1. More on Duration & Convexity
Class 7, Chap 9 - Appendix B

2. Lecture Outline
Purpose: Gain a deeper understanding of duration and its properties and weaknesses
Properties of duration
Hedging with duration
Weaknesses of duration
Convexity

3. Properties of Duration
Duration increases with maturity but at a decreasing rate
Duration decreases as the yield to maturity increases
Duration decreases as the coupon payments or interest rate increases

4. 1. Duration increases with maturity at a decreasing rate
Calculate the duration for bonds of several maturities with an 8% coupon paid semiannually,
$1,000 face value and yield to maturity of 12%.
Adding a year means:
The big payment occurs 1 year later
Adds 1 year to the weighted average
Because there is not a lot of discounting, the weight on the additional year is large
So when you go from one year to two years the largest PV and thus the largest weight is still on the last payment. So when it comes to duration, you are adding another year and you are giving that year the largest weight.
925.60
37.74
35.60
33.58
823.78
When we add a year to a long maturity bond it changes the duration much less than when we add a year to a short maturity bond

5. 1. Duration increases with maturity at a decreasing rate
Calculate the duration for bonds of several maturities with an 8% coupon paid semiannually,
$1,000 face value and yield to maturity of 12%.
Adding a year means:
The big payment occurs 1 year later
Adds 1 year to the weighted average
There is a lot of discounting so the weight on the additional year is small compared to other years
On the longer end of the curve, you can see that the present value of the later payments are relatively small. This means that most of the duration weight is given to earlier years (payment periods) and that the duration is already well established. Therefore, increasing the maturity by one year will not change the duration much.
Remember duration tell us, on average, when do we receive the full value of the bond. At these larger maturities, the later payments are highly discounted. Therefore, their values are relatively small. The interpretation is that we have already received most of the bond value early on. So, playing with these later payments does not affect the bond duration very much because they represent only a small fraction of the bond value.
37.74
35.60
1.53
33.42
1.44
37.74
35.60
1.36
31.53
1.29
When we add a year to a long maturity bond it changes the duration much less than when we add a year to a short maturity bond

6. 1. Duration increases with maturity at a decreasing rate
Lets just look at what happens to the present value of cash flows as the maturity increases
1,040 40
Time to Maturity = 5 years
Duration = 4.14

7. 1. Duration increases with maturity at a decreasing rate
Lets just look at what happens to the present value of cash flows as the maturity increases
1,040 40
Time to Maturity = 10 years
Duration = 6.61

8. 1. Duration increases with maturity at a decreasing rate
Lets just look at what happens to the present value of cash flows as the maturity increases
1,040 40
Time to Maturity = 15 years
Duration = 7.91

9. 1. Duration increases with maturity at a decreasing rate
Lets just look at what happens to the present value of cash flows as the maturity increases
1,040 40
Time to Maturity = 20 years
Duration = 8.53

10. 1. Duration increases with maturity at a decreasing rate
Lets just look at what happens to the present value of cash flows as the maturity increases
Duration = 8.53
1,040 40
Time to Maturity = 20 years
A large percent of the bond value has been received early-on !!!
86% of my value has already been received by year 19.5 and that is not going to change. So, 86% of my duration is also fixed and will not change. So, if I extend it out a little more (one more year) the best I can do is try to play with the remaining 14%
Total weight (sum) = 86%
Total weight (sum) = 75%
Total weight (sum) = 48%

11. 1. Duration increases with maturity at a decreasing rate
Conclusion:
Duration increases with maturity but at a decreasing rate because of two effects:
Increasing the maturity adds more years to the bond, which increases duration
As we increase the time to maturity (TTM), a smaller and smaller fraction of bond value is being received at a later date. This is because later payments are highly discounted. As a result, a large fraction of bond value is received early on, which stabilizes the duration.

