This document describes a framework for constructing efficient frontiers for fixed income portfolios under China's CROSS (China Risk Oriented Solvency System) regulatory framework. The framework uses quadratic programming to optimize portfolios to meet expected yield targets while staying within regulatory capital limits. A simulation case examines efficient frontiers with and without duration constraints. It finds that holding long-duration corporate bonds to maturity uses less regulatory capital than trading them. The framework allows insurance firms to maximize returns within capital limits by providing optimal asset allocations.

1. Efficient Frontier Searching of Fixed Income Portfolio under
China Solvency II (CROSS) Framework
Sun Zhi1
Abstract: With the enforcement of CROSS, regulatory capital optimization under the
CROSS (China Risk Oriented Solvency System) framework is becoming strategic for
insurance firms in China, Efficient frontier construction considering regulatory capital charge
is now an indispensable part of the decision process of annual strategic asset allocation. This
paper provides a systematic framework in constructing efficient frontiers for fixed income
portfolios through the application of quadratic programming theory. In this framework,
expected yield requirements, regulatory credit risk capital requirements and targeted interest
rate risk offsetting effect measured by modified duration can be fulfilled in a high level of
accuracy.
Key words: CROSS; Efficient frontier; Credit risk regulatory capital; Expected yield;
Duration
1. Introduction
China Risk oriented solvency system(CROSS) is a critical regulatory reform and key
innovation put forward by China Insurance Regulatory Commission(CIRC), in order to
regulate booming Chinese insurance industry in a more proactive and dynamic way. CROSS
is mainly comprised of three pillars with the first pillar impose minimum regulatory capital
based on the risk characteristics of various kinds of asset and liabilities and pillar two stresses
on the importance of enterprise wide risk management through add-on capital linked to the
quality of risk management conducts in insurance firms. Pillar three of CROSS establish a
systematic mechanism for regulatory reporting and market disclosure, strengthening the
influence of external supervision and market discipline on the governance of insurance firms.
The formal enacting and implementation of CROSS will inevitably exert great impact on the
entire insurance industry, millions of insurance policy holders as well as the real economy.
Under CROSS framework, the level of risk assumed by an insurance firm’s balance sheet
directly determines the amount of regulatory capital that needs to be set aside. Against the
backdrop of interest rate sliding down in China, more insurance firms are taking aggressive
asset driven expansion strategy. These firms tend to invest significantly in more risky
financial products with higher expected yield but significant higher risk. As a result, the
regulatory capital imposed on these firms is much higher than regulatory capitals required for
their counterparts following traditional insurance operation models.
1
Risk Management Department, Hua Xia Life Insurance Limited, Beijing 100086, China.
Email:sunzhi822@yahoo.com.sg. This paper represents only the personal views of the author.

2. With CROSS pending full implementation, regulatory capital optimization under the CROSS
framework is becoming a strategic task in virtually all insurance firms in China. In addition,
efficient frontier construction considering regulatory capital charge is now an essential step of
the decision process of annual strategic asset allocation in many sophisticated insurance firms.
This paper provides a systematic approach in searching for the optimal fixed income portfolio
where the regulatory credit risk capital limit is not breached and the expected yield reaches
the highest level possible, while achieving targeted interest rate risk offsetting effect
measured by modified duration.
2. Optimization Problem Formulation
Without loss of generality, in this paper, the investment choices of insurance firms are limited
to high-grade corporate bonds and risk free government bonds with modified duration from
one year to ten years. As CIRC only allow insurance firms to invest in AA class or above
corporate bonds, choices of corporate bonds are further limited to AAA, AA+, AA, AA-
corporate bonds. In CROSS framework corporate bonds that are classified as available for
sale or trading in accounting treatment are subject to credit spread risk charge only. As for
corporate bonds classified as hold to maturity in accounting treatment, these bonds are subject
to default risk charge. Government bonds are free of any credit related regulatory capital
charge in CROSS. With investment choices limited as above, both credit spread risk charge
and default risk charge can be represented as simple numerical vectors.
The following variables are defined for the ease of exposition:
x is a nonnegative vector, denoting the vector of fixed income investment choices, defined as
below.
x1 …x10: AAA corporate bonds classified as available for sale or trading in accounting with
duration from one year to ten years in ascending order.
x11 …x20: AA+ corporate bonds classified as available for sale or trading in accounting with
duration from one year to ten years in ascending order.
x21 …x30: AA corporate bonds classified as available for sale or trading in accounting with
duration from one year to ten years in ascending order.
x31 …x40: AA- corporate bonds classified as available for sale or trading in accounting with
duration from one year to ten years in ascending order.
x41 …x50: Government bonds classified as available for sale or trading in accounting with
duration from one year to ten years in ascending order.
x51…x100 are defined in the same fashion for AAA, AA+, AA, AA- rated corporate bonds and
Government bonds but classified as hold to maturity in accounting treatment.
y: vector of expected yields for AAA, AA+, AA, AA- rated corporate bonds and Government
bonds corresponding to durations from one year to ten years.

