Transcript 02/09/2001

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Quantitative Analysis 1
Zvi Wiener
1
FRM 2000
• Capital Markets Risk Management 20
• Legal, Accounting and Tax
6
• Credit Risk Management
36
• Operational Risk Management
8
• Market Risk Management
35
• Quantitative Analysis
23
• Regulation and Compliance
12
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Quantitative Analysis
Jorion, Value-at-Risk.
Jorion, …
Hull, Options, Futures and Other Derivatives.
Fabozzi F., Bond Markets: Analysis and Strategies.
Fabozzi F., Fixed Income Mathematics.
Golub B., Risk Management.
Crouchy, Galai, Mark, Risk Management.
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Quantitative Analysis
• Bond fundamentals
• Fundamentals of probability
• Fundamentals of Statistics
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Bond Fundamentals
CT
Discounting, Present Value PV 
T
(1  y)
$100
$55.84 
10
(1  0.06)
FV  PV (1  y)
Future Value
t
$179.08  (1  0.06)  $100
10
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Compounding
US Treasuries market uses semi-annual
compounding.
PV 
CT

y 
1 

2 

s
2T
Continuous compounding
PV  CT e
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 y cT
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A bond pays $100 in ten years and its price is
$55.9126. This corresponds to an annually
compounded rate of 6% using
PV=CT/(1+y)10, or (1+y) = (CT/PV)0.1.
This rate can be transformed into semiannual
compounded rate, using (1+ys/2)2 = (1+y),
or ys = ((1+0.06)0.5-1)*2 = 5.91%.
It can be transformed into a continuously
compounded rate exp(yc) = 1+y, or
yc = ln(1+0.06) = 5.83%.
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Note that as we increase the frequency of the
compounding the resulting rate decreases.
Intuitively, since our money works harder
with more frequent compounding, a lower
rate will achieve the same payoff.
Key concept: For a fixed present and final values, increasing
the frequency of the compounding will decrease the
associated yield.
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FRM-99, Question 17
Assume a semi-annual compounded rate of
8% per annum. What is the equivalent
annually compounded rate?
A. 9.2%
B. 8.16%
C. 7.45%
D. 8%
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FRM-99, Question 17
(1 + ys/2)2 = 1 + y
(1 + 0.08/2)2 = 1.0816 ==> 8.16%
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FRM-99, Question 28
Assume a continuously compounded interest
rate is 10% per annum. What is the
equivalent semi-annual compounded rate?
A. 10.25% per annum.
B. 9.88% per annum.
C. 9.76% per annum.
D. 10.52% per annum.
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FRM-99, Question 28
(1 + ys/2)2 = ey
(1 + ys/2)2 = e0.1
1 + ys/2 = e0.05
ys = 2 (e0.05 - 1) = 10.25%
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Price-Yield Relationship
T
Here
Ct
P
t
(
1

y
)
t 1
Ct is the cashflow
t - number of periods to each payment
T number of periods to maturity
y - the discount factor.
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Face value, nominal.
Bond that sells at face value is called par bond.
A bond has a 8% annual coupon and IRR of 8%.
What is the price of the bond?
Is this always true?
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Price-yield Relationship
$
Price of a straight bond as a function of yield
y
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FRM-98, Question 12
A fixed rate bond, currently priced at 102.9,
has one year remaining to maturity and is
paying an 8% coupon. Assuming that the
coupon is paid semiannually, what is the
yield of the bond?
A. 8%
B. 7%
C. 6%
D. 5%
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FRM-98, Question 12
$4
$104
$102.9 

s
2
s
y


y
1
1  
2
2 

ys = 5%
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Taylor Expansion
To measure the price response to a small
change in risk factor we use the Taylor
expansion.
Initial value y0, new value y1, change y:
y1  y0  y
1
2
f ( y1 )  f ( y 0 )  f ' ( y 0 )y  f " ( y 0 )y  
2
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F(x)
Derivatives
x
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Properties of derivatives
 f ( x)  g ( x) '  f ' ( x)  g ' ( x)
 f ( x)  g ( x) '  f ' ( x)  g ( x)  f ( x)  g ' ( x)
 f ( x)  f ' ( x) g ( x)  f ( x) g ' ( x)

 
2
g ( x)
 g ( x) 
d
f ( g ( x))  f ' ( g ( x))g ' ( x)
dx
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Bond Price Derivatives
D* - modified duration, dollar duration is
the negative of the first derivative:
dP
f ' ( y0 ) 
  D *  P0
dy
Dollar convexity = the second derivative,
C - convexity.
d 2P
f " ( y0 )  2  C  P0
dy
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Duration of a portfolio
N
P( y )   xi Pi ( y )
i 1
dPi ( y )
dP N
  xi
dy i 1
dy
N
dP
*
*
  D  P   xi Di Pi
dy
i 1
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Macaulay Duration
T
Ct
P( y )  
t
(
1

y
)
i t
T
tCt
1
DM  
t
P t 1 (1  y)
Modified duration
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DM
D 
1 y
*
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Bond Price Change
1
P   D  P  y  C  P  y 2  
2
*
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Example
10 year zero coupon bond with a semiannual
yield of 6%
100
P
(1  0.03)
20
 $55.368
The duration is 10 years, the modified duration is:
10
D 
 9.71
(1  0.03)
*
1 d 
100

C
P dy 2  (1  0.5 y) 20
2
The convexity is
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
  98.97

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Example
If the yield changes to 7% the price change is
P 
 9.71 $55.37 0.01 98.97 $55.37 0.012  0.5 
 $5.375 $0.274  $5.101
100
100
P 

