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Yield Curve Modeling Modeling The Tax-Specific Yield Curve Copyright © 1996-2006 Investment Analytics
Modeling the Tax-Specific Yield Curve The impact of taxes ¾ Implication for bond valuation ¾ Bond Efficiency ¾ Tax Arbitrage ¾ Building an After-Tax Yield Curve ¾
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 2
Tax Effects Tax effects in most bond markets ¾ Tax effects are important even for tax-exempt investors ¾ Tax effects typically create arbitrage opportunities, depending on: ¾
9frictions (spreads) 9short-selling constraints 9asymmetric tax treatment of long and short positions
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 3
Summary of Analysis Without Taxes ¾
Price = Present value for each bond 9 All bonds, all investors)
Each investor prepared to hold any bond ¾ PRICE = COUPON X ANNUITY FACTOR + 100 X DISCOUNT FACTOR ¾
n
P =
∑
C × Di + 100 × Dn
1 n
P = C ×
∑
Di + 100 × Dn
1
P = C × AF + 100 × Dn Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 4
Price
The Price-Coupon Relationship with No Taxes
Coupon (%) Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 5
The Problem of Tax Effects ¾
Investors in different tax brackets will not agree on relative value of bonds Coupon 3% 9% 15% V15/V3
After Tax Cash Flows T = 0% T = 30% T = 60% 103 102.1 101.2 109 106.3 103.6 115 110.5 106 115.0/103.0 110.5/102.1106.0/101.2 = 1.117 = 1.082 = 1.047
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 6
Tax Analysis of Bonds ¾
Suppose we live a world with 3 coupon bonds • All three are 1 year bonds • Coupons are 3%, 9% and 15% • The 1 yr. spot rate is 10%
¾
Suppose we have 3 groups of tax-payers • 0%, 30% and 60%
¾
What are the bond prices? • What are the post-tax cash flows? • What are the post-tax yields?
¾
Use worksheet: Bonds & Taxes
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 7
Solution: Tax Analysis of Bonds 1 3% 93.64
BOND 2 3 9% 15% 99.09 104.55
103.0 102.1 101.2
109.0 106.3 103.6
10% 9.0% 8.1%
10% 7.3% 4.6%
¾
Coupon Price After-tax cash flows Tax rate 0% Tax rate 30% Tax rate 60%
115.0 110.5 106.0
After-tax yield Tax rate 0% Tax rate 30% Tax rate 60% Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
10% 5.7% 1.4% Slide: 8
Implications of Tax Analysis ¾
A Zero Tax Payer • All the bonds offer the same yield • Hence, s/he will hold any of the bonds
¾
A Tax Payer • Will receive higher yield on low coupon bond • Hence, will prefer a low coupon bond to high coupon bond
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 9
Efficient Bonds Bond Price > NPV of cash flows for some investors ¾ An Efficient Bond: ¾
• Price = NPV ¾
Example: • The 3% coupon bond is efficient • For both 30% and 60% taxpayers
¾
Next Issue: What is the spot rate?
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 10
Post-Tax Spot Rate ¾
Post-Tax Yields: Non-Taxpayer • For the 0% tax payer, the yield on all bonds is 10% • Hence pre-tax spot rate = post-tax spot rate = 10%
¾
Post-Tax Yields: Taxpayers • Yields vary for each bond • Which yield is the post-tax spot rate?
