Why FOAKs Don't Get Funded: A Mathematical Analysis

The Core Argument

First-of-a-kind projects produce negative risk-adjusted returns, placing them below the capital allocation line.

The finding: FOAKs generate Sharpe ratios of -0.5 to -1.6, indicating expected returns below the risk-free rate.

The implication: Standard risk-return frameworks do not recommend assets with negative Sharpe ratios.

The Math

Part 1: Compound Probability

FOAKs must pass through 3-5 sequential gates. Each stage has reasonable success probability (40-95%), but these multiply:

Concept:         40% success
Pre-Feasibility: 50% success  
Feasibility:     65% success
Construction:    80% success
Operation:       95% success

Final success = 0.40 × 0.50 × 0.65 × 0.80 × 0.95 = 9.9%

Result: 90% of projects fail before reaching commercial operation.

Part 2: Return Asymmetry

Physical assets face structural return caps that software ventures don't:

Result: Success returns of 25-30% IRR cannot offset 90% failure probability.

Part 3: Expected Return Calculation

Probability tree outcomes:
├─ Success (9.9%):           +25% IRR → +1.25x return
├─ Partial recovery (14.6%): +10% IRR → Recovery varies by stage
└─ Total loss (75.5%):       -100% IRR → 0x return

Expected return = (0.099 × +25%) + (0.146 × +10%) + (0.755 × -100%)
                = +2.5% + 1.5% - 75.5%
                = -71.5%

Part 4: Risk-Adjusted Return

Sharpe Ratio = (Expected Return - Risk-Free Rate) / Standard Deviation
             = (-71.5% - 3%) / 52%
             = -1.43

For comparison:

The Probability Tree Structure

===== PROBABILITY TREE =====

Root (P=1.000)
└─● Stage 0: Concept (P=1.000)
  └─● Investment: -$500k (t=0.0)
    ├─● Success to Stage 1 (P=0.400) [Path P=0.400]
    │ └─● Stage 1: Pre-Feasibility (P=1.000)
    │   └─● Investment: -$750k (t=1.5)
    │     ├─● Success to Stage 2 (P=0.500) [Path P=0.200]
    │     │ └─● Stage 2: Feasibility (P=1.000)
    │     │   └─● Investment: -$2.0M (t=3.0)
    │     │     ├─● Success to Stage 3 (P=0.650) [Path P=0.130]
    │     │     │ └─● Stage 3: Construction (P=1.000)
    │     │     │   └─● Investment: -$15M (t=4.0)
    │     │     │     ├─● Success to Stage 4 (P=0.800) [Path P=0.104]
    │     │     │     │ └─● Stage 4: Operation (P=1.000)
    │     │     │     │   └─● Investment: -$30M (t=7.0)
    │     │     │     │     ├─● Success to Exit (P=0.950) [Path P=0.099]
    │     │     │     │     │ └─● Exit: +$60M (t=8.0) ★
    │     │     │     │     ├─● Partial Recovery (P=0.043) ★
    │     │     │     │     └─● Total Loss (P=0.007) ★
    │     │     │     ├─● Partial Recovery (P=0.130) ★
    │     │     │     └─● Total Loss (P=0.070) ★
    │     │     ├─● Partial Recovery (P=0.123) ★
    │     │     └─● Total Loss (P=0.227) ★
    │     ├─● Partial Recovery (P=0.075) ★
    │     └─● Total Loss (P=0.425) ★
    ├─● Partial Recovery (P=0.057) ★
    └─● Total Loss (P=0.543) ★

Total probability: 1.000 ✓

Outcome distribution:
├─ Success: 9.9%
├─ Partial recovery: 14.6%
└─ Total loss: 75.5%

Key observation: Path probability decays from 1.000 → 0.099 through multiplication of stage success rates.

The Cash Flow Structure

===== CASH FLOW TABLE =====

| TIME  | EVENT                  | NOMINAL    | PATH PROB | EXPECTED VALUE |
|-------|------------------------|------------|-----------|----------------|
| t=0.0 | Initial Investment     | -$500k     | 1.000     | -$500k         |
| t=1.5 | Follow-on Investment   | -$750k     | 0.400     | -$300k         |
| t=3.0 | Follow-on Investment   | -$2.0M     | 0.200     | -$400k         |
| t=4.0 | Follow-on Investment   | -$15M      | 0.130     | -$1,950k       |
| t=7.0 | Follow-on Investment   | -$30M      | 0.104     | -$3,120k       |
|       | Total Investment   | -$48.25M  |           | -$6.27M        |
|       |                        |            |           |                |
| t=8.0 | Success Exit           | +$60M      | 0.099     | +$5,940k       |
| t=0-7 | Partial Recoveries     | +$54.6M    | 0.146     | +$394k         |
|       | Total Return       | +$114.6M  |           | +$6.33M        |
|       |                        |            |           |                |
|       | NET                | +$66.4M   |           | +$64k          |

Expected IRR: 0.4% over 8 years

Key observation: Investment occurs at high probability (certain if stage reached), return occurs at 9.9% probability.