 $5.111
20
20
(1  0.035)
(1  0.03)
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Duration-Convexity
$
Price of a straight bond as a function of yield
y
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Effective duration
P( y0  y )  P( y0  y )
D 
2yP
E
Effective convexity
P( y0  y)  2 P( y0 )  P( y0  y)
C 
2
Py 
E
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Effective Duration and Convexity
Consider a 30-year zero-coupon bond with a
yield of 6%. With semi-annual
compounding its price is $16.9733.
We can revalue this bond at 5% and 7%.
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100
P
 16 .9733
60
1.03
100
P (5%) 
 22.7284
60
1.025
100
P (7%) 
 12.6934
60
1.035
22.7284  12.6934
D 
 29.56  29.13
16.9733  0.02
E
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22.7284 2  16.9733 12.6934
C 
2
16.9733 0.01
E
 869.11  862.48
5%
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6%
7%
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Coupon Curve Duration
If IR decrease by 100bp, the market price of a
6% 30 year bond will go up close to the
price of a 30 years 7% coupon bond.
Thus we associate a higher coupon with a
drop in yield equal to the difference in
coupons.
This approach is useful for mortgages.
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FRM-98, Question 20
Coupon curve duration is a useful method to
estimate duration from market prices of
MBS. Assume that the coupon curve of
prices for Ginnie Maes is as follows: 6% at
92, 7% at 94, 8% at 96.5. What is the
estimated duration of the 7s?
A. 2.45
B. 2.4
C. 2.33
D. 2.25
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FRM-98, Question 20
P  P
96.5  92
D 

 2.4
2 Py
2  94  0.01
E
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FRM-98, Question 21
Coupon curve duration is a useful method to
estimate duration from market prices of
MBS. Assume that the coupon curve of
prices for Ginnie Maes is as follows: 6% at
92, 7% at 94, 8% at 96.5. What is the
estimated convexity of the 7s?
A. 53
B. 26
C. 13
D. -53
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FRM-98, Question 21
P  2P  P 96.5  2  94  92
C 

 53
2
2
Py
94  0.01
E
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Duration of a coupon bond
T
Ct
P( y )  
t
(
1

y
)
t 1
 tCt
dP( y)
D


P
t 1
dy
1 y
t 1 (1  y )
T
T
tCt
1
D 
P t 1 (1  y) t
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Exercise
Find the duration and convexity of a consol
(perpetual bond).
Answer: (1+y)/y.
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Convexity
d P( y) T t (t  1)Ct

2
t 2
dy
t 1 (1  y )
2
t (t  1)Ct
1
C 
P t 1 (1  y ) t  2
T
Exercise: compute duration and convexity of
a 2-year, 6% semi-annual bond when IR=6%.
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FRM-99, Question 9
A number of terms in finance are related to
the derivative of the price of a security with
respect to some other variable. Which pair
of terms is defined using second
derivatives?
A. Modified duration and volatility
B. Vega and delta
C. Convexity and gamma
D. PV01 and yield to maturity
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FRM-98, Question 17
A bond is trading at a price of 100 with a yield
of 8%. If the yield increases by 1 bp, the
price of the bond will decrease to 99.95. If
the yield decreases by 1 bp, the price will
increase to 100.04. What is the modified
duration of this bond?
A. 5.0
B. -5.0
C. 4.5
D. -4.5
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FRM-98, Question 17
DModified
P  P 100.04  99.95


 4.5
2 Py
100 0.0002
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FRM-98, Question 22
What is the price of a 10 bp increase in yield
on a 10-year par bond with a modified
duration of 7 and convexity 0f 50?
A. -0.705
B. -0.700
C. -0.698
D. -0.690
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FRM-98, Question 22
y
P   D  y 
CP 
2
2
0.001
 7  $100 0.001
50  $100  0.6975
2
2
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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
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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.
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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
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Portfolio Duration and Convexity
N
D Pp   D x Pi
*
p
i 1
*
i i
Portfolio weights
N
D   wi D
*
p
i 1
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*
i
xi Pi
wi 
Pp
N
C   wi C
*
p
i 1
*
i
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Example
Construct a portfolio of two bonds: A and B to
match the value and duration of a 10-years,
6% bond with value $100 and modified
duration of 7.44 years.
A. 1 year zero bond - price $94.26
B. 30 year zero - price $16.97
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100  x1  94.26  x2  16.97

7.44  100  0.97  x1  94.26  29.13 x2  16.97
 x1  1.021

 x 2  0.221
Barbel portfolio consists of very short and
very long bonds.
Bullet portfolio consists of bonds with similar
maturities.
Which of them has higher convexity?
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FRM-98, Question 18
A portfolio consists of two positions. One is long
$100 of a two year bond priced at 101 with a
duration of 1.7; the other position is short $50
of a five year bond priced at 99 with a duration
of 4.1. What is the duration of the portfolio?
A. 0.68
B. 0.61
C. -0.68
D. -0.61
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FRM-98, Question 18
P1
P2
D  D1  D2

P
P
101
 49.5
1.7
 4.1
 0.61
101 49.5
101 49.5
Note that $100 means notional amount
and can be misunderstood.
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Duration Gap
A - L = C, assets - liabilities = capital
A
L
DC  D A  DL
C
C
D gap
L
 D A  DL
A
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D gap
A
 DC
C
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Useful formulas
N 1
1 a
1 a  a  a  a 
1 a
2
3
N
cF
cF
cF
F
P




2
T
T
1  y (1  y )
(1  y )
(1  y )
c 
1
 F 1 
T
y  (1  y )
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
F

T
 (1  y )
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