¾
The Post-Tax Spot Rate: the highest yield • The yield on the efficient bond
¾
Example: • For 30% taxpayers, post-tax spot rate is 9% • For 60% taxpayers, post-tax sport rate is 8.1%
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 11
Tax Specific Yield Curves Investors in different tax brackets will see the same price for different after-tax cashflows ¾ No one investor will set prices for all bonds ¾ Different investors will therefore have different after-tax discount factors and yield curves ¾ Investors in different tax brackets may or may not agree to hold the same bond ¾
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 12
Post-Tax Bond Valuation Spot rates are different for different tax brackets ¾ So they will not agree on bond values ¾ Example: and 8% coupon 1 year bond ¾
• Use the Bonds & taxes worksheet • Use the post-tax spot rates • Find the NPV of the post-tax cash flows
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 13
Example: Bond Valuation Value of an 8% 1-year Bond ¾ Tax Post-Tax Post-Tax Rate Spot rate Cash Flow ¾
0% 30% 60%
10.0% 9.0% 8.1%
Copyright © 1996-2006 Investment Analytics
108.0 105.6 103.2
Modeling Tax-Specific Yield Curves
Present Value 98.18 96.88 95.47
Slide: 14
Tax Clientele Hypothesis ¾
No Tax Effects • All bonds priced so they can be held optimally by a 0% tax payer • Price coupon relationship linear • No arbitrage
¾
Clientele Effects • Different bonds prices so they can be held optimally by different tax• •
clienteles of investors Price coupon relationship non-linear Arbitrage
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 15
Tax Implication Assume different tax rates ¾ Bonds with different coupons ¾ Then there will be a tax arbitrage for at least one tax bracket ¾
• Note: Green & Oedegaard reject the no-tax hypothesis in formal test of US market
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 16
Tax Arbitrage: Non-linear PriceCoupon Curve Cost saving{
B
C
Price
B* A
Coupon Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 17
Rich/Cheap Analysis & Relative Value Trading Select the appropriate tax-rate ¾ Identify the tax-efficient bonds ¾ Plot the spot tax-yield curve using the efficient bonds ¾ Identify the issues which are low yield (‘rich’) or high yield (‘cheap’) relative to the curve ¾ Initiate duration-weighted trade ¾
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 18
Rich/Cheap Graphical Analysis
Yield
Cheap Issue
Rich Issues
Maturity Modeling Tax-Specific Yield Curves
Copyright © 1996-2006 Investment Analytics
Slide: 19
Example: Bond Replication ¾
Bond
¾
A 3% 103 B 5% 105 C 8% 108 Replicate Bond B, using Bonds A & C: • • • •
Coupon
Cash Flows Price 93.64 96.00 98.18
Create B* = W1 A + W2 C Require W1 and W2 so that W1 (103) + W2 (108) = 105 W1 + W2 = 1
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 20
Example: Bond Replication ¾
Create replicating Bond B* • B* = 3/5 A + 2/5 C
¾
Check cash flows: • 3/5(103) + 2/5(108) = 105
¾
Cost Saving • Price of Bond B Cost of B* = 3/5(93.64 ) + 2/5 (98.18 ) Cost Saving
¾
96.00 95.45 0.55
Arbitrage Trade • Sell 10 x Bond B • Buy 6 x Bond A and 4 x Bond C • Riskless profit of $5.50
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 21
Tax Arbitrage: Basic Idea ¾
Use LP to find bond portfolio which replicates CF’s of target bond, more cheaply. 9Will identify & use efficient bonds
Pick a target bond & replicate test for each tax rate. ¾ Restrictions on arbitrage: ¾
9Transaction costs 9Short sale restrictions
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 22
Linear Programing ¾
Maximize (or minimize) an objective function 9Subject to a set of constraints
¾
Objective function and constraints are linear 9Maximize x1p1 + x2p2 9Subject to: • x1a11 + x2a12 <= b1 • x1a21 + x2a22 <= b2
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 23
Linear Programing x1 x1a11 + x2a12 = b1
x1a21 + x2a22 = b2 Solution space
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
x2
Slide: 24
Summary: Tax Effects Taxes important for both tax paying and taxexempt investors ¾ Strong evidence of tax-clienteles ¾ Taxes will influence investors choice of bonds ¾ A non-linear price-coupon relationship implies arbitrage opportunities ¾
• Studies in Germany, Japan, UK & USA all show that prices reflect non-zero income and capital gains taxes
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 25
Building After-Tax Yield Curves ¾
Use LP to find efficient bonds: 9Bonds that minimize the cost of a given cash flows
¾
Build a yield curve from the efficient bonds 9Use regression and basis splines
Curve will be different for each tax bracket ¾ NB- Assumptions: ¾
9No short sales allowed 9No transaction costs
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 26
LP Formulation for Cost Minimization ¾
Optimal portfolio: 9 Provides cash flows at a minimum cost 9 Minimize cost of a given set of cashflows from a portfolio of bonds m
¾
Notation: • • • • • • •
j = 1, . . .,T Periods in time I = 1, . . ., m Bonds in portfolio x: Bond holding p: Bond price s: Cash flows from portfolio d: Discount factor a: After tax payment from bond
Min ∑ x i p i i =1
Subject to :
∑ a ij x i ≥ s j i
xi ≥ 0 Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 27
LP Formulation for Cashflow Maximization ¾
Maximize discounted cashflows from given portfolio 9 Subject to a yield curve generated from efficient bonds
¾
Notation: • • • • • • •
T
j = 1, . . .,T Periods in time I = 1, . . ., m Bonds in portfolio x: Bond holding p: Bond price s: Cash flows from portfolio d: Discount factor a: After tax payment from bond
M ax ∑ s jd j j =1
S u b ject to : T
∑ a ij d j ≤ p i j =1
d j ≥0 Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 28
LP & Basis Splines ¾
Use weighted sum of basis splines to represent discount function: L
d (t j ) = ∑ αl f l (t j ) 1 L
T
Max ∑ ωlαl
ωl = ∑ s j f l ( t j )
l =1
j =1
Subject to: L
T
∑ αl ∑ aij fl ( t j ) ≤ l =1
j =1
Copyright © 1996-2006 Investment Analytics
pi
Modeling Tax-Specific Yield Curves
Slide: 29
Portfolio Cashflows ¾
Simple version: 9Set all cashflows = 1
¾
More advanced: 9Make objective function equally sensitive to changes in yield curve at all points • Set cashflows: Sj = [jd(tj)]-1 • Iterate to compute the values of sj and identify efficient bonds.
