Transcript Com 4FJ3 - McMaster University

Business F723
Fixed Income Analysis
Week 4
Measuring Price Risk
Basics of Price Risk
• As YTM changes, bond prices change
• Bond prices move in the opposite direction
to the change in yield
• Not all bonds react the same amount to a
given change in yield
• For large changes in yield, an increase has a
higher change than a decrease
2
Time to Maturity Effect
For a 10% Coupon Bond
$2,000
$1,750
Price
$1,500
$1,250
$1,000
$750
$500
0%
5%
10%
15%
20%
YTM
1-year bond
10-year bond
par
3
Time to Maturity
• All else held constant the longer the time to
maturity the larger the price volatility of a
bond with respect to changing yields
• Intuition; if I am paying a premium to lock
in an above average current yield, I am
willing to pay more to lock it in for a longer
period of time
4
Coupon Rate
• Consider two bonds, A and B;
• $1,000 face value
• YTM = 9%
• Coupon rate; A = 5%
maturity = 10 years
B = 10%
• Initial Prices = PVcoupons + PVface
• A = 325 + 415 = $740
• B = 650 + 415 = $1,065
• What % change in price if YTM ↓ 8%?
5
Coupon Rate
• Price A, 8% = 340 + 456 = $796
– Change = (796 - 740) / 740 = 7.6%
• Price B, 8% = 680 + 456 = $1136
– Change = (1,136 - 1,065) / 1,065 = 6.7%
• In general, the larger the coupon payment,
the less the change in price with a change in
yield.
6
Effect of YTM
Effect of a 100 basis point increase on price
YTM
2%
4%
6%
8%
10%
12%
14%
16%
Price
$1,541
$1,327
$1,149
$1,000
$875
$771
$682
$607
New Price
$1,429
$1,234
$1,071
$935
$821
$725
$643
$574
% Decrease
7.28%
7.02%
6.76%
6.50%
6.24%
5.98%
5.72%
5.45%
for 8% coupon, 10 year bond
7
Level of YTM
• As the level of interest rates rise, the
sensitivity of bond prices to changes in the
yield falls
• Intuition; a change from 2% to 2.1% is
much more significant than a change from
16% to 16.1% as a fraction of total return
8
Price Value of a Basis Point
• One measure of price change is the dollar
change in the price of a bond for a 1 basis
point increase in the required yield
• Also known as dollar value of an 01
• Stated based on the pricing convention of
quotes per $100 of face value
• p63; 5 year 9% coupon par bond, 3.96¢
9
Yield Value of a Price Change
• Pricing conventions used to quote prices in
32nds or 8ths of a point (fraction of a dollar
per $100 of face value)
• This measure converts the minimum price
change into the effective change to YTM
– 5 year 9% par bond; -⅛ = $99.875
– New YTM = 9.032%
– Yield value of an 8th = 3.2 basis points
10
Macaulay’s Duration
• First published in 1938
• A bond can be considered to be a package
of zero coupon bonds
• By taking a weighted average of the
maturity of those zero coupon bonds, you
can approximate the price sensitivity of the
portfolio that the bond represents
11
Macaulay’s Duration
• The average time that you wait for each
payment, weighted by the percentage of the
price that each payment represents.
• Captures the effect of maturity, coupon rate
and yield on interest rate risk.
• The higher the duration the greater the level
of interest rate risk in an investment.
12
Duration Calculation
• Two bonds;
– YTM = 8%
– Maturity = 3 years
– Coupon rate
• A = 6%
• B = 10%
– Face Value = $1,000
• Find the duration.
Period Payment
1
$30
2
$30
3
$30
4
$30
5
$30
6
$1,030
PV
% % x time
$28.85
3.04%
0.0304
$27.74
2.93%
0.0585
$26.67
2.81%
0.0844
$25.64
2.71%
0.1083
$24.66
2.60%
0.1301
$814.02 85.91%
5.1543
$947.58 100.00%
5.5661
duration =
2.7831
Period Payment
1
$50
2
$50
3
$50
4
$50
5
$50
6
$1,050
PV
% % x time
$48.08
4.57%
0.0457
$46.23
4.39%
0.0879
$44.45
4.22%
0.1267
$42.74
4.06%
0.1624
$41.10
3.90%
0.1952
$829.83 78.85%
4.7310
$1,052.42 100.00%
5.349
duration =
2.6745
13
Price Elasticity
• Using calculus on the price equation
C
C
C
C
M



...


