Description
The PPT explaining about Convexity Approach with ways to measure interest rate risk.
1. Interest Rate Risk
2. Ways of measuring Interest Rate Risk
3. Full valuation approach
? Example
4. Duration/Convexity Approach
? Duration
? What is duration?
? Calculation
? Properties
? Application
? Limitations
? Convexity
? Introduction
? Predicting percentage price change using convexity
? Examples
? Convexity for callable and putable bonds
? Applications
?Changes in Bond prices due to interest rates
fluctuations
? Formula of Bond Value calculation.
? Where
? Po=value of bond
? c1,c2,…,cn=cash flows expected from bond over ‘n’
periods
? r=discount rate
?Full Valuation Approach
?Duration - Convexity Approach
?The full valuation approach to measuring the
interest rate risk involves using a pricing
model to value individual bonds and can be
used to find the price impact of any scenario
of interest rate/yield curve changes.
?Adv/Disadv. of this approach:
? Precision
? Flexibility
? Complex
The Full Valuation Approach
Scenario Yield ? Bond X
(in millions)
Bond Y
(in millions)
Portfolio Portfolio
Value ?%
Current +0 bp ($108.42) ($81.78) ($190.21)
1 +50 bp ($106.23) ($77.93) ($184.17) -3.18%
2 +100 bp ($104.10) ($74.32) ($178.42) -6.20%
-2.02%
-3.99%
-4.71%
-9.12%
Bond X Bond Y
Coupon 8% 5%
Maturity (yr) 5 15
FV($) 100 100
YTM 6.00% 7.00%
?This approach provides an approximation of
the actual interest rate sensitivity of a bond
or a bond portfolio.
?Its main advantage is its
? Simplicity
? Quick
1. Approximate percentage change in price
for 1% change in yield
2. Duration is the slope of the price-yield
curve of bonds current YTM
3. Weighted average of time until each cash
flow would be received. Weights are
proportions of total bond value that each
cash flow presents
?Example: 20yr, 8% coupon PV=908, YTM=9%
? If yield decreases by 50 bps, price=952.3
? If yield increases by 50 bps, price=866.8
1. Coupon Rate: Lower Coupon rate means
Higher duration
2. Maturity: Longer Maturity means Higher
duration
The Full Valuation Approach
Scenario Yield ? Bond X
(in millions)
Bond Y
(in millions)
Portfolio Portfolio
Value ?%
Current +0 bp ($108.42) ($81.78) ($190.21)
1 +50 bp ($106.23) ($77.93) ($184.17) -3.18%
2 +100 bp ($104.10) ($74.32) ($178.42) -6.20%
-2.02%
-3.99%
-4.71%
-9.12%
3. Higher market yield means lower duration.
? Price Yield Curve for an option free bond
? 8% coupon rate, 20 year bond, YTM=8%, FV=100
$110.59
$100.00
$90.87
$85.00
$90.00
$95.00
$100.00
$105.00
$110.00
$115.00
Price Yield Curve
Price Vs Yield
?It is a good measure of sensitivity of a
portfolio, and can be used to reduce or
increase the exposure to a particular term
interest rate risk
?Large changes in interest rates
?This approach is applicable for a portfolio of
bonds with only parallel yield curve shifts.
? Measures how much a bond’s price-yield curve deviates from a
straight line
? Second derivative of price with respect to yield divided by
bond price
? BENEFIT: Allows us to improve the duration approximation for
bond price changes
? Duration underestimates actual prices
? Previous Example: 20yr, 8% coupon PV=908, YTM=9%
? Duration : 9.42
? Now if yield changes by 1% we can see changes as seen in diagram:
?Recall approximation using only duration:
?The predicted percentage price change
accounting for convexity is:
AP
P
×100 = ÷D
m
*
× Ay ×100
AP
P
×100 = ÷D
m
*
× Ay ×100
( )
+
1
2
× Convexity× (Ay)
2
×100
|
\
|
.
FIN 509 Class session 2 20
?Consider a 20-year 9% coupon bond selling at
$134.6722 to yield 6%. Coupon payments are
made semiannually.
?D
m
= 10.66
?The convexity of the bond is 164.106.
