??"? Duration

1
??"? Duration

• ?"? ??? ????
• ??? ?????? ??????
• ??? ???? ????? ?????
• ??????????? ?????? ????????

2
Orange County

• ??? ??????, ??? ?????
• ??? ?????? ?? 7.5 ??????? ???? (???? ????-???,
  ???? ?????)
• ???? 12.5 ??????? ????, ????? ????? ???? ??
  ????? ???"? ?-5 ????.
• ???? ?????? ??? ???? ??????
• ????? ????? ?????? ????? ?????
• ????? ????? ?? 1.64 ??????? ????

3
???

• ???? 1,000 ??????.
• ???????? ??????? ?? 1-3 ??????, ?? ????? ?? ??
  1,000.
• ??? ????? ???? ?????? ??? M1 ????, ????? ???? 7
  ????? ?????.
• ??? ???????? ????????
• ???? ???? ????? ????? ?? ???????

4
???? ??? ?????
????
?????
5
?????? ?????
????
0 t D
6
??"? ??? ???????
7
?????
y10, ?y0.5
T P0 P1 ?P 1 90.91 90.50 -0.45 2 82.64 81.90 -0
.98 10 38.55 36.84 -4.43
8
?????
y10, ?y1
T P0 P1 ?P 1 90.91 90.09 -0.90 2 82.64 81.16 -1
.79 10 38.55 35.22 -8.65
9
????? 1
??????? ?? ??"? ????? (??? ???????) ???? ??????
????? ?????? ??????. ?????? ????? (?-) ????
???"? ??? ??????.
10
????? 2
???"? ?????, ???? ?? ????? ???? ?????? ?????
?????? ??? ???? (?? ?????? ?????) ?????. ?? ??
???? ??????? ????????
11
????? 3
???"? ?????, ???? ?? ????? ????? ?????? ?????
???????? ????? (????????? ??????) ???????. ?? ??
???? ??????? ????????
12
????? 4
???"? ?????, ???? ?? ????? ???? ??????, ?????
????? ??????? ???? ????? ?????? ????? ???????.
?? ?? ???? ??????? ????????
13
??????? ???????? ?? ???????? ?????
????? ????? ????? ??? ?????? ????? ??????
14
???????? ????? ??"?
???? ??? ?????? ????? ???? ????? ??? ?????? ????
???? ???????? ????? ???? ????? ???? ????
???????? ????? ???? ??"? ?????? ????? ????????
????? ??? ????? ????? ?? ??????? ?????? ???
????? ?? ????? ???? (Price value of a Basis
Point (PVBP)) ?????? ????? ????? ?????? ??
0.01 ??????.
15
??"? ???? ????? ?????
1 2 3 4 5 6
16
)Duration??"? (
F. Macaulay (1938) ??? ??? ???? ????
??????. ????? ?????? ?? ?? ???????? ?? ????
??????.

• Bond Price Sum CFt/(1y)t
• ??????? ??? ?????
• wt CFt/(1y)t /Bond Price
• ?wt ?? ???? ?? ?

17
)Duration??"? (
????????? ????? ???"? ??? ????? ???"? ???. ?? ??
???"? ??? ??? ??????? ?????.
18
??"? ?????? )Modified Duration(
?????? ????? ???"? ??????? ??? ????? ?? ???"?
??????? ?????? ?????? ??????
19
)Duration??"? (
20
)Duration??"? (
21
)Duration??"? (
22
??? ??????
??? ?????? ???? ?????? ?????? ?? ????? ??? ?????
????? ?? ?????. y0 ??? ???? 0 y1 ??? ????
?????? ?y ??????
23
??????
F(x)
x
24
????? ?????? ?????
25
?????? ??????
?????? ?????? ?? ??"? ????? ?????? ????
26
Macaulay Duration

• ????? ???"?

27
Macaulay Duration
???? ?????? ?? ?????? ?????? ?? ?? ????????.
?? ???"? ?? ??"? ??? ????????
28
?????? ???"?
29
????? ??????
r
T
30
?????? ??? ??? ??"?
??"? ??? ????? ?? ??"? ??? ???? ?? ??? ?????? ??
1.8853 ????. ?????? ?? 5 ?????
???? (1,000/1.053.7706)831.9623 ?? ???? ??
?????? ?-5.01 ????? ??? ? (1,000/1.05013.7706)
831.66 ?????, ??? ?- 0.359.
??"? ?????? ????? ??? ???? ?? ??"? 1.8853. ?????
??? 5 ??? 964.5405. ?? ???? ?? ?????? ??????
???? 1 (?-5.01) ????? ??? ? 964.1942, ?????,
??? ?- 0.359.
31
??"?
??"?
Zero coupon bond
15 coupon, YTM 15
??? ??????
0 3m 6m 1yr 3yr 5yr 10yr 30yr
32
?????
??"? ?? 30 ??? ??????. ????? 8 ????? ?? 6
??????.

• YTM 9
• P0 897.26
• D 11.37 Yrs
• if YTM 9.1, what will be the price?
• ?P/P - ?y D
• ?P -(?y D)P -9.36
• P 897.26 - 9.36 887.90

33
??????? ???????? ?? ???"?
???"? ?? ??"? ??? ??????? ???? ????
??????. ?????? ???? ??? ??????, ???"? ???? ???
????????? ?????. ????? ??????? ????, ???"? ????
??? ??????? ?????? ????. ???"? ?? perpetuity ???
(1y)/y
34
??????? ???????? ?? ???"?
?????? ??????? ????, ???"? ???? ???? ???? ??????
???? ??????.
35
??????
36
?????