1  y 1  y 2 1  y 3
1  y n 1  y n
dP
C
 2C
 3C
 nC
 nM



 ... 

2
3
4
n 1
dy 1  y  1  y  1  y 
1  y 
1  y n1
P
dP  1  C
2C
3C
nC
nM 



 ... 


2
3
n
dy 1  y 1  y 1  y  1  y 
1  y  1  y n 
dP 1
 1  1C
2C
3C
nC
nM  1



 ... 


2
3
n
dy P 1  y 1  y 1  y  1  y 
1  y  1  y n  P
14
Modified Duration
• From the last line of the previous equation,
the right hand side is
•
•
•
•
•
•
-1/(1+y) x Macaulay’s duration
The negative of this is called Modified Duration
Modified duration = Macaulay’s duration/(1+y)
Often used to approximate percentage price changes
Duration in years = D in six month periods/2
use 6 month rate for (1+y) in modified duration
15
Alternate Method
• From the annuity formula for the price of a
bond we can get a formula for modified
duration instead of calculating weighted
average (per $100 of face value)

C
n100 
y
C 
1 

1

n
2 
y  1  y  
1  y n1
modifiedduration 
P
16
Properties of Duration
• Increases with time to maturity
• Increases as coupon rate decreases to a
maximum of time to maturity for a zero
coupon bond
• Decreases as YTM increases due to face
value having less weight in portfolio
Modified Duration is similar, but lower max
17
Approximate Price Change
• The change in price for a given change in
yield can be calculated using modified
duration (a.k.a. volatility)
• The approximate percentage change in price
= - modified duration x change in yield
• Given MD = 7.66, calculate change in price for a 50
basis point increase in yield
• DP% = -7.66 x 0.5% = -3.83%
18
Dollar Price Change
• The approximate dollar price change is
simply the approximate percent price
change times the price
• Given the bond on the previous slide, if the
initial price was $102.5 the decrease in
value is 3.83%
• In dollar terms, $3.926
19
How Close is This?
• For small yield changes, the approximation
is reasonable, p. 70 example, for a 1 basis
point increase on a 25 year 6% coupon bond
with an initial yield of 9%, the forecast
change is -$0.0747 actual is -$0.0746
• For large changes it is not as good
• Reason: duration is a linear approximation
of the price/yield relationship
20
Portfolio Duration
• Since duration is simply a weighted average
of the time to the coupon payments and face
value, portfolio duration is simply the
weighted average of the durations of the
individual bonds
• Portfolio managers look at the contribution
to portfolio duration to assess their interest
rate risk of a single bond issue
21
Convexity
For a 10% Coupon Bond
$2,000
$1,750
Price
$1,500
$1,250
$1,000
Tangent line for
estimated price
$750
$500
0%
5%
10%
15%
20%
YTM
1-year bond
10-year bond
par
22
Convexity
• Due to the shape of the yield curve, the
predicted price will always be lower than
the actual price
• How close the approximation is depends on
how convex the price/yield relationship is
for a given bond
23
Measuring Convexity
• Convexity is based on the rate of change of
slope in the price/yield relationship
• That means that we need the second
derivative of the price of a bond
• This is the dollar convexity
d 2 P n t t  1C nn  1M