FIN 509 Class session 2 21
? If yields increase instantaneously from 6% to 8%, the
percentage price change of this bond is given by:
? First approximation (Duration):
–10.66 × .02 × 100 = –21.32
? Second approximation (Convexity)
0.5 × 164.106 × (.02)
2
× 100 = +3.28
Total predicted % price change: –21.32 + 3.28 = –18.04%
(Actual price change = –18.40%.)
FIN 509 Class session 2 22
? What if yields fall by 2%?
? If yields decrease instantaneously from 6% to 4%, the
percentage price change of this bond is given by:
? First approximation (Duration):
–10.66 × –.02 × 100 = 21.32
? Second approximation (Convexity)
0.5 × 164.106 × (–.02)
2
× 100 = +3.28
Total predicted price change: 21.32 + 3.28 = 24.60%
Note that predicted change is NOT SYMMETRIC.
?Comparison of bonds:
? A bond with greater convexity is less affected by
interest rates
? A bond with greater convexity will have higher
prices than bonds with lower convexity,
regardless of interest rate rise or fall
? With a callable or prepayable debt, the upside price
appreciation in response to decreasing yields is limited.
? When the price begins to rise at a decreasing rate in
response to further decreases in yield, the price yield
curve bends over to the left and exhibits negative
convexity.
? Compared to an option-free bond, a
putable bond will have less price
volatility at higher yields.
?Convexity is a interest rate risk management
tool, which is used in managing bond
portfolios
?If the combined convexity and duration of a
trading book is high, so is the risk. However,
if the combined convexity and duration are
low, the book is hedged, and little money
will be lost even if fairly substantial interest
movements occur.
?Fixed Income, Derivatives, and Alternative
Investments – CFA L 1, Kaplan Schweser
?Valuation of Financial Assets – A.S.Ramasastri
?http://www.investopedia.com/terms/c/conv
exity.asp
?http://www.investopedia.com/articles/bond
s/08/duration-convexity.asp?viewed=1
?http://www.investopedia.com/university/ad
vancedbond/advancedbond6.asp
doc_797172332.pptx
The PPT explaining about Convexity Approach with ways to measure interest rate risk.
1. Interest Rate Risk
2. Ways of measuring Interest Rate Risk
3. Full valuation approach
? Example
4. Duration/Convexity Approach
? Duration
? What is duration?
? Calculation
? Properties
? Application
? Limitations
? Convexity
? Introduction
? Predicting percentage price change using convexity
? Examples
? Convexity for callable and putable bonds
? Applications
?Changes in Bond prices due to interest rates
fluctuations
? Formula of Bond Value calculation.
? Where
? Po=value of bond
? c1,c2,…,cn=cash flows expected from bond over ‘n’
periods
? r=discount rate
?Full Valuation Approach
?Duration - Convexity Approach
?The full valuation approach to measuring the
interest rate risk involves using a pricing
model to value individual bonds and can be
used to find the price impact of any scenario
of interest rate/yield curve changes.
?Adv/Disadv. of this approach:
? Precision
? Flexibility
? Complex
The Full Valuation Approach
Scenario Yield ? Bond X
(in millions)
Bond Y
(in millions)
Portfolio Portfolio
Value ?%
Current +0 bp ($108.42) ($81.78) ($190.21)
1 +50 bp ($106.23) ($77.93) ($184.17) -3.18%
2 +100 bp ($104.10) ($74.32) ($178.42) -6.20%
-2.02%
-3.99%
-4.71%
-9.12%
Bond X Bond Y
Coupon 8% 5%
Maturity (yr) 5 15
FV($) 100 100
YTM 6.00% 7.00%
?This approach provides an approximation of
the actual interest rate sensitivity of a bond
or a bond portfolio.