12. 2. Duration decreases as YTM ↑
Lets just look at what happens to the present value of cash flows as the YTM increases
1,000

13. 2. Duration decreases as YTM ↑
Lets just look at what happens to the present value of cash flows as the YTM increases
1,000
YTM = 10%
Duration = 4.18

14. 2. Duration decreases as YTM ↑
Lets just look at what happens to the present value of cash flows as the YTM increases
1,000
YTM = 30%
Duration = 3.74

15. 2. Duration decreases as YTM ↑
Lets just look at what happens to the present value of cash flows as the YTM increases
1,000
YTM = 50%
Duration = 3.23

16. 2. Duration decreases as YTM ↑
Lets just look at what happens to the present value of cash flows as the YTM increases
1,000
YTM = 70%
Duration = 2.71

17. 2. Duration decreases as YTM ↑
Lets just look at what happens to the present value of cash flows as the YTM increases
1,000
YTM = 90%
Duration = 2.26

18. 2. Duration decreases as YTM ↑
Lets just look at what happens to the present value of cash flows as the YTM increases
1,000
As we increase the yield to maturity, the present value (and as a result the duration weights) of the earlier payments increase relative to the PV (duration weights) of the later payments
That is, the percentage of value [PV(future cash flows)] received early in the bond’s life increases – so the later payments (more interest rate sensitive) are not as important

19. 3. Duration Decreases as the coupon ↑
Again, this has to do with how the present value of cash flows is distributed over time and how that changes when we change the coupon rate.
1,000
Coupon = 10%
Duration = 4.04

20. 3. Duration Decreases as the coupon ↑
Again, this has to do with how the present value of cash flows is distributed over time and how that changes when we change the coupon rate.
1,000
Coupon = 40%
Duration = 3.29

21. 3. Duration Decreases as the coupon ↑
Again, this has to do with how the present value of cash flows is distributed over time and how that changes when we change the coupon rate.
1,000
Coupon = 70%
Duration = 3.07

22. 3. Duration Decreases as the coupon ↑
Again, this has to do with how the present value of cash flows is distributed over time and how that changes when we change the coupon rate.
1,000
Coupon = 100%
Duration = 2.97

23. 3. Duration Decreases as the coupon ↑
Again, this has to do with how the present value of cash flows is distributed over time and how that changes when we change the coupon rate.
As we increase the coupon rate the present value of early cash flows (duration weights) increases relative to later payments
That is, the percentage of value [PV(future cash flows)] received early in the bonds life increases – so the later payments (more interest rate sensitive) are not as important

24. What do you need to know
Duration increases with maturity but at a decreasing rate
Duration decreases as the yield to maturity increases
Duration decreases as the coupon payments increase
You need to have a basic understanding of why duration behaves this way

25. Hedge With Duration

26. Duration and interest rate risk
We have seen that duration measures the sensitivity of assets to changes in interest rates
Now lets see how we can use that to manage interest rate risk
Basic idea: by taking an offsetting position in an asset/liability with a matched duration an investor can hedge interest rate risk

27. Hedging a single asset Suppose a company has 5 years left on a loan:
The company wants to pay back the loan today but there are stiff prepayment penalties. So, the company decides to offset the loan with another asset.
The loan is a balloon payment loan - it is paid back in one lump sum payment in five years – no interim interest payments
Current value of the loan is $1,000 at 8% = $ due in 5 years
The company wants to hedge against changes in interest rates and can choose from the following instruments:
A 3 year 3% coupon bond with $1,000 face value
A five year zero coupon bond with an 8% YTM and face value = 1000
A six year bond with an 8% coupon paid annually and face value = 1,000 and YTM = 8%

28. How to choose?
The company can manage its interest rate risk by matching durations
The duration of the 3 year bond will definitely be too short
The five year zero coupon bond has a duration of 5 years
The six year coupon can not be ruled out so we need to calculate the duration

