Risk-Adjusted Costs Formula

Quick Reference

Costi espressi in unita di Sharpe Ratio - permette confronto fair tra strumenti con volatilita diversa.

Formula Principale

Total Risk-Adj Cost = Holding Costs + Transaction Costs

In unita di Sharpe Ratio (SR units).

Holding Costs

HC_risk = HC_annual / σ%

Variabili

• HC_annual: Costo annuale di holding (%)
• σ%: Annual standard deviation (%)

Esempio

US 2-year bond future: - HC_annual = 0.033% - σ% = 0.54% - HC_risk = 0.033 / 0.54 = 0.061 SR units

Transaction Costs

Cost Per Trade

TC_ratio = (Spread / 2 + Commission) / Price
TC_risk = TC_ratio / σ%

Annual Transaction Costs

TC_annual = TC_risk × Turnover

Dove: - Turnover: Numero volte all'anno che tradi average position - Non include rolling (counted separately)

Esempio S&P 500 Micro

Per trade: - Spread = 0.25 points = $1.25 - Commission = $0.25 - Total per contract = $1.50 - Price = 4500, Multiplier = 5 - Exposure = $22,500 - TC_ratio = 1.50 / 22,500 = 0.0067% - σ% = 16% - TC_risk = 0.0067% / 16% = 0.00004 SR units

Annual (turnover = 5×): - TC_annual = 0.00004 × 5 = 0.0002 SR units

Rolling Costs

Per dated products:

Rolling_annual = TC_risk × (Rolls per year × 2)

×2 perche ogni roll = close + open (2 trades)

Esempio

Quarterly futures (4 rolls/year): - TC_risk = 0.0001 SR units - Rolling_annual = 0.0001 × 4 × 2 = 0.0008 SR units

Total Cost Formula Completa

C_total = (HC_risk) + (TC_risk × Turnover) + (TC_risk × Rolls × 2)

Esempio Completo: S&P 500

• HC_risk = 0 (no holding cost for equity index future)
• TC_risk = 0.00004
• Turnover = 5
• Rolls = 4 (quarterly)

C_total = 0 + (0.00004 × 5) + (0.00004 × 4 × 2)
        = 0 + 0.0002 + 0.00032
        = 0.00052 SR units

Interpretazione

C_total = 0.00052 significa:

Se pre-cost SR = 0.5000, allora: Post-cost SR = 0.5000 - 0.00052 = 0.4995

Praticamente gratis!

Speed Limit Rule

Max Acceptable Cost = Expected SR / 3

Esempio: - Expected SR = 0.30 - Max cost = 0.10 SR units

Se C_total > 0.10 → strumento troppo costoso.

Maximum Cost Per Trade

Dato expected SR e turnover:

Max TC_risk = (SR / 3) / (Turnover + Rolls × 2)

Esempio: - Expected SR = 0.30 - Max acceptable total cost = 0.10 - Turnover = 3 - Rolls = 4

Max TC_risk = 0.10 / (3 + 4×2)
            = 0.10 / 11
            = 0.0091 SR units per trade

Comparing Instruments

Instrument A (S&P 500): - TC_risk = 0.00004 - C_total (5× turnover) = 0.0005

Instrument B (2-year bond): - TC_risk = 0.024 - C_total (5× turnover) = 0.24

Conclusion: Bond è 480× piu costoso!

Anche se spread assoluto sembra simile, risk-adjusted e drammaticamente diverso.

Leverage Invariance

Risk-adjusted costs NON cambiano con leverage:

Scenario A (low leverage): - Capital = $100,000 - Risk target = 10% - Position = 2 contracts - Annual cost = $20 - Cost / Risk = $20 / $10,000 = 0.002 = 0.2% - Risk-adj = 0.2% / 10% = 0.02 SR units

Scenario B (high leverage): - Capital = $100,000 - Risk target = 40% - Position = 8 contracts - Annual cost = $80 - Cost / Risk = $80 / $40,000 = 0.002 = 0.2% - Risk-adj = 0.2% / 40% = 0.02 SR units

Stesso risk-adjusted cost!

From Risk-Adjusted to Dollar Costs

Reverse calculation:

Dollar Cost = C_total × Risk Target × Capital

Esempio: - C_total = 0.05 SR units - Risk target = 20% - Capital = $50,000

Dollar Cost = 0.05 × 0.20 × 50,000
            = 0.05 × 10,000
            = $500 per year

Turnover Calculation

Turnover = Total Contracts Traded / Average Position

Esempio anno 2021: - Total contracts traded = 60 - Average position = 9.3 - Turnover = 60 / 9.3 = 6.5×

Usa media multi-anno per stima piu stabile.

Variable vs Fixed Risk

Fixed risk (strategy 2): - Lower turnover (~2-3×) - Lower transaction costs

Variable risk (strategy 3): - Higher turnover (~5-6×) - Higher transaction costs - Ma: better risk targeting

Trade-off da valutare.

Cost Ranking

Strumenti tipici (order of magnitude):

Instrument Type	TC_risk (SR units)
Equity index futures (micro)	0.0001-0.001
Liquid FX futures	0.001-0.005
Commodity futures	0.01-0.05
Short-duration bonds	0.02-0.10
Agricultural futures	0.05-0.15

Cost Decomposition

Breakdown typical total cost:

Low-cost instrument (S&P 500): - Transaction: 40% - Rolling: 60% - Holding: 0%

Medium-cost (Crude oil): - Transaction: 50% - Rolling: 30% - Holding: 20%

High-cost (Agricultural): - Transaction: 70% - Rolling: 20% - Holding: 10%

Reducing Costs

1. Lower Turnover

Trade less frequently: - Turnover 10× → 3× saves 70%! - But: reduces responsiveness

2. Use Cheaper Instruments

Micro/mini contracts often cheaper risk-adjusted

3. Limit Orders

Can save ~50% of spread cost

4. Avoid High-Cost Instruments

If C_total > speed limit, don't trade

Errori Comuni

• Comparing absolute costs: Ignore volatility difference
• Forgetting rolling: Dated products have 2× trades per roll
• Using wrong turnover: Include rolling in turnover (wrong!)
• Ignoring holding costs: Especially relevant for bonds
• Not accounting for leverage: Dollar costs scale, risk-adj don't

Concetti Correlati

• [[Sharpe Ratio]] - unita di misura per costs
• [[Volatility]] - denominatore della formula
• [[Transaction Costs]] - componente dei total costs
• [[Position Sizing]] - determines dollar costs