3. y1 …y50 represents expected yields of different types of bonds and durations similar to the
way x1 …x50 is defined. Accounting classification has no impact on the expected yields.
sc: credit spread charge vector given in CROSS.
dc: default charge vector in CROSS.
ρ: correlation factor used for aggregating credit spread risk charge and default risk charge.
The objective function of the optimization problem can be constructed as below:
Minimization of the square of regulatory capital assumed
Where: A = scT
*sc +dcT
*dc+2* ρ *scT
*dc
Based on the real data from CROSS regulation, it is easy to verify that A is an indefinite
matrix and hence RC(x) 2
is a non-convex quadratic function.
The following constraints are set up to factor in key criteria in making fixed income asset
allocation decisions including expected yield.
Constraint 1: investment size limit
∑
Where: Z denotes total investment size.
Constraint 2: expected yield target
∑ ∑
Where: Y denotes the expected yield target of the fixed income portfolio
Constraint 3: percentage limit on bonds classified as hold to maturity in accounting treatment
∑
Where: R is the maximum ratio of bonds that can classified as hold to maturity in accounting
treatment.
Constraint 4: portfolio duration target
Duration measures interest rate sensitivity of the fixed income portfolio and is a relatively
accurate measure in reflecting the amount of interest rate risk exposure originating from the
liability side of insurance firms that can be offset.

4. ∑ ∑
Where: D represents the overall fixed income portfolio duration.
By now, the questing of efficient frontier under CROSS for fixed income portfolio has been
formulated as a quadratic programming problem with linear constraints. Since the objective
function is not a convex function, there is no guarantee of a unique globally optimized
solution, which is available in closed formula if the objective function is convex. In this case,
the solution of this optimization problem has to resort to numerical methods.
3. Overview of the Numerical Methods
Quadprog function in Matlab’s Optimization Toolbox is a widely adopted local solver
implementation for quadratic problems. It is able to find a locally optimal solution to
quadratic optimization problems in a fast and reliable way. In this paper, a Multi-start
technique is used in conjunction with Matlab’s Quadprog function, which implements an
active-set strategy. The goal is to seek the best local optimal solution in the collection of local
optimal solutions arrived from different starting points located on a uniform grid with a
relatively large size of dimension.
A brief review of the key principles of the active-set strategy in solving quadratic
programming problem is given as below:
The optimization problem can be posed more generically as below:
The feasibility region is denoted as .
The active-set strategy is mainly composed of the following two steps:
Step 1: if the starting point is not feasible, a feasible starting point is calculated
through solving a linear programming problem.
Step 2: an active set matrix Sk is maintained as an estimate of the active constraints at the
solution point. Sk is updated at each iteration k. The search direction dk that minimizes the
objective function while remaining on active constraint boundaries is calculated.
Next, Lagrange multipliers, λk, are calculated. If all elements of λk are positive, xk is the
optimal solution of the quadratic programming problem. However, if any component of λk is
negative, and the component does not correspond to an equality constraint, then the
corresponding element is deleted from the active set and a new iteration point is sought.

5. 4. Key Simulation Results and the Projected Efficient Frontier
In order to provide more insights on the impact of CROSS regulatory framework on fixed
income investment decisions, a comprehensive simulation case is constructed mimicking the
real investment decision context of Chinese insurance firms.
As CIRC only allow insurance firms to invest in AA class or above corporate bonds, the
investment choices in this simulation case are limited to AAA, AA+, AA, AA- corporate
bonds and risk free government bonds. The yield curves for each type of the bonds are freely
available on www.chinabond.com.cn and these curves retrieved are used to construct the
following yield table with entries associated with different ratings and modified durations.
Modified Duration/
Risk Rating
AAA AA+ AA AA-
Government
Bond
1 3.334793 3.715183 4.075118 5.280336 2.598444
2 3.689818 4.06381 4.406722 6.108481 2.743498
3 3.752241 4.167423 4.534621 6.225571 2.880162
4 3.892922 4.375252 4.757626 6.447706 2.964192
5 4.030127 4.658369 5.061125 6.867294 3.071944
6 4.045612 4.731762 5.214426 6.995973 3.172192
7 4.10264 4.829489 5.38639 7.18747 3.167761
8 4.165817 4.956765 5.570569 7.367911 3.145756
9 4.235908 5.046077 5.684212 7.483511 3.209506
10 4.305998 5.135389 5.797856 7.613261 3.304508
Table 1: Bond yield in percentages as of 19th
of Nov 2015
Default charge table in CROSS is given as below:
Risk Rating Default Charge Factor
AAA 0.015
AA+ 0.036
AA 0.045
AA- 0.049
Table 2: CROSS default risk charge table for corporate bonds
Credit risk spread charge table in CROSS is as follows:
Risk Rating Modified Duration
Credit Spread Risk Charge
Factor
AAA
0<D≤5 D (-0.0015 D+0.0175)
D>5 D 0.010
AA+
0<D≤5 D (-0.0014 D+0.018)
D>5 D 0.011
AA
0<D≤5 D (-0.0013 D+0.0195)
D>5 D 0.013
AA-
0<D≤5 D (-0.0012 D+0.022)
D>5 D 0.016