29. Duration of the 6 year bond
Step#1 Find the coupon
Coupon = (1,000)*.08 = $80
Draw the cash flows
Step#2 Find present values
Step#4 Find duration weights
Step#5 Find duration
1,000
The 6 year bond is also a viable option for the hedge

30. Hedge with the zero coupon bond
The company will owe in 5 years
So the company wants to receive $ (for sure) in 5 years to be completely hedged
Each bond pays 1000 in 5 years so they need to buy /1000 = zero coupon bonds
Cost:
The price of the zero coupon = 1000/(1.08)5 =
The company needs of them so the total cost is ( )(680.58) = $1,000
The full amount of their loan

31. Hedge with the zero coupon bond
The company is perfectly hedged!!!!
After purchasing the zero coupon bonds, the company has locked-in a positive cash flow in five years no matter what interest rates do!!!
1,469.33
- 1,469.33
Bond
Loan

32. Hedging with the 6 year bond
We saw that the 6 year bond had a duration of 5 years so lets try using it to hedge.
To hedge the company can buy one 5 year duration bond for a cost of $1,000
Consider three cases :
The YTM stays at 8%
The YTM instantaneously increases to 9%
The YTM instantaneously decreases to 7%
Why 1 bond? – if we find the value of all cash flows at time 5 years (1000)(1.085) =$1,
If this was not the case, we would need to buy more or less than one bond
But if this was not the case the bond would not have a 5 year duration

33. Hedging with the 6 year bond
Base case: Show that the company is hedged if the YTM = 8%
The company will hold the bond for 5 years
The coupon will be reinvested at the YTM
Reinvest for 4 years
Reinvest for 3 years
Reinvest for 2 years
Reinvest for 1 years
Collect coupon & sell bond
1,000

34. Hedging with the 6 year bond
Case YTM increases to 9%
The company will hold the bond for 5 years
The coupon will be reinvested at the YTM
Reinvest for 4 years
Reinvest for 3 years
Reinvest for 2 years
Reinvest for 1 years
Collect coupon & sell bond
1,000

35. Hedging with the 6 year bond
Case YTM decreases to 7%
The company will hold the bond for 5 years
The coupon will be reinvested at the YTM
Reinvest for 4 years
Reinvest for 3 years
Reinvest for 2 years
Reinvest for 1 years
Collect coupon & sell bond
1,000

36. What kind of risk would the coupons be subject to?
What does that all mean?
If the company offsets its assets or liabilities with an instrument of the same duration the position will be immune to changes in interest rates
Do you think this really works?
It could, but we run into two problems
The duration of the bond (used to hedge) will change
The YTM of the bond used to hedge could change
What kind of risk would the coupons be subject to?

37. Difficulties with duration (preview)
1. Duration Change
Lets calculate the duration of the bond right after the second coupon is paid – there are four years (coupons) left. Assume they YTM = 8%
Weights:
Duration
Loan: the loan still has 3 years to maturity so the durations no longer match – this is ok as long as the coupons have and can continue to be reinvested at 8%

38. Difficulties with duration (preview)
2. Reinvestment risk
Suppose that after the first two payments the interest rate increases to 9%
So what’s the point? This seems really ineffective – why am I not teaching you how to fully resolve this problem?
1,000
The company no longer has enough money to repay its loan of $
IT IS HARD!!!

39. Difficulties with Duration

40. Difficulties with Duration
Reallocating large quantities of assets or liabilities to attain the needed durations for assets and liabilities can be very costly
Immunization is a dynamic problem
Every time the interest rate changes the hedging portfolio must be rebalanced
One decision managers have to make is how often to rebalance and weigh the cost of doing so
Convexity- Duration only works for small changes in the interest rate
For large changes in rates duration will not accurately predict the percent change in the price of a security

41. Price a 20 year bond with coupon of 30% and semiannual payments
Convexity
Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch
Price a 20 year bond with coupon of 30% and semiannual payments

42. Price a 20 year bond with coupon of 30% and semiannual payments
Convexity
Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch
Price a 20 year bond with coupon of 30% and semiannual payments

