data-optimizer-presentation.md

1. Why Optimization After Causal Modeling?

The Core Problem

Causal models provide: $P(\text{renew} | \text{price}, X)$ for 200 discrete price points

Causal models DON'T provide:

• Optimal decisions under portfolio constraints
• Segment-level pricing (operational requirement)
• Multi-term coordination
• Volume guarantees

Solution: Two-stage framework

Stage 1: Causal Model → Predictions
Stage 2: Optimization → Decisions under constraints

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2. Mathematical Formulation

Notation

• $\mathcal{S}$: Segments (indexed by $s$)
• $\mathcal{T}$: Terms (indexed by $t$, e.g., ${1,2,3,5}$ years)
• $\mathcal{P}$: Price options (200 discrete margins)
• $b_i$: Balance of mortgage $i$
• $c_i$: Cost of funds for mortgage $i$
• $P_{i,t,p}$: Renewal probability (from causal model)
• $V_{\min}$: Minimum required volume

Decision Variables

$$x_{s,t} \in \mathcal{P} \quad \text{(margin for segment } s \text{, term } t \text{)}$$

Objective Function

Expected profit for mortgage $i$: $$\pi_{i,t,p} = (p - c_i) \times b_i \times t \times P_{i,t,p}$$

Total portfolio profit: $$\max_{x_{s,t}} \sum_{s \in \mathcal{S}} \sum_{t \in \mathcal{T}} \sum_{i \in s} \pi_{i,t,x_{s,t}}$$

Constraints

Volume constraint: $$\sum_{s,t} \sum_{i \in s} P_{i,t,x_{s,t}} \times b_i \geq V_{\min}$$

Price bounds: $$p_{\min} \leq x_{s,t} \leq p_{\max}$$

Segment-Level Aggregation

Segment average expected profit: $$\bar{\pi}{s,t,p} = \frac{1}{|s|} \sum{i \in s} (p - c_i) \times b_i \times t \times P_{i,t,p}$$

Reformulated objective: $$\max_{x_{s,t}} \sum_{s,t} |s| \times \bar{\pi}{s,t,x{s,t}}$$

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3. Numerical Example

Setup

Portfolio: 3,500 mortgages, 3 segments, 3-year term, 5 price options

Segment	#	Avg Balance	Avg CoF
S1 (Shoppers)	1,000	$300K	1.5%
S2 (Aware)	2,000	$250K	1.6%
S3 (Loyal)	500	$400K	1.4%

Renewal Probabilities:

Price	S1	S2	S3
3.29%	0.92	0.91	0.94
3.49%	0.88	0.89	0.93
3.69%	0.75	0.85	0.92
3.89%	0.60	0.78	0.91
3.99%	0.50	0.73	0.90

Expected Profit Calculation

S1 @ 3.49%: [(0.0349 - 0.015) \times 300K \times 3 \times 0.88 = $15,761] per mortgage → Segment total: $$15.8M$

S2 @ 3.89%: [(0.0389 - 0.016) \times 250K \times 3 \times 0.78 = $13,397] → Segment total: $$26.8M$

S3 @ 3.99%: [(0.0399 - 0.014) \times 400K \times 3 \times 0.90 = $27,972] → Segment total: $$14.0M$

Unconstrained optimal: $$56.6M$ profit, $834M$ volume (80% retention)

Problem: Violates 85% volume constraint ($886M$ required)

Constrained solution: Lower S2 to 3.49% → $$55.0M$ profit, $889M$ volume ✓

Trade-off: Sacrifice $$1.6M$ profit to meet volume target

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4. Implementation Pipeline

# Pseudocode
def optimize_portfolio(df_predictions, config):
    # Step 1: Expand for terms
    df = expand_terms(df_predictions, terms=[1,2,3,5])
    
    # Step 2: Calculate profit
    df['profit'] = (df['margin'] - df['cof']) * df['balance'] * df['term']
    df['expected_profit'] = df['profit'] * df['renewal_prob']
    
    # Step 3: Segment by elasticity
    df['segment'] = assign_segments(df, method='elasticity_quantiles')
    
    # Step 4: Smooth curves (isotonic + spline)
    df['renewal_prob_smooth'] = smooth_curves(df, group_by=['segment','term'])
    
