

120

130

140

150

1% 3% 5% 7% 9% 11% 13%


Price Yield Curve

70

80

90

100

110

120

1% 3% 5% 7% 9% 11% 13%

5 year 5% annual coupon 7% yield


Price Yield Curve

0

25

50

75

100

125

150

175

200

1% 3% 5% 7% 9% 11% 13% 15%

20 year 6% semi-annual coupon 8% yield


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



Base Price: 62/180 days Accrued Interest: Invoice Price:

YTM: 5.41186%

$10,200.00 $103.33

$10,303.33

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%


Semi-Annual Bonds

1 1/2 year 6% Semi-Annual Coupon Bond at 8%

T CashFlow

NPV NPV/P DurationT*NPV/P

ConvexityD*(T+1)

1

2

3

4

5


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


22 y

2y

1

Convexity21

y

2y

1

DurationPP

ΔΔΔ

Volatility

Taylor Expansion:

Modified Duration Modified Convexity

Yield at which duration was calculated


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 yrs2

Modified C: 6.922 = 6.400(1.04)2


Δ Yield +200 basis points

Duration only Convexity Correction

Total

-2.355 (+.02) + 1 6.922 (+.02)2 =

(1.04) 2 (1.04)2


Δ Yield -200 basis points

Duration only Convexity Correction

Total

-2.355 (-.02) + 1 6.922 (-.02)2 =

(1.04) 2 (1.04)2


Price Yield Curve

0

25

50

75

100

125

150

175

200

1% 3% 5% 7% 9% 11% 13% 15%

Convexity corrections are always positive

Price effect is asymmetric


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


Spreadsheet Exercise

15-1 15-2

Bonds IV© Oltheten & Waspi 2012

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

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