43. Price a 20 year bond with coupon of 30% and semiannual payments
Convexity
Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch
Price a 20 year bond with coupon of 30% and semiannual payments

44. Price a 20 year bond with coupon of 30% and semiannual payments
Convexity
Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch
Price a 20 year bond with coupon of 30% and semiannual payments

45. Price a 20 year bond with coupon of 30% and semiannual payments
Convexity
Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch
Price a 20 year bond with coupon of 30% and semiannual payments

46. Price a 20 year bond with coupon of 30% and semiannual payments
Convexity
Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch
Price a 20 year bond with coupon of 30% and semiannual payments

47. Price a 20 year bond with coupon of 30% and semiannual payments
Convexity
Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch
Price a 20 year bond with coupon of 30% and semiannual payments

48. Price a 20 year bond with coupon of 30% and semiannual payments
Convexity
Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch
Price a 20 year bond with coupon of 30% and semiannual payments

49. Convexity
Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch

50. Convexity What does duration say about this relation?
Duration is the derivative of the bond pricing formula with respect to the interest rate at a specific point on the graph
What does the derivative look like on the graph?

51. Duration is the derivative. It is the slope of the tangent line
Convexity
Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch
Duration is the derivative. It is the slope of the tangent line
This is what duration says the graph (relationship) should look like
When we do the duration calculation, we find a point on this line
13%

52. Convexity & accuracy of duration
Calculate the duration of the bond if the YTM is 13%
D = 7.23 years
If the YTM dropped to 3% what price would the duration predict?
What is the actual price?

53. Convexity & accuracy of duration
Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch 3%
13%

54. Convexity & accuracy of duration
Calculate the duration of the bond if the YTM is 13%
D = 7.23 years
If the YTM jumped to 23% what price would the duration predict?
What is the actual price?

55. Convexity & accuracy of duration
Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch
Asymmetric Pricing Errors!
23% 3%
13%

56. Convexity & accuracy of duration
The larger the convexity the more curvature there is in the line
Duration will work better for bonds with low convexity
We will calculate convexity next

57. Take away
Duration is only accurate for small changes in interest rates
Duration will predict lower than actual values
The under prediction error is greater when interest rates fall then when they increase
Duration will change depending on the interest rate!!!!!!!!!

58. Calculate Convexity Calculate the duration weights
Multiply the weights by the time period squared plus and the same time period
Sum values and divide by (1+ YTM)2 to get convexity

59. Convexity Calculation Example
Consider a 6 year bond with an 8% coupon paid annually the YTM is 6%. Face value of Calculate the convexity of the bond
1000
Step #1 find the present value of payments
Step #2 calculate weights

60. Convexity Calculation Example
Consider a 6 year bond with an 8% coupon paid annually the YTM is 6%. Face value of Calculate the convexity of the bond
Step #3 calculate the convexity
Measures the curvature of the YTM bond price relationship – larger values = more curvature

61. Calculate the convexity of a 1
Calculate the convexity of a 1.5 year 4% coupon bond with semiannual payments and face value of 5,000 if the risk free rate is currently 5% and the YTM is 9%

62. So what good is convexity?
We can use it to adjust the accuracy of the duration calculation!!
Example:
Estimate the expected percent change in the price of the bond from the previous example (FV = 5000, coupon = 4%, TTM = 1.5yrs, semiannual compounding) if interest rates are expected to increases from 9% to 11.4% (the duration of the bond 1.47yrs).

63. Convexity Adjustment Price implied by Duration
Price implied by Duration & Convexity

64. Lecture Summary 3 properties of duration Hedging by matching duration
Duration increases with maturity but at a decreasing rate
Duration decreases as the yield to maturity increases
Duration decreases as the coupon payments or interest rate increases
Hedging by matching duration
The hedge is only perfect if YTM remains constant over the life of the hedge
Convexity
Concept
Calculation