?Its main advantage is its
? Simplicity
? Quick
1. Approximate percentage change in price
for 1% change in yield
2. Duration is the slope of the price-yield
curve of bonds current YTM
3. Weighted average of time until each cash
flow would be received. Weights are
proportions of total bond value that each
cash flow presents
?Example: 20yr, 8% coupon PV=908, YTM=9%
? If yield decreases by 50 bps, price=952.3
? If yield increases by 50 bps, price=866.8
1. Coupon Rate: Lower Coupon rate means
Higher duration
2. Maturity: Longer Maturity means Higher
duration
The Full Valuation Approach
Scenario Yield ? Bond X
(in millions)
Bond Y
(in millions)
Portfolio Portfolio
Value ?%
Current +0 bp ($108.42) ($81.78) ($190.21)
1 +50 bp ($106.23) ($77.93) ($184.17) -3.18%
2 +100 bp ($104.10) ($74.32) ($178.42) -6.20%
-2.02%
-3.99%
-4.71%
-9.12%
3. Higher market yield means lower duration.
? Price Yield Curve for an option free bond
? 8% coupon rate, 20 year bond, YTM=8%, FV=100
$110.59
$100.00
$90.87
$85.00
$90.00
$95.00
$100.00
$105.00
$110.00
$115.00
Price Yield Curve
Price Vs Yield
?It is a good measure of sensitivity of a
portfolio, and can be used to reduce or
increase the exposure to a particular term
interest rate risk
?Large changes in interest rates
?This approach is applicable for a portfolio of
bonds with only parallel yield curve shifts.
? Measures how much a bond’s price-yield curve deviates from a
straight line
? Second derivative of price with respect to yield divided by
bond price
? BENEFIT: Allows us to improve the duration approximation for
bond price changes
? Duration underestimates actual prices
? Previous Example: 20yr, 8% coupon PV=908, YTM=9%
? Duration : 9.42
? Now if yield changes by 1% we can see changes as seen in diagram:
?Recall approximation using only duration:
?The predicted percentage price change
accounting for convexity is:
AP
P
×100 = ÷D
m
*
× Ay ×100
AP
P
×100 = ÷D
m
*
× Ay ×100
( )
+
1
2
× Convexity× (Ay)
2
×100
|
\
|
.
FIN 509 Class session 2 20
?Consider a 20-year 9% coupon bond selling at
$134.6722 to yield 6%. Coupon payments are
made semiannually.
?D
m
= 10.66
?The convexity of the bond is 164.106.
FIN 509 Class session 2 21
? If yields increase instantaneously from 6% to 8%, the
percentage price change of this bond is given by:
? First approximation (Duration):
–10.66 × .02 × 100 = –21.32
? Second approximation (Convexity)
0.5 × 164.106 × (.02)
2
× 100 = +3.28
Total predicted % price change: –21.32 + 3.28 = –18.04%
(Actual price change = –18.40%.)
FIN 509 Class session 2 22
? What if yields fall by 2%?
? If yields decrease instantaneously from 6% to 4%, the
percentage price change of this bond is given by:
? First approximation (Duration):
–10.66 × –.02 × 100 = 21.32
? Second approximation (Convexity)
0.5 × 164.106 × (–.02)
2
× 100 = +3.28
Total predicted price change: 21.32 + 3.28 = 24.60%
Note that predicted change is NOT SYMMETRIC.
?Comparison of bonds:
? A bond with greater convexity is less affected by
interest rates
? A bond with greater convexity will have higher
prices than bonds with lower convexity,
regardless of interest rate rise or fall
? With a callable or prepayable debt, the upside price
appreciation in response to decreasing yields is limited.
? When the price begins to rise at a decreasing rate in
response to further decreases in yield, the price yield
curve bends over to the left and exhibits negative
convexity.
? Compared to an option-free bond, a
putable bond will have less price
volatility at higher yields.
?Convexity is a interest rate risk management
tool, which is used in managing bond
portfolios
?If the combined convexity and duration of a
trading book is high, so is the risk. However,
if the combined convexity and duration are
low, the book is hedged, and little money
will be lost even if fairly substantial interest
movements occur.
?Fixed Income, Derivatives, and Alternative
Investments – CFA L 1, Kaplan Schweser
?Valuation of Financial Assets – A.S.Ramasastri
?http://www.investopedia.com/terms/c/conv
exity.asp
?http://www.investopedia.com/articles/bond
s/08/duration-convexity.asp?viewed=1
?http://www.investopedia.com/university/ad
vancedbond/advancedbond6.asp
doc_797172332.pptx