• 10 year zero coupon bond with a semiannual yield
  of 6

The duration is 10 years, the modified duration
is
The convexity is
37
?????
If the yield changes to 7 the price change is
38
FRM-98, Question 17
??"? ????? ????? 100 ?? ????? ?? 8. ?? ??????
???? ?????? ???? 1, ???? ???"? ??? ? 99.95. ??
?????? ????? ?????? ???? 1, ????? ????
?-100.04. ??? ???"? ??????? ???"??

• A. 5.0
• B. -5.0
• C. 4.5
• D. -4.5

39
FRM-98, Question 17
40
FRM-98, Question 22
???? ????? ???? ?-10 ?????? ???? ?? ???? ??"?
?????? ????? 100 ?-10 ???? ?? ??"? ?????? ?? 7
??????? ?? 50?

• A. -0.705
• B. -0.700
• C. -0.698
• D. -0.690

41
FRM-98, Question 22
42
??"? ?? ???
???? ???"? ?? ??"? ?????, ?? ?????? ???? ???
????? ???? ????.
43
Bond Price Derivatives

• D - modified duration, dollar duration is the
  negative of the first derivative

Dollar convexity the second derivative, C -
convexity.
44
Duration of a portfolio
45
ALM Duration

• Does NOT work!
• Wrong units of measurement
• Division by a small number

46
Duration Gap

• A - L C, assets - liabilities capital

47
ALM Duration

• A similar problem with measuring yield

48
??? ??????? ???"? ???? ?? ???!!
49

• Key rate duration
• Principal component duration
• Partial duration

50
???? ????!
?????? ????? ???? ??? ????? ????? ???? ??????
????? ????? ???? ???? ???? ????? (??? ???) ???
???"??
51
????? ????
?? ???"? ?? floater? ?? ???"? ?? inverse
floater? ???? ??????? ?????? ?????? ?? ???"?? ??
???????? ?? ??"? ?????? ????? ??"? Macaulay? ????
???? ?????? ???"? ????? ??????
52
??"? / ??????
?? ???? ???"? ???? ?? ????? ?????? ??? ???????
????? ????? ?????? ???"? ???? callable? ???
??????? ????? ????? ?????? ???"? ????puttable ?
53
Callable ??"? ????
???? ??"? ?? ??? ?????? ?????? ?????? ???"?
?????. ???????? ?????? ????... ??????? ??????
?????? ???? ???? ???? ????? ?????? ????? ????.
54
?????? call ?????
r
55
Puttable ??"? ????
???? ???"? ???? ????? ???? ?? ???????. ????????
?????? ????... ??????? ????? ?????? ????? ??????
??????????.
56
?????? put ?????
??"? ?????
r
57
??"? ?????
?????
????
58
???? ???????

• European
• American
• Bermudian
• Lock up time

59
????? ??? ?? ???
???????? ?????
???
???????? ????
10,000
???? ?????????
0 8 r
60
???
61
Macaulay Duration
Modified duration
62
Bond Price Change
63
Duration of a coupon bond
64
Exercise

• Find the duration and convexity of a consol
  (perpetual bond).
• Answer (1y)/y.

65
FRM-98, Question 29

• A and B are perpetual bonds. A has 4 coupon, and
  B has 8 coupon. Assume that both bonds are
  trading at the same yield, what can be said about
  duration of these bonds?
• A. The duration of A is greater than of B
• B. The duration of A is less than of B
• C. They have the same duration
• D. None of the above

66
FRM-97, Question 24

• Which of the following is NOT a property of bond
  duration?
• A. For zero-coupon bonds Macaulay duration of the
  bond equals to time to maturity.
• B. Duration is usually inversely related to the
  coupon of a bond.
• C. Duration is usually higher for higher yields
  to maturity.
• D. Duration is higher as the number of years to
  maturity for a bond selling at par or above
  increases.

67
FRM-99, Question 75

• You have a large short position in two bonds with
  similar credit risk. Bond A is priced at par
  yielding 6 with 20 years to maturity. Bond B has
  20 years to maturity, coupon 6.5 and yield of
  6. Which bond contributes more to the risk of
  the portfolio?
• A. Bond A
• B. Bond B
• C. A and B have similar risk
• D. None of the above

68
??"? ??????? ????

• ???????

69
?????
?????? ??? ???? ??"?, A ?-B, ?????? ?????? 10
???? ??????, 6, ??? 100 ???"? ?????? ?? 7.44
????.

• A. 1 year zero bond - price 94.26
• B. 30 year zero - price 16.97

70

• Barbel portfolio consists of very short and very
  long bonds.
• Bullet portfolio consists of bonds with similar
  maturities.
• Which of them has higher convexity?

71
FRM-98, Question 18
??? ????? ???? ????????. ??????? ???????? long,
????? ????? ?? ??"? ???????, ??? ???? 100, ????
101 ???"? 1.7. ????? ???????? short, ????? ?????
?? ??"?, ??? ???? 50, ?-5 ????, ????? 99 ???"?
4.1. ?? ??"? ?????

• A. 0.68
• B. 0.61
• C. -0.68
• D. -0.61

72
FRM-98, Question 18
Note that 100 means notional amount and can be
misunderstood.
73
Useful formulas
74
8 Coupon Bond
Zero Coupon Bond