Allfineworld > Analytical Compounded Return Explorer
GBM log-growth per year (Normal); Total multiplier (Lognormal)
Inputs
Example: 7 means 7% average simple return per year; GBM drift uses ln(1+r_arith_simple).
Chart inputs; the table grid uses ranges below. σ is the GBM log-return volatility (std dev of ln growth per year); the implied std dev of simple annual returns is √(e^σ²−1)·(1+r), slightly larger.
GBM assumption: per-year log-growth g ~ N( ln(1+r_arith_simple) − ½σ², σ² ). Over T years, ln(X) ~ N( (ln(1+r_arith_simple) − ½σ²)·T, σ²·T ).
Overview
Median total multiplier (T)
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Mean total multiplier (T)
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Prob gain (>0%)
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Prob lose half (≤ -50%)
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Prob > arithmetic mean
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Zero-σ annual r (comp.)
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Zero-σ annual r (simple)
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Multiplier threshold at 80%
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Annual distribution (GBM, Normal)
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Total multiplier PDF (Lognormal GBM)
Median Return Scenario Analysis (rows: σ, cols: r)
Kelly Criterion analysis
Kelly fraction f*
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Median (K vs 100%)
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Mean (K vs 100%)
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Prob gain (K vs 100%)
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Prob ≤ −50% (K vs 100%)
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P80 threshold (K vs 100%)
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Expected log-growth/yr (K vs 100%)
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Portfolio multiplier PDF: Kelly vs 100% risky (r=0)
Grid (rows: σ, cols: r) — Kelly fraction
Derivation (with risk-free):
Let a = ln(1+r) be the log of the risky asset's expected gross return per year (its instantaneous GBM drift — the expected log-growth per year is a − ½σ², not a), σ its log-return volatility. Let a_f = ln(1+r_free) be the risk-free log-rate. If fraction f is invested in the risky asset and (1−f) in risk-free (rebalanced continuously), the portfolio expected log-growth per year is:
G(f) = a_f + f(a − a_f) − ½ f² σ².
Maximize G(f): dG/df = (a − a_f) − f σ² = 0 ⇒ f* = (a − a_f)/σ². Long-only policy: f* = 0 when a ≤ a_f (no edge ⇒ hold the risk-free asset, which still grows at (1+r_free)^T); f* is capped at the max fraction above. The Kelly curve and grid here use this f* and recompute all values when r_free changes.
Let a = ln(1+r) be the log of the risky asset's expected gross return per year (its instantaneous GBM drift — the expected log-growth per year is a − ½σ², not a), σ its log-return volatility. Let a_f = ln(1+r_free) be the risk-free log-rate. If fraction f is invested in the risky asset and (1−f) in risk-free (rebalanced continuously), the portfolio expected log-growth per year is:
G(f) = a_f + f(a − a_f) − ½ f² σ².
Maximize G(f): dG/df = (a − a_f) − f σ² = 0 ⇒ f* = (a − a_f)/σ². Long-only policy: f* = 0 when a ≤ a_f (no edge ⇒ hold the risk-free asset, which still grows at (1+r_free)^T); f* is capped at the max fraction above. The Kelly curve and grid here use this f* and recompute all values when r_free changes.
Utility Maximization Analysis (CRRA)
CRRA f* vs Kelly vs 100%
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Median (U vs K vs 100%)
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Mean (U vs K vs 100%)
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Prob gain (U vs K vs 100%)
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P80 threshold (U vs K vs 100%)
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Prob ≤ −50% (U vs K vs 100%)
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Zero-σ annual r (comp.)
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Zero-σ annual r (simple)
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CRRA utility (Utility vs multiplier x)
Total multiplier PDF: 100% risky (dashed), Kelly (dashed), Max-Utility (solid)
Grid (rows: σ, cols: r) — Max-Utility fraction
CRRA utility: U(W) = (W^{1−γ} − 1)/(1−γ); for γ=1, U(W)=ln W. Under GBM, W_T is lognormal. For a split f in risky and 1−f in risk-free — where a = ln(1+r) and a_f = ln(1+r_free) are the logs of expected gross returns (instantaneous drifts) and σ is the log-return volatility — terminal ln(W_T) ~ N(m_p, v_p). Expected utility is E[U(W_T)] = (exp((1−γ)m_p + ½(1−γ)^2 v_p) − 1)/(1−γ); when γ=1, E[ln W_T] = m_p.
Optimal fraction: f*_CRRA = (a − a_f)/(γ σ²). Long-only policy: 0 when a ≤ a_f; capped at the max fraction above. γ=1 recovers Kelly; γ=0 gives risk-agnostic linear utility.