IDM Calculation

Quick Reference

Instrument Diversification Multiplier: fattore per cui puoi aumentare posizioni individuali grazie a correlazione imperfetta.

Formula Semplificata

Con N instruments, correlazione media ρ:

IDM = √[N / (1 + (N-1) × ρ)]

Casi Limite

Perfect Correlation (ρ = 1.0)

IDM = √[N / (1 + (N-1) × 1)]
    = √[N / N]
    = 1.0

No diversification benefit.

Zero Correlation (ρ = 0.0)

IDM = √[N / (1 + 0)]
    = √N

Maximum possible diversification.

Partial Correlation (ρ = 0.5)

Esempio 4 instruments:

IDM = √[4 / (1 + 3 × 0.5)]
    = √[4 / 2.5]
    = √1.6
    = 1.26

Formula Completa (Correlation Matrix)

Con matrice correlazione C:

Portfolio Variance = w' × C × w

Dove w = vector di pesi instruments

IDM implicito:

IDM = √[Σw_i² / Portfolio Variance]

Assumendo equal risk allocation.

Calcolo Pratico

Step 1: Correlation Matrix

Esempio 3 instruments A, B, C:

       A     B     C
A   1.00  0.50  0.30
B   0.50  1.00  0.40
C   0.30  0.40  1.00

Step 2: Average Correlation

ρ_avg = (0.50 + 0.30 + 0.40) / 3 = 0.40

Step 3: IDM Formula

IDM = √[3 / (1 + 2 × 0.40)]
    = √[3 / 1.8]
    = √1.67
    = 1.29

Step 4: Position Scaling

Senza diversificazione (1 instrument): - Capital = $100k - Target = 20% - Position risk = 20% / 1 = 20%

Con diversificazione (3 instruments, IDM 1.29): - Position risk each = 20% × 1.29 / 3 = 8.6% - Total portfolio risk = 20% (grazie a IDM)

IDM Values per Asset Classes

Within Asset Class

Equity indices (10 countries): - Average ρ ≈ 0.70 - N = 10 - IDM ≈ 1.8

FX (8 pairs): - Average ρ ≈ 0.50 - N = 8 - IDM ≈ 2.0

Commodities (12 futures): - Average ρ ≈ 0.30 - N = 12 - IDM ≈ 2.5

Across Asset Classes

Diversified portfolio (4 asset classes, 30 instruments): - Intra-class ρ ≈ 0.60 - Inter-class ρ ≈ 0.20 - IDM ≈ 3.5-4.0

Much higher!

Diminishing Returns

Benefit per additional instrument:

N Instruments	IDM (ρ=0.5)	Incremental Gain
1	1.00	-
2	1.41	+41%
4	1.79	+27%
8	2.18	+22%
16	2.51	+15%
32	2.72	+8%

After ~10-15 instruments: Diminishing returns steep.

Dynamic IDM

Correlations change over time:

Normal markets: - ρ_equity ≈ 0.50 - IDM_equity ≈ 2.0

Crisis markets (2008, 2020): - ρ_equity → 0.90+ - IDM_equity → 1.2 - Diversification collapses!

IDM e Risk Contribution

Equal risk contribution:

Risk per instrument = Portfolio Target × IDM / N

Esempio (Portfolio 20% target, 8 instruments, IDM 2.0):

Risk each = 20% × 2.0 / 8 = 5%

Verifica (assumendo ρ = 0.5):

Portfolio Var = 8 × 5%² × (1 + 7×0.5) / 8
              = 8 × 25 × 4.5 / 8
              = 112.5
Portfolio σ = √112.5 ≈ 10.6%

Hmm, non exactly 20%... perche formula semplificata è approximation!

Forecast Diversification Multiplier (FDM)

Analogo per trading rules:

Multiple rules invece di multiple instruments:

FDM = √[M / (1 + (M-1) × ρ_forecast)]

Dove M = numero trading rules.

Esempio 3 rules (trend, carry, breakout), ρ = 0.3:

FDM = √[3 / (1 + 2×0.3)]
    = √[3 / 1.6]
    = 1.37

Handcrafting IDM

Se non hai correlation matrix:

Conservative estimate:

IDM = √N × 0.7

Esempio 10 instruments:

IDM ≈ √10 × 0.7 = 3.16 × 0.7 = 2.2

Assumes average ρ ≈ 0.5, safety factor.

Using IDM in Practice

Position Sizing

Position_i = (Capital × Target × IDM) / (N × Price × σ%_i)

Invece di:

Position_i = (Capital × Target / N) / (Price × σ%_i)

IDM aumenta sizes proporzionalmente.

Minimum Capital

Min Capital per instrument = Base Min / IDM

Esempio (need $50k per instrument standalone):

With 3 instruments, IDM 1.5:

Min Capital = 3 × $50k / 1.5 = $100k

Instead of $150k (3 × $50k).

Limitations

1. Assumes Normality

IDM formula assumes returns Gaussian.

Reality: Fat tails, tail correlation higher than normal correlation.

2. Static Correlations

Uses historical average ρ.

Reality: Correlations spike during stress.

3. Equal Weighting Assumption

Formula assumes equal risk weights.

Reality: Might want different weights.

Errori Comuni

• Assuming √N always: Ignoring correlations
• Using sample correlation: Small sample, noisy estimates
• Static IDM: Not updating as correlations change
• Ignoring crisis correlation: "Diversification works... until it doesn't"
• Too many instruments: Beyond ~20, marginal benefit tiny
• Wrong correlation matrix: Usando data contaminated/outdated

Concetti Correlati

• [[Diversification Multiplier]] - concetto generale
• [[Position Sizing]] - usa IDM nel calcolo
• [[Correlation]] - input principale per IDM
• [[Portfolio Construction]] - IDM central to allocation