    # Step 5: Aggregate to segment level
    segment_data = df.groupby(['segment','term','margin']).agg({
        'expected_profit': 'mean',
        'renewal_prob_smooth': 'mean'
    })
    
    # Step 6: Optimize per term
    results = {}
    for term in [1,2,3,5]:
        results[term] = optimize_lp(segment_data[term], config.min_volume)
    
    return format_recommendations(results)

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5. Key Technical Components

Smoothing: Two-Stage Approach

from scipy.interpolate import UnivariateSpline
from sklearn.isotonic import IsotonicRegression

def smooth_curve(margins, renewal_probs, s=0.03):
    # Stage 1: Enforce monotonicity
    iso = IsotonicRegression(increasing=False, out_of_bounds='clip')
    probs_mono = iso.fit_transform(margins, renewal_probs)
    
    # Stage 2: Spline smoothing
    spline = UnivariateSpline(margins, probs_mono, s=s, k=3)
    return np.clip(spline(margins), 0, 1)

Why: Causal predictions are noisy at extremes; smoothing ensures monotonicity and stability

Segmentation: Elasticity-Based

Price elasticity: $\varepsilon = \frac{\Delta P(\text{renew}) / P(\text{renew})}{\Delta \text{price} / \text{price}}$

def create_segments(df, n_segments=20):
    # Calculate elasticity for each mortgage
    elasticities = calculate_arc_elasticity(df)
    
    # Quantile-based segmentation
    df['segment'] = pd.qcut(elasticities, q=n_segments, labels=False)
    
    return df

Benefit: Reduces from O(millions) to O(thousands) of variables

Optimization: Three Approaches

1. Discrete (Differential Evolution):

• Select from exact 200 price points
• Global search over discrete grid
• Use for non-smooth objectives

2. Continuous (SLSQP):

• Interpolate between prices (cubic splines)
• Smooth optimization landscape
• Faster convergence

3. Linear Programming (Recommended):

• Formulate as discrete choice (MIP)
• Provably optimal solution
• Fastest for large-scale problems

from ortools.linear_solver import pywraplp

def optimize_lp(segment_data, min_volume):
    solver = pywraplp.Solver.CreateSolver('GLOP')
    
    # Binary variables: y[s,p] = 1 if segment s chooses price p
    y = {(s,p): solver.BoolVar(f'y_{s}_{p}') 
         for s in segments for p in prices}
    
    # Each segment chooses exactly one price
    for s in segments:
        solver.Add(sum(y[s,p] for p in prices) == 1)
    
    # Objective: maximize profit
    solver.Maximize(sum(y[s,p] * profit[s,p] for s,p in y))
    
    # Volume constraint
    solver.Add(sum(y[s,p] * volume[s,p] for s,p in y) >= min_volume)
    
    solver.Solve()
    return extract_solution(y)

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6. Performance Metrics

Computational Complexity:

• Without segmentation: O(millions of mortgages) - infeasible
• With segmentation: O(10 segments × 4 terms × 200 prices) ≈ 8K variables
• Solve time: <1 minute

Typical Results:

• Profit lift: 8-15% vs one-size-fits-all
• Volume control: Guaranteed 85-95% retention
• Segmentation efficiency: 90% of personalization value at 1% computational cost

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7. Validation Strategy

Backtesting

for month in historical_months:
    train = data[data.month < month]
    test = data[data.month == month]
    
    recommendations = optimizer.run(train)
    actuals = test.merge(recommendations)
    
    metrics[month] = {
        'predicted_profit': recommendations.profit.sum(),
        'actual_profit': actuals.profit.sum(),
        'error': abs(predicted - actual) / actual
    }

8. Key Takeaways

Aspect	Causal Model Alone	With Optimization
Predictions	✓	✓
Portfolio constraints	✗	✓
Segment pricing	✗	✓
Multi-term coordination	✗	✓
Volume guarantees	✗	✓
Actionable decisions	✗	✓

Bottom Line: Causal models predict elasticity to the price/rate; optimization prescribes strategy under real-world constraints.