Utility Optimization¶

This guide covers the utility-optimal spending and allocation framework based on Merton's continuous-time portfolio optimization theory.

Overview¶

Traditional retirement planning often uses rules of thumb like the "4% rule." While simple, these approaches don't account for individual preferences about risk and the trade-off between spending now versus later.

Utility optimization provides a rigorous framework for finding the optimal spending rate and asset allocation based on:

Risk aversion (gamma): How much you dislike uncertainty
Time preference: How much you prefer spending now vs. later
Subsistence floor: Minimum spending you need to survive

Key Formulas¶

Optimal Equity Allocation¶

The Merton formula gives the optimal fraction to invest in risky assets:

\[ k^* = \frac{\mu - r}{\gamma \cdot \sigma^2} \]

Where:

\(\mu\) = expected stock return
\(r\) = bond/risk-free return
\(\gamma\) = risk aversion coefficient
\(\sigma\) = stock volatility

Certainty Equivalent Return¶

The guaranteed return that provides the same utility as the risky portfolio:

\[ r_{CE} = r + k^*(\mu - r) - \frac{\gamma \cdot (k^*)^2 \cdot \sigma^2}{2} \]

Optimal Spending Rate¶

For an infinite horizon:

\[ c^* = r_{CE} - \frac{r_{CE} - \rho}{\gamma} \]

Where \(\rho\) is your time preference (discount rate).

Quick Start¶

from fundedness import (
    calculate_merton_optimal,
    MarketModel,
    UtilityModel,
)

# Define market assumptions
market = MarketModel(
    stock_return=0.05,      # 5% expected real return
    bond_return=0.015,      # 1.5% risk-free rate
    stock_volatility=0.16,  # 16% annual volatility
)

# Define your preferences
utility = UtilityModel(
    gamma=3.0,              # Risk aversion (typical: 2-5)
    subsistence_floor=30000,  # Minimum annual spending
    time_preference=0.02,   # 2% discount rate
)

# Calculate optimal policy
result = calculate_merton_optimal(
    wealth=1_000_000,
    market_model=market,
    utility_model=utility,
    remaining_years=30,
)

print(f"Optimal equity allocation: {result.optimal_equity_allocation:.1%}")
print(f"Wealth-adjusted allocation: {result.wealth_adjusted_allocation:.1%}")
print(f"Optimal spending rate: {result.optimal_spending_rate:.1%}")
print(f"Year 1 spending: ${1_000_000 * result.optimal_spending_rate:,.0f}")

Wealth-Adjusted Allocation¶

Near the subsistence floor, you can't afford to take risk. The wealth-adjusted allocation accounts for this:

\[ k_{adj} = k^* \cdot \frac{W - F}{W} \]

Where \(W\) is wealth and \(F\) is the subsistence floor.

As wealth approaches the floor, allocation approaches zero. As wealth rises far above the floor, allocation approaches the unconstrained optimal.

from fundedness import wealth_adjusted_optimal_allocation

# Near the floor: low allocation
k_low = wealth_adjusted_optimal_allocation(
    wealth=50_000,  # Just above $30k floor
    market_model=market,
    utility_model=utility,
)
print(f"Allocation at $50k: {k_low:.1%}")  # ~18%

# Well above floor: approaches optimal
k_high = wealth_adjusted_optimal_allocation(
    wealth=2_000_000,
    market_model=market,
    utility_model=utility,
)
print(f"Allocation at $2M: {k_high:.1%}")  # ~44%

Spending Rate by Age¶

Optimal spending rate increases with age as the remaining horizon shortens:

from fundedness import optimal_spending_by_age

rates = optimal_spending_by_age(
    market_model=market,
    utility_model=utility,
    starting_age=65,
    end_age=95,
)

for age in [65, 75, 85, 95]:
    print(f"Age {age}: {rates[age]:.1%}")

Example output:

Age 65: 3.8%
Age 75: 5.2%
Age 85: 7.8%
Age 95: 100.0%

Using Merton Policies in Simulation¶

Merton Spending Policy¶

from fundedness.withdrawals import MertonOptimalSpendingPolicy
from fundedness.allocation import MertonOptimalAllocationPolicy
from fundedness import run_simulation_with_utility, SimulationConfig

# Create policies
spending_policy = MertonOptimalSpendingPolicy(
    market_model=market,
    utility_model=utility,
    starting_age=65,
    end_age=95,
)

allocation_policy = MertonOptimalAllocationPolicy(
    market_model=market,
    utility_model=utility,
)

# Run simulation with utility tracking
config = SimulationConfig(
    n_simulations=5000,
    n_years=30,
    market_model=market,
)

result = run_simulation_with_utility(
    initial_wealth=1_000_000,
    spending_policy=spending_policy,
    allocation_policy=allocation_policy,
    config=config,
    utility_model=utility,
)

print(f"Expected lifetime utility: {result.expected_lifetime_utility:.2e}")
print(f"Certainty equivalent: ${result.certainty_equivalent_consumption:,.0f}/year")
print(f"Success rate: {result.success_rate:.1%}")

Key Insights¶

Spending rate increases with age - As the horizon shortens, you should spend more
Allocation decreases near the floor - You can't afford risk when close to subsistence
Risk aversion (gamma) is critical - Higher gamma means lower allocation and spending
The 4% rule is often suboptimal - Merton optimal typically starts lower and ends higher

Choosing Your Risk Aversion¶

Risk aversion (gamma) is the most important input. Guidelines:

Gamma	Risk Tolerance	Profile
1-2	High	Comfortable with volatility
2-3	Moderate	Typical investor
3-5	Low	Prefers stability
5+	Very low	Strong loss aversion

References¶

Merton, R. C. (1969). Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case. The Review of Economics and Statistics, 51(3), 247-257.
Haghani, V., & White, J. (2023). The Missing Billionaires: A Guide to Better Financial Decisions. Wiley.