6. Table 3: CROSS credit risk spread charge table for corporate bonds
With the above population of key variables, the simulation is conducted for various expected
yield targets with asset duration and without asset duration constraint. The percentage limit on
bonds classified as hold to maturity in accounting treatment is set as 20%. In each
optimization run, Matlab’s Quadprog function is executed with maximum evaluation run of
20,000, in order to achieve an accurate local optimal point. To ensure a reasonable level of
accuracy of the global convergence of the results, a multi-start strategy is deployed with a
9 100 uniform grid composed of different starting points. In the simulation results, the
converging trend of optimization results is reasonably good, and it improves further with
yield target getting higher.
The following figure depicts the simulated efficient frontier without restriction on the target
asset duration:
Figure 1: efficient frontier with no asset duration constraint
As shown from the figure above, there is a monotone increasing relationship between
expected yield and credit risk regulatory capital required in the efficient frontier. This testifies
the key principle behind CROSS regulatory framework, which impose more regulatory
capital for more risky assets with higher expected yield. In addition, the slope of increasing is
getting flatter as expected yield goes up. Namely, when expected yield reaches a certain level,
the marginal gain on expected yield is coupled with a significant increase in regulatory capital
charge. For example, in the figure above, an increase in expected yield from 7% to 7.5% will
result in an additional credit risk regulatory capital charge close to 5% of investment size,
which may not be economical when insurance firms’ own capital is scarce.
In most cases, insurance firms are facing significant interest rate risk arising predominantly
from the liability end due to its massive size and relatively long duration. Asset duration is a
key measure of the amount of interest rate exposure that can be mitigated through proper
investment in fixed income instruments. Target asset duration constraint can be easily
managed in the optimization framework given in this paper.
0.00%
1.00%
2.00%
3.00%
4.00%
5.00%
6.00%
7.00%
8.00%
0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00% 14.00%
Expected Yield
Minimum Credit Risk Capital Required/Portfolio Size
No Duration Constraint

7. The figure below is a contrast of two efficient frontiers with no asset duration constraint and
target asset duration of 10 years:
Figure 2: efficient frontiers with no asset duration constraint and target asset duration of 10
years
Based on the current values of CROSS credit risk spread charge factors and default risk
charge factors, hold to maturity corporate bonds with long duration tends to consume less
credit risk capital than the same type of bond but classified as trading or available for sale.
This simulation study does verify that hold to maturity corporate bonds of long duration are
preferred over the same type of bonds classified as trading or available for sale if credit risk
regulatory capital is the sole concern.
The figure below is a contrast of two efficient frontiers having hold to maturity asset ratio as
20% and 40% respectively, without asset duration constraint.
0.00%
1.00%
2.00%
3.00%
4.00%
5.00%
6.00%
7.00%
8.00%
0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00% 14.00%
Expected Yield
Minimum Credit Risk Capital Required/Portfolio Size
Target Duration = 10
No Duration Constraint
0.00%
1.00%
2.00%
3.00%
4.00%
5.00%
6.00%
7.00%
8.00%
0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00% 14.00%
Expected Yield
Minimum Credit Risk Capital Required/Portfolio Size
No Duration
Constraint,HTM Asset
Ratio=40%
No Duration
Constraint,HTM Asset
Ratio=20%

8. 5. Conclusion
Risk based capital regulatory regime is becoming a consensus among regulators from
different regions and key industry players in both banking and financial industry. CROSS is a
revolutionary initiative in China’s insurance regulation, which will have profound and long-
lasting effects on the operation of insurance firms in the Chinese market.
This paper establishes a comprehensive, easy to implement and efficient framework in the
form of a non-convex quadratic programming problem with linear constraints for the purpose
of fixed income investment decision. With this aid of this framework, insurance firms can
achieve significantly superior investment return without incurring additional regulatory
capital. As a natural product of the quadratic optimization problem, this framework provides
the investment choices across the fixed income instrument spectrum to achieve the minimum
regulatory capital with fixed expected yield target. Taking advantage of the monotone
increasing relationship between regulatory capital consumed and expected yield on the
efficient frontier, this framework is also able to provide investment choices achieving optimal
expected yield with given credit risk regulatory capital limit.
In addition to expected yield and regulatory capital, other key measures of significance to
fixed income investment such as asset duration and the ratio of hold to maturity assets are
also nicely fitted into the optimization framework of this paper. This provides more flexibility
for deriving optimal fixed income investment decisions. The targeted asset duration constraint
is especially critical as it measures interest rate risk mitigation effect. The interest rate risk
mitigation effect can be combined with credit risk regulatory capital to provide a more
complete picture on the impact of fixed income portfolio on the overall regulatory capital. It
is also found out that hold to maturity corporate bonds of long duration are preferred over the
same type of bonds classified as trading or available for sale if credit risk regulatory capital is
the sole concern.
This optimization framework is also highly scalable in two directions. Firstly, other specific
investment decision constrains can always be added as linear constraints with little impact on
performance. In addition, the choices of bonds can be extended to bonds with durations that
are not integer through enlarging the size of the investment choice vector.

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