The Asymmetry

Investments are front-loaded and certain:

• t=0.0: Deploy $500k with path probability = 1.000
• t=1.5: Deploy $750k with path probability = 0.400 (certain if reached)
• t=3.0: Deploy $2.0M with path probability = 0.200 (certain if reached)

Returns are back-loaded and uncertain:

• t=8.0: Receive $60M with path probability = 0.099 (uncertain)

Expected value reality:

• Expected investment: $6.27M
• Expected return: $6.33M
• Net expected gain: $64k (1% over 8 years)

The Capital Allocation Line Test

The Test

An asset is investable if:

Sharpe Ratio ≥ 0.3 (typical threshold)

This ensures the asset offers adequate return per unit of risk compared to alternatives.

FOAK Performance

Traditional binary model:
Sharpe = (-82% - 3%) / 55% = -1.59

Recovery-enhanced model:
Sharpe = (-44% - 3%) / 48% = -1.01

Both are deeply negative. Even with maximum recovery optimization, FOAKs remain below the investable threshold.

Comparison

What Management Can Control

Fixed Variables (Cannot Change)

Success probability = 9.9%

Determined by:

• Technical risk: Will the technology work at commercial scale?
• Regulatory risk: Will permits be approved?
• Market risk: Will customers adopt?

These probabilities are structurally determined by technical, regulatory, and market factors. Management execution affects outcomes within constrained ranges.

Variable Parameters (Can Improve)

Recovery rate: 25% → 65%

Controllable through:

• Asset structuring: Equipment with secondary market value
• IP protection: Patents retain value independent of project success
• Modular design: Components redeployable across portfolio
• Governance discipline: Early termination if failing

Impact:

Binary model (poor recovery):
Expected return: -82%
Sharpe ratio: -1.59

Recovery-enhanced (good recovery):
Expected return: -44%
Sharpe ratio: -1.01

Improvement: +38 percentage points

This improves Sharpe from -1.59 to -1.01. Both remain below zero, indicating expected returns below the risk-free rate.

The Returns Cap Problem

Why Physical Assets Can't Return 10x

Software venture math works because:

• Near-zero marginal cost: Serve user 1M same cost as user 1,000
• Exponential scaling: 100 users → 10,000 users overnight
• Pivot capability: Failed dating app becomes B2B tool in 6 months
• Power law returns: 1 winner at 100x covers 20 failures

FOAK math fails because:

• High marginal cost: Each unit requires materials, energy, labor
• Linear scaling: Double production requires double capital
• No pivot capability: $50M chemical plant has one use
• Capped returns: Success returns 1.25x-1.5x

Example comparison:

With 100x returns, software ventures can sustain 2% success rates. With 1.5x returns, FOAKs require success rates above 80% for positive expected returns.

Physical asset characteristics constrain return potential independent of management quality.

Calculation Summary

Summary

1. Compound probability:
   0.40 × 0.50 × 0.65 × 0.80 × 0.95 = 9.9% success

2. Expected return:
   (0.099 × +25%) + (0.146 × +10%) + (0.755 × -100%) = -71.5%

3. Risk-adjusted return:
   Sharpe = (-71.5% - 3%) / 52% = -1.43

4. Position:
   Below capital allocation line (Sharpe < 0)

The Comparison to Treasury Bonds

The Conclusion

FOAKs have negative expected risk-adjusted returns at standalone project scale because:

1. Compound probability drives success to 10-20%; management quality has limited impact on compound probability
2. Physical asset constraints cap success returns at 1.25x-1.5x (cannot pivot or scale exponentially)
3. Cash flow asymmetry requires investing at high probability while returning at low probability
4. Result: Expected returns of -44% to -82% with volatility of 48-55%
5. Risk-adjusted position: Sharpe ratios of -1.0 to -1.6, below capital allocation line

Recovery optimization improves Sharpe from -1.59 to -1.01 but does not change the negative risk-adjusted return profile.

Portfolio diversification can achieve Sharpe of +0.3 at $500M-2B scale but requires institutional capacity most FOAK developers lack.

This analysis explains the $2.5 trillion FOAK financing gap. Capital markets are rejecting assets with negative risk-adjusted returns.

The model demonstrates three characteristics:

1. Explicit probability accounting: Shows all outcome paths and compound failure
2. Cash flow transparency: Demonstrates investment-return asymmetry
3. Risk-adjusted measurement: Quantifies position relative to capital allocation line

This is not a market failure—it is market pricing of risk-return tradeoffs.

For technical model documentation, see PROBABILISTIC_MODEL.md. For implementation details, see RESTART_GUIDE.md.