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 30
Curve Building Procedure ¾
Use LP to select the efficient bonds for given tax bracket 9This will select a small number of bonds
¾
Include some slightly less efficient bonds to fill out the curve 9Use bonds which are “relatively efficient”
¾
Apply basis splines & fit regression model
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 31
Relative Efficiency ¾
Define the Relative Efficiency of a bond for a given tax bracket as: 9NPV of Cash flows / Bond Price
Define a tolerance level (e.g. 99%) ¾ Include bonds with: Relative Efficiency > Tolerance ¾
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 32
Lab: Pencoa Fund Management ¾
Scenario: 9Running private client bond fund 9Customers in different tax brackets (0%, 25%, 50%)
Analyze market and recommend suitable purchases for different tax clienteles ¾ Worksheet: Pencoa Fund Management ¾
9See Lab and Solution Notes
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 33
Solution: Pencoa Fund Management 25% tax rate 12% 10% 8% 6% 4% 2% 0% 0
500
1000
LP Spot Regression Spot
Copyright © 1996-2006 Investment Analytics
1500
2000
2500
3000
LP Forw ard Regression Forw ard
Modeling Tax-Specific Yield Curves
Slide: 34
Summary: Modeling the Tax-Specific Yield Curve The impact of taxes ¾ Implication for bond valuation ¾ Bond Efficiency ¾ Tax Arbitrage ¾ Building an After-Tax Yield Curve ¾
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 35
Modeling the Tax-Specific Yield Curve The impact of taxes ¾ Implication for bond valuation ¾ Bond Efficiency ¾ Tax Arbitrage ¾ Building an After-Tax Yield Curve ¾
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 2
Tax Effects Tax effects in most bond markets ¾ Tax effects are important even for tax-exempt investors ¾ Tax effects typically create arbitrage opportunities, depending on: ¾
9frictions (spreads) 9short-selling constraints 9asymmetric tax treatment of long and short positions
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 3
Summary of Analysis Without Taxes ¾
Price = Present value for each bond 9 All bonds, all investors)
Each investor prepared to hold any bond ¾ PRICE = COUPON X ANNUITY FACTOR + 100 X DISCOUNT FACTOR ¾
n
P =
∑
C × Di + 100 × Dn
1 n
P = C ×
∑
Di + 100 × Dn
1
P = C × AF + 100 × Dn Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 4
Price
The Price-Coupon Relationship with No Taxes
Coupon (%) Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 5
The Problem of Tax Effects ¾
Investors in different tax brackets will not agree on relative value of bonds Coupon 3% 9% 15% V15/V3
After Tax Cash Flows T = 0% T = 30% T = 60% 103 102.1 101.2 109 106.3 103.6 115 110.5 106 115.0/103.0 110.5/102.1106.0/101.2 = 1.117 = 1.082 = 1.047
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 6
Tax Analysis of Bonds ¾
Suppose we live a world with 3 coupon bonds • All three are 1 year bonds • Coupons are 3%, 9% and 15% • The 1 yr. spot rate is 10%
¾
Suppose we have 3 groups of tax-payers • 0%, 30% and 60%
¾
What are the bond prices? • What are the post-tax cash flows? • What are the post-tax yields?
