1. V: Bonds
15: Duration

2. Duration
Concept
Calculation
Duration and Price Volatility

3. Fundamental Risk Reinvestment Risk: Price 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.

4. Duration
Duration: Weighted by Net Present Value average term to maturity.
Duration can be calculated on any cash flow structure.

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

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

7. How much of my investment faces a reinvestment risk every year?
A Tale of Two Bonds
How much of my investment faces a reinvestment risk every year?
$50
$1050
$90
$1090
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8. Calculating Duration I
5 year 5% Annual Coupon Bond at 7% T
Cash Flow
NPV
NPV/P
1 year
$50
$46.73
5.09%
2 years
$43.67
4.76%
3 years
$40.81
4.45%
4 years
$38.14
4.16%
5 years
$1050
$748.64
81.55%
Total NPV =$918.00
100.00%
© Oltheten & Waspi 2012

9. A Tale of Two Bonds
How much of each bond must be reinvested after 1,2,3,4 and 5 years?
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10. 5 year 5% Annual Coupon Bond at 7%
Calculation
5 year 5% Annual Coupon Bond at 7% T
Cash Flow
NPV
NPV/P
Duration T*NPV/P
Convexity D*(T+1)
1 year
$50
$46.73
5.09%
2 years
$43.67
4.76%
3 years
$40.81
4.45%
4 years
$38.14
4.16%
5 years
$1050
$748.64
81.55%
Total NPV =$918.00
100.00%
4.523 yrs
yrs2
© Oltheten & Waspi 2012

11. A Tale of Two Bonds
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12. Duration & Price Risk
Volatility: Change in the price of the bond due to a change in market yield.

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

14. Price Yield Curve

15. Price Yield Curve
5 year 5% annual coupon
7% yield

16. Price Yield Curve
20 year 6% semi-annual coupon
8% yield

17. Calculating Duration II
Calculate the duration and convexity of a semi-annual bond
$10,000
6% coupon
December 31, 2017
Settles March 2, 2014

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

19. Calculating Duration II
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20. Exercise Calculate the duration and convexity of a semi-annual bond
$1000
6% coupon
2.5 years to maturity
Priced to yield 8%

21. 1 1/2 year 6% Semi-Annual Coupon Bond at 8%
Semi-Annual Bonds
1 1/2 year 6% Semi-Annual Coupon Bond at 8% T
Cash Flow
NPV
NPV/P
Duration T*NPV/P
Convexity D*(T+1) 1 2 3 4 5
© Oltheten & Waspi 2012

22. Volatility Duration: First derivative of the Price Yield Curve
.D = dP/dY
Modified duration is the slope of the Yield Curve
Convexity: Second derivative of the Price Yield Curve
.C = dP2/d2Y
Curvature of the Yield Curve

23. Volatility Taylor Expansion: Modified Duration Modified Convexity
Yield at which duration was calculated
Modified Duration
Modified Convexity

24. Volatility $1000 Duration: 2.355 yrs 6% semi-annual coupon
2 ½ years to maturity
Priced to Yield 8%
Duration: yrs
Modified D: = (1.04)
Convexity: yrs2
Modified C: = (1.04)2
© Oltheten & Waspi 2012

25. Δ Yield +200 basis points Duration only Convexity Correction Total
-2.355
(+.02) + 1
6.922
(+.02)2 =
(1.04) 2
(1.04)2
© Oltheten & Waspi 2012

26. Δ Yield -200 basis points Duration only Convexity Correction Total
-2.355
(-.02) + 1
6.922
(-.02)2 =
(1.04) 2
(1.04)2
© Oltheten & Waspi 2012

27. Price Yield Curve Convexity corrections are always positive
Price effect is asymmetric

28. Volatility Yields increase by 2% - 4.5288% + 0.128% = - 4.4009%
Yields decrease by 2% % % = %
Convexity corrections are always positive
Price effect is asymmetric

29. Spreadsheet Exercise
15-1
15-2

30. Bonds IV
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