Factor-Based Asset Return Simulation Model

Factor-Based Asset Return Simulation Model: An Integrated Approach to Multi-Dimensional Risk Factors

This is foundational research for our analysis. Various models can be set up for asset class paths, and since Minami Lab conducts quantitative analysis based on multi-factor models, we have made the following settings as a new approach.

In modern financial markets, asset returns have a complex structure driven by multiple systematic risk factors. To reflect realistic characteristics that cannot be captured by simple multivariate normal distribution models, this study proposes a comprehensive factor-based simulation model. By comprehensively considering common factors, seasonality effects, policy cycles, market environment shocks, and time-varying correlation structures, this model reproduces more realistic stochastic processes of asset returns.

Theoretical Framework of the Model

Basic Model Structure

The return $R_{i,t}$ of asset $i$ at time $t$ is expressed by the following comprehensive factor model:

R_{i,t} = \sum_{j=1}^{3} \beta_{i,j} \cdot F_{j,t} + S_{i,t} + P_t + \text{Shock}_{i,t}(\text{regime}_t) + \epsilon_{i,t}

Each component is defined as follows:

Symbol	Definition
$\beta_{i,j}$	Sensitivity of asset $i$ to factor $j$ (factor loading)
$F_{j,t}$	Return of factor $j$ at time $t$
$S_{i,t}$	Seasonality effect
$P_t$	Policy cycle effect
$\text{Shock}_{i,t}$	Market environment-dependent shock
$\epsilon_{i,t}$	Idiosyncratic risk (residual term)

Factor Structure

This model adopts three common factors widely supported by empirical research. Each factor is assumed to follow an independent normal distribution $F_{j,t} \sim \mathcal{N}(\mu_j, \sigma_j^2)$ .

Factor	Expected Return	Volatility	Economic Meaning
Growth	0.5%/month	3.0%/month	Growth stock factor
Value	0.4%/month	2.5%/month	Undervalued stock factor
Quality	0.3%/month	2.0%/month	High-quality company factor

Factor Sensitivities

The sensitivity of each asset to factors is set as follows:

Asset	Growth	Value	Quality
Equity	0.7	0.2	0.1
Bonds	-0.1	0.3	0.6
REIT	0.3	0.5	0.2
Commodities	0.2	0.1	0.1
Cash	0.0	0.0	0.0

Seasonality Effect

The seasonality effect $S_{i,t}$ reflects monthly patterns observed in empirical research:

S_{i,t} = \alpha_{t \bmod 12}

Month	Jan	Feb	Mar	Apr	May	Jun
Adjustment	+0.3%	+0.1%	0.0%	-0.1%	-0.2%	-0.3%

Month	Jul	Aug	Sep	Oct	Nov	Dec
Adjustment	-0.2%	-0.1%	0.0%	+0.1%	+0.2%	+0.3%

This setting considers known seasonal patterns such as the January effect, "Sell in May" effect, and year-end rally.

Policy Cycle Effect

The policy cycle effect $P_t$ is expressed as a sine wave with a 3-year (36-month) cycle:

P_t = 0.002 \times \sin\left(\frac{2\pi \cdot (t \bmod 36)}{36}\right)

This setting reflects the medium-term impact of political and policy cycles on financial markets.

Market Environment Shocks

Asset-specific shocks occur depending on the market environment $\text{regime}_t$ :

Market Environment	Equity	Bonds	REIT	Commodities
Rate Hike Period	-0.2%	-1.0%	0%	0%
Inflation Period	0%	0%	+0.2%	+1.0%
Recession Period	-1.2%	+0.3%	-0.8%	0%
Geopolitical Risk Period	-0.5%	0%	0%	+0.8%

Time-Varying Correlation Structure

Idiosyncratic risk $\epsilon_t$ follows a multivariate normal distribution with correlation structure dependent on market environment:

\boldsymbol{\epsilon}_t \sim \mathcal{N}(\boldsymbol{0}, \boldsymbol{\Sigma}_{\text{regime}_t})

Normal Period Correlation Matrix $\boldsymbol{C}_{\text{normal}}$ :

	Equity	Bonds	REIT	Commodities	Cash
Equity	1.0	-0.2	0.5	0.3	0.0
Bonds	-0.2	1.0	-0.1	-0.2	0.1
REIT	0.5	-0.1	1.0	0.2	0.0
Commodities	0.3	-0.2	0.2	1.0	0.0
Cash	0.0	0.1	0.0	0.0	1.0

Crisis Period Correlation Matrix $\boldsymbol{C}_{\text{crisis}}$ :

	Equity	Bonds	REIT	Commodities	Cash
Equity	1.0	0.4	0.7	0.6	0.0
Bonds	0.4	1.0	0.2	0.3	0.1
REIT	0.7	0.2	1.0	0.5	0.0
Commodities	0.6	0.3	0.5	1.0	0.0
Cash	0.0	0.1	0.0	0.0	1.0

During crisis periods, inter-asset correlations generally increase, reproducing the phenomenon of reduced diversification effects.

Idiosyncratic Risk

The idiosyncratic volatility of each asset is set as follows:

Asset	Idiosyncratic Volatility
Equity	1.5%/month
Bonds	0.5%/month
REIT	2.0%/month
Commodities	3.0%/month
Cash	0.1%/month

Probability Distribution of Market Environments

Market environments switch stochastically according to the following probability distribution:

Market Environment	Probability
Normal Period	75%
Crisis Period	5%
Rate Hike Period	5%
Inflation Period	5%
Recession Period	5%
Geopolitical Risk Period	5%

Model Features and Advantages

Realistic Assumptions

Feature	Description
Factor Structure	Adopts growth, value, and quality factors supported by empirical research
Time-Varying Correlation	Reproduces correlation increase phenomenon during crisis periods
Seasonality	Reflects seasonal patterns observed in actual markets
Policy Cycle	Simulates medium-term policy impacts in 3-year cycles

Diversity of Risk Factors

Risk Layer	Description
Common Factors	Systematic factors affecting the entire market
Idiosyncratic Risk	Asset-specific variation factors
Market Environment Shocks	Additional impacts under specific market conditions
Correlation Changes	Dynamic changes in inter-asset correlations according to market conditions

Implementation Considerations

Item	Notes
Inter-Factor Correlation	Factor returns are generated independently, but may actually be correlated
Market Environment Transitions	Market environment transitions are determined independently, but may actually have Markov properties
Parameter Calibration	Estimation from historical data is required, and estimation errors must be considered

Conclusion

The factor-based asset return simulation model proposed in this study provides a framework that comprehensively captures the complex structure of modern financial markets. By integrating multi-layered risk factors, time-varying correlation structures, and stochastic switching of market environments, it enables simulations that reflect realistic characteristics difficult to reproduce with simple multivariate normal distribution models.

This model can serve as a useful tool for a wide range of financial applications including portfolio optimization, risk management, stress testing, and derivative valuation. Future research is expected to incorporate dynamic parameter estimation using machine learning methods and more detailed market microstructure.