¾
Use worksheet: Bonds & Taxes
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 7
Solution: Tax Analysis of Bonds 1 3% 93.64
BOND 2 3 9% 15% 99.09 104.55
103.0 102.1 101.2
109.0 106.3 103.6
10% 9.0% 8.1%
10% 7.3% 4.6%
¾
Coupon Price After-tax cash flows Tax rate 0% Tax rate 30% Tax rate 60%
115.0 110.5 106.0
After-tax yield Tax rate 0% Tax rate 30% Tax rate 60% Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
10% 5.7% 1.4% Slide: 8
Implications of Tax Analysis ¾
A Zero Tax Payer • All the bonds offer the same yield • Hence, s/he will hold any of the bonds
¾
A Tax Payer • Will receive higher yield on low coupon bond • Hence, will prefer a low coupon bond to high coupon bond
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 9
Efficient Bonds Bond Price > NPV of cash flows for some investors ¾ An Efficient Bond: ¾
• Price = NPV ¾
Example: • The 3% coupon bond is efficient • For both 30% and 60% taxpayers
¾
Next Issue: What is the spot rate?
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 10
Post-Tax Spot Rate ¾
Post-Tax Yields: Non-Taxpayer • For the 0% tax payer, the yield on all bonds is 10% • Hence pre-tax spot rate = post-tax spot rate = 10%
¾
Post-Tax Yields: Taxpayers • Yields vary for each bond • Which yield is the post-tax spot rate?
¾
The Post-Tax Spot Rate: the highest yield • The yield on the efficient bond
¾
Example: • For 30% taxpayers, post-tax spot rate is 9% • For 60% taxpayers, post-tax sport rate is 8.1%
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 11
Tax Specific Yield Curves Investors in different tax brackets will see the same price for different after-tax cashflows ¾ No one investor will set prices for all bonds ¾ Different investors will therefore have different after-tax discount factors and yield curves ¾ Investors in different tax brackets may or may not agree to hold the same bond ¾
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 12
Post-Tax Bond Valuation Spot rates are different for different tax brackets ¾ So they will not agree on bond values ¾ Example: and 8% coupon 1 year bond ¾
• Use the Bonds & taxes worksheet • Use the post-tax spot rates • Find the NPV of the post-tax cash flows
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 13
Example: Bond Valuation Value of an 8% 1-year Bond ¾ Tax Post-Tax Post-Tax Rate Spot rate Cash Flow ¾
0% 30% 60%
10.0% 9.0% 8.1%
Copyright © 1996-2006 Investment Analytics
108.0 105.6 103.2
Modeling Tax-Specific Yield Curves
Present Value 98.18 96.88 95.47
Slide: 14
Tax Clientele Hypothesis ¾
No Tax Effects • All bonds priced so they can be held optimally by a 0% tax payer • Price coupon relationship linear • No arbitrage
¾
Clientele Effects • Different bonds prices so they can be held optimally by different tax• •
clienteles of investors Price coupon relationship non-linear Arbitrage
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 15
Tax Implication Assume different tax rates ¾ Bonds with different coupons ¾ Then there will be a tax arbitrage for at least one tax bracket ¾
• Note: Green & Oedegaard reject the no-tax hypothesis in formal test of US market
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 16
Tax Arbitrage: Non-linear PriceCoupon Curve Cost saving{
B
C
Price
B* A
Coupon Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 17
Rich/Cheap Analysis & Relative Value Trading Select the appropriate tax-rate ¾ Identify the tax-efficient bonds ¾ Plot the spot tax-yield curve using the efficient bonds ¾ Identify the issues which are low yield (‘rich’) or high yield (‘cheap’) relative to the curve ¾ Initiate duration-weighted trade ¾
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 18
Rich/Cheap Graphical Analysis
Yield
Cheap Issue
Rich Issues
Maturity Modeling Tax-Specific Yield Curves
Copyright © 1996-2006 Investment Analytics
Slide: 19
Example: Bond Replication ¾
Bond
¾
A 3% 103 B 5% 105 C 8% 108 Replicate Bond B, using Bonds A & C: • • • •
Coupon
Cash Flows Price 93.64 96.00 98.18
Create B* = W1 A + W2 C Require W1 and W2 so that W1 (103) + W2 (108) = 105 W1 + W2 = 1
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 20
Example: Bond Replication ¾
Create replicating Bond B* • B* = 3/5 A + 2/5 C
¾
Check cash flows: • 3/5(103) + 2/5(108) = 105
¾
Cost Saving • Price of Bond B Cost of B* = 3/5(93.64 ) + 2/5 (98.18 ) Cost Saving
¾
96.00 95.45 0.55
Arbitrage Trade • Sell 10 x Bond B • Buy 6 x Bond A and 4 x Bond C • Riskless profit of $5.50
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 21
Tax Arbitrage: Basic Idea ¾
