BOND PORTFOLIO MANAGEMENT:
INCORPORATING DEFAULT
PROBABILITIES
MS&E 348 Final Presentation
Assignment Overview
Design a system for optimal bond management
Analyze impact of the default risk of corporate
bonds
Specific objectives:
1)
2)
3)
4)
5)
Create an appropriate utility function
Select and calibrate interest rate model
Implement two-stage stochastic optimization model
Develop method of incorporating default risk
Conduct sensitivity analysis
Model Overview
Generate Scenarios
Two-Stage Stochastic Model
Max , + , Subject to:
Budget:
% & ∗ & ≤ )
Inventory Balance:
= & + − ( )
Cash Balance:
At each node:
Interest Rate
Curves
Bond Prices
And Cash
Flows
& + (1 − #) = 1 + # ( )
Where –
= = = = !"
= "
# = !$"$"
Piece-wise Utility Function
4 pieces
./ corresponds to annual
returns of:
3% ($1.06m)
5% ($1.10m)
7% ($1.14m)
GAMS Implementation:
% !
" " = !!$
!
". " = ∆./
= % !
" ∗ 2
CIR Interest Rate Model
Cox, Ingersoll, Ross model for short rates
Incorporates
Where:
mean reversion
5
!34
= !35 + 6 7 − !35 8 + 9 !35 :35
5
!34
= new short rate
5
!3 = old short rate
6, 7 = CIR model parameters
8 = model time step
5
:3
= N(0, 8)
CIR Simulated Zero-Curves
Old CIR Parameters
Revised CIR Parameters
Getting the Input – Pricing Bonds
Matlab was used to generate all interest rate
scenarios, price bonds, and write to GAMS
Bond pricing:
Calculated
NPV of bond cash flows under CIR yield
curve
Coupon payments and bond maturity – cash
accumulation under small risk-free rate (r = .01)
Adding Corporate Bond Default
Two methods for incorporating bond default
probabilities
1)
2)
Straight-to-Default – probability of bond default given
current rating and bond age
Rating Transition – constant probability of changing
bond ratings each period (ex. AA to B, BBB to default)
From/To
AAA
AA
A
BBB
BB
B
CCC/C
AAA
92%
1%
*
*
*
*
*
AA
7%
90%
2%
*
*
*
*
A
*
8%
91%
4%
*
*
*
BBB
*
*
6%
90%
6%
*
*
BB
*
*
*
5%
83%
5%
1%
B
*
*
*
*
9%
83%
11%
CCC/C
*
*
*
*
*
5%
53%
Default
*
*
*
*
1%
6%
34%
Corporate Bonds – Credit Spread
For each bond rating, a premium is added to the
bond’s annual yield and the bond is re-priced
8.0%
7.0%
6.0%
5.0%
Yield
BBB
A
AA
AAA
TSY
4.0%
3.0%
2.0%
1.0%
0.0%
5
10
15
20
Years To Maturity
25
30
Mechanics of Bond Transition
Current Rating =
A
2%
AA
Price = $102
C.S. = +87 bp
91%
A
Price = $100
C.S. = +117 bp
6%
BBB
Price = $97
C.S. = +198 bp
• For each bond, a uniform random distribution is sampled to determine bond transition
path
Number of Defaults Generated
Straight-to-Default Probability Approach:
Bond Rating
AAA
AA
A
BBB
# of Defaults
7
9
47
297
A
448
1234
11
BBB
1015
1054
134
Rating Transition Approach:
Upgrade
Downgrade
Defaults
AAA
0
1727
1
AA
59
2027
8
Number of Scenarios
Because default probabilities are low, a large
number of scenarios are needed to accurately
account for defaults
Tradeoff due to computational capacity –
# of scenarios/trial vs. # of trials run
Current runs = 100 x 100 =10,000 total scenarios
GAMS Optimization has:
31k
single equations
63k single variables
237k non-zero elements
Initial Results – US Treasury Bonds
10 bond sample (short to long term)
1st period – 100% on bond with highest yield
2nd period – 100% on bond that matures at the end
of period
Optimization
does not want to take on curve risk
Places money in high yield to start with, then puts in
“sure-thing” bond
Conclusions –
Factor
away curve risk by only optimizing bonds with-in
same asset classes (short-term vs. long-term)
Results with Corporate Bonds
Tested only two bond asset classes – short term (<5
yrs) and long term (25-30 yrs)
For each asset class, we tested:
~10
corporate bonds of various ratings and sectors
~10 corporates + relevant maturity Treasury bonds
Types of corporate bonds –
Finance, Utility, Industrial sectors
Treasuries
Corporate
Corporate Bonds – Short term (<5 yrs)
Rating
AAA
AA
AA
AA
A
A
A
BBB
BBB
BBB
BBB
TSY
TSY
TSY
TSY
Coupons Maturity
5.45
4
4.375
2
4.75
5
6.25
3m
4.25
5
6.4
7m
7.25
2
6.45
3
6.95
4
6.15
5
6.25
5
4.5
6m
4.875
1
5
3
4.25
5
Treasury + Corporate
Stage 1
Stage 2
0%
0%
21%
1%
15%
79%
7%
17%
15%
16%
28%
0%
0%
0%
0%
Treasuries
Corporate
Corporate Bonds – Short term (<5 yrs)
Rating
AAA
AA
AA
AA
A
A
A
BBB
BBB
BBB
BBB
TSY
TSY
TSY
TSY
TSY
TSY
TSY
Coupons
5.45
4.375
4.75
6.25
4.25
6.339
7.25
6.45
6.95
6.15
6.25
3.875
4.25
4.5
4.875
5
5
4.375
Maturity
4
2
5
3m
5
7m
2
3
4
5
5
5
5
6m
1
3
3
4
Corporate Only
Stage 1
Stage 2
1%
1%
6%
N/A
3%
100%
N/A
0%
17%
21%
24%
25%
Treasury + Corporate
Stage 1 Stage 2
1%
0%
4%
N/A
5%
100%
N/A
7%
24%
17%
19%
23%
0%
0%
0%
0%
0%
0%
0%
TSY
Corporate
Corporate Bonds – Long term (25-30 yrs)
Rating
AAA
AA
A
A
A
BBB
BBB
BBB
BBB
TSY
TSY
TSY
Coupons
5.95
5.55
5.375
6.5
6.125
6.65
6.75
6.8
7
4.5
4.75
4.375
Maturity
29
29
27
29
30
28
29
30
30
28
29
30
Corporate Only
Stage 1
Stage 2
2%
7%
11%
100%
2%
6%
14%
20%
19%
20%
Treasury and Corporate
Stage 1
Stage 2
8%
5%
9%
100%
3%
11%
17%
18%
16%
14%
0%
0%
0%
Bond Correlations
Treasury Correlations
1
2
3
4
5
6
1
2
3
1
1
1
1
1
1
1
1
1
1
1
1
0.999 1
1
0.999 0.999 0.999
4
1
1
1
1
1
1
5
6
0.999 0.999
1 0.999
1 0.999
1
1
1
1
1
1
Corporate Correlations
TSY
AAA
AA
A
BBB
BBB
TSY
1.00
0.92
0.78
0.65
0.75
0.23
AAA
0.92
1.00
0.73
0.60
0.71
0.20
AA
0.78
0.73
1.00
0.48
0.55
0.15
A
0.65
0.60
0.48
1.00
0.45
0.46
BBB
0.75
0.71
0.55
0.45
1.00
0.14
BBB
0.23
0.20
0.15
0.46
0.14
1.00
Sensitivity Analyses
Adjusted utility function to highly penalize downside
Results in allocation to cash instead of Treasury bonds
Does result in slight diversification (2 bonds in stage 1)
Adjusted credit spreads
Increased spreads do not result in additional diversification
Mismatch between current (empirical) price data and
simulated price data
Possible solution to diversification problems:
Impose position limits
TSY
Corporate
Corporate Bonds – Long term (25-30 yrs)
Rating
AAA
AA
A
A
A
BBB
BBB
BBB
BBB
TSY
TSY
TSY
Coupons Maturity
5.95
29
5.55
29
5.375
27
6.5
29
6.125
30
6.65
28
6.75
29
6.8
30
7
30
4.5
28
4.75
29
4.375
30
Treasury and Corporate
Stage 1
Stage 2
4%
4%
38%
8%
62%
4%
5%
18%
20%
19%
19%
0%
0%
0%
Further Considerations
Credit spreads are constant across time, maturity,
and sector
Treasury
bonds have inferior Sharpe ratios
Precludes “Flight To Quality” scenarios such as 2008
Transition probabilities are constant across time
Utility function
Empirical
to model-based pricing
Adjust gradients and wealth levels in each period
Conclusions
Optimized portfolio is highly sensitive to:
Bond
pricing (Interest rate models and credit spreads)
Default probability
Diversification is good, but is highly dependent on:
Risk-aversion
factor of investor
Credit spreads (risk-aversion factor of market)
Generating scenarios and optimizing bond
performance has a high computational cost –
Need
to narrow down bond list to a small number
before running optimization
Infeasible
for entire market of bonds
Questions for Future Research
How does the implementation of a different interest
rate model affect optimization outcomes?
How much does improved fitting of the interest rate
model to the current yield environment affect actual
bond performance?
How does the optimized outcome fare in a historical
back-test against bond index funds?