t 2
n2
2
dy




1

y
1

y
t 1
24
Convexity Measure
• The convexity measure is the second
derivative of the price divided by the price
d 2P 1
 convexitymeasure
2
dy P
convexitymeasurein years
convexitymeasurein m periods/year
m2
dP 1
2
% pricechange 
 convexitymeasuredy
P 2
25
Convexity Example
Coupon rate
Maturity
YTM
Price
9%
5
9%
100
Period
1
2
3
4
5
6
7
8
9
10
CF t(t+1)CF
1/(1+y)^(t+2) combined
4.5
9.0 0.876297
7.9
4.5
27.0 0.838561
22.6
4.5
54.0 0.802451
43.3
4.5
90.0 0.767896
69.1
4.5
135.0 0.734828
99.2
4.5
189.0 0.703185
132.9
4.5
252.0 0.672904
169.6
4.5
324.0 0.643928
208.6
4.5
405.0 0.616199
249.6
104.5 11,495.0 0.589664 6,778.2
Second derivative = 7,781.0
Convexity measure (half years)=
77.81
Convexity measure (years)= 19.4526
26
Price Change Example
• Given 25 year, 6% bond yielding 9%
• Required yield increases to 11%
– Mod. Duration = 10.62
• change due to duration = -10.62 x 2%=-21.24%
– Convexity in years = 178
• change due to convexity = 1/2 x 178 x 0.022=3.66%
• Forecast change = -21.24 + 3.66 = -17.58%
• Actual change = -18.03%
27
Alternate Calculation
• We could also take the second derivative of
the annuity based price formula

C




n
n

1
100


2
y
d P 2C 
1 
2Cn

 3 1 
 2

n
n 1
2
dy
y  1  y   y 1  y 
1  y n2
• Divide by price for convexity measure
• Divide by m2 to convert to years
28
Note on Convexity
• Different writers compute the convexity
measure differently
• One method moves the ½ into the measure
1 d 2P 1
 convexitymeasure
2
2 dy P
convexitymeasurein m periods/year
convexitymeasurein years
m2
dP
2
% pricechange 
 convexitymeasuredy
P
29
Price
Value of Convexity
Bond A
Bond B
Yield
30
Value of Convexity
• Two bonds offering the same duration and
yield, but with different convexity
• Bond A will outperform bond B if the
required yield changes
• Bond A should have a higher price
• Increase in value of A over B should be
related to the volatility of interest rates
31
Positive Convexity
• As required yields increase convexity will
decrease
• As yields increase the slope of the tangent
line will become flatter
• Implication
– as yield increases, prices fall and duration falls
– as yield decreases, prices rise and duration rises
32
Properties of Convexity
• For a given yield and maturity, the lower
the coupon rate, the higher the convexity
• For a given yield and modified duration, the
higher the coupon rate, the higher the
convexity
• Although coupon rate has an impact on the
convexity it has a bigger impact on duration
33
Effective Duration
• If a bond has embedded options, that will change
the bond’s price sensitivity to changes in required
yields
• The value of a call option on the bond decreases as
yields increase, and increases as yields decrease
• Effective duration can be calculated to account for
the fact that expected cash flows may change in
yields change
34
Duration vs. Time
• With plain vanilla bonds, duration can be
seen as a measure of time
• With more complex instruments, this link is
broken
• Modified duration is a measure of the
bond’s price volatility with respect to
changes in the required yield
35
Duration of Floaters
• A floating rate bond usually trades near par since
the coupon rate adjusts to changes in interest rates
• Therefore a floater’s duration is near zero
• An inverse floater has a high duration (possibly
greater than its maturity) since, when interest rates
go up its coupon payments go down, exaggerating
the impact of a change in yields
• A double floater could have a negative duration
36
Approximating Duration
• Instead of using duration to approximate
price changes, we can use price changes to
approximate duration
• Potentially useful for complex instruments
as a measure of price volatility
approximate duration 
P  P
2 P0 Dy 
P-= price if yield down
P+= price if yield up
P0= original price
37
Approximating Convexity
• We can also approximate convexity using a
similar method
P  P  2P0
approximate convexitymeasure 
2
P0 Dy 
P-= price if yield down
P+= price if yield up
P0= original price
38
Changing Yield Curve
• What happens if the shape of the yield curve
changes?
• It is possible that prices on 30 year bonds could
change while short term rates are stable
• Duration calculations can change to; key rate
durations, duration vectors, partial durations, etc.
• Key rate durations are illustrated in the text; they
are calculated using the approximation formula
39