Use LP to find bond portfolio which replicates CF’s of target bond, more cheaply. 9Will identify & use efficient bonds
Pick a target bond & replicate test for each tax rate. ¾ Restrictions on arbitrage: ¾
9Transaction costs 9Short sale restrictions
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 22
Linear Programing ¾
Maximize (or minimize) an objective function 9Subject to a set of constraints
¾
Objective function and constraints are linear 9Maximize x1p1 + x2p2 9Subject to: • x1a11 + x2a12 <= b1 • x1a21 + x2a22 <= b2
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 23
Linear Programing x1 x1a11 + x2a12 = b1
x1a21 + x2a22 = b2 Solution space
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
x2
Slide: 24
Summary: Tax Effects Taxes important for both tax paying and taxexempt investors ¾ Strong evidence of tax-clienteles ¾ Taxes will influence investors choice of bonds ¾ A non-linear price-coupon relationship implies arbitrage opportunities ¾
• Studies in Germany, Japan, UK & USA all show that prices reflect non-zero income and capital gains taxes
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 25
Building After-Tax Yield Curves ¾
Use LP to find efficient bonds: 9Bonds that minimize the cost of a given cash flows
¾
Build a yield curve from the efficient bonds 9Use regression and basis splines
Curve will be different for each tax bracket ¾ NB- Assumptions: ¾
9No short sales allowed 9No transaction costs
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 26
LP Formulation for Cost Minimization ¾
Optimal portfolio: 9 Provides cash flows at a minimum cost 9 Minimize cost of a given set of cashflows from a portfolio of bonds m
¾
Notation: • • • • • • •
j = 1, . . .,T Periods in time I = 1, . . ., m Bonds in portfolio x: Bond holding p: Bond price s: Cash flows from portfolio d: Discount factor a: After tax payment from bond
Min ∑ x i p i i =1
Subject to :
∑ a ij x i ≥ s j i
xi ≥ 0 Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 27
LP Formulation for Cashflow Maximization ¾
Maximize discounted cashflows from given portfolio 9 Subject to a yield curve generated from efficient bonds
¾
Notation: • • • • • • •
T
j = 1, . . .,T Periods in time I = 1, . . ., m Bonds in portfolio x: Bond holding p: Bond price s: Cash flows from portfolio d: Discount factor a: After tax payment from bond
M ax ∑ s jd j j =1
S u b ject to : T
∑ a ij d j ≤ p i j =1
d j ≥0 Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 28
LP & Basis Splines ¾
Use weighted sum of basis splines to represent discount function: L
d (t j ) = ∑ αl f l (t j ) 1 L
T
Max ∑ ωlαl
ωl = ∑ s j f l ( t j )
l =1
j =1
Subject to: L
T
∑ αl ∑ aij fl ( t j ) ≤ l =1
j =1
Copyright © 1996-2006 Investment Analytics
pi
Modeling Tax-Specific Yield Curves
Slide: 29
Portfolio Cashflows ¾
Simple version: 9Set all cashflows = 1
¾
More advanced: 9Make objective function equally sensitive to changes in yield curve at all points • Set cashflows: Sj = [jd(tj)]-1 • Iterate to compute the values of sj and identify efficient bonds.
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 30
Curve Building Procedure ¾
Use LP to select the efficient bonds for given tax bracket 9This will select a small number of bonds
¾
Include some slightly less efficient bonds to fill out the curve 9Use bonds which are “relatively efficient”
¾
Apply basis splines & fit regression model
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 31
Relative Efficiency ¾
Define the Relative Efficiency of a bond for a given tax bracket as: 9NPV of Cash flows / Bond Price
Define a tolerance level (e.g. 99%) ¾ Include bonds with: Relative Efficiency > Tolerance ¾
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 32
Lab: Pencoa Fund Management ¾
Scenario: 9Running private client bond fund 9Customers in different tax brackets (0%, 25%, 50%)
Analyze market and recommend suitable purchases for different tax clienteles ¾ Worksheet: Pencoa Fund Management ¾
9See Lab and Solution Notes
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 33
Solution: Pencoa Fund Management 25% tax rate 12% 10% 8% 6% 4% 2% 0% 0
500
1000
LP Spot Regression Spot
Copyright © 1996-2006 Investment Analytics
1500
2000
2500
3000
LP Forw ard Regression Forw ard
Modeling Tax-Specific Yield Curves
Slide: 34
Summary: Modeling the Tax-Specific Yield Curve The impact of taxes ¾ Implication for bond valuation ¾ Bond Efficiency ¾ Tax Arbitrage ¾ Building an After-Tax Yield Curve ¾
Copyright © 1996-2006 Investment Analytics
Modeling Tax-Specific Yield Curves
Slide: 35
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