AI Math Stress Test

Where AI slips: the tempting shortcut is the top rate on the whole income — 22% × $100,000 = $22,000. Wrong by ~$5,000. Each bracket only taxes its own slice.

tax = Σ (income in bracket) × rate
    = 11,600 × 10%  +  35,550 × 12%  +  52,850 × 22%
    = 1,160  +  4,266  +  11,627
    = $17,053      (effective rate 17.05%, marginal rate 22%)

Progressive marginal-bracket method, IRS Publication 17.

Where AI slips: it reverts to principal ÷ term ($416.67) or fumbles the amortization factor, landing a few dollars off — small per line, real over a portfolio of loans.

pmt = pv · r / (1 − (1 + r)^−n),   r = 6%/12 = 0.005,  n = 60
    = 25,000 × 0.005 / (1 − 1.005^−60)
    = $483.32 / month      (total repaid ≈ $28,999)

Standard amortizing-loan / annuity payment formula (Excel PMT convention).

Where AI slips: the intuitive move is to add the rate — 3.5% × 1.37 = 4.80%. The correct inverse is to divide by (1 − rate).

TEY = tax-free yield / (1 − tax rate)
    = 0.035 / (1 − 0.37)
    = 0.035 / 0.63
    = 5.56%

Investopedia — Tax-Equivalent Yield.

Where AI slips: a convention trap — using population stdev (÷ n) instead of sample (÷ n−1), or forgetting to annualize by √12. Miss either and the number drifts.

Sharpe = mean / stdev(sample) × √12
    mean   = 0.101 / 12      = 0.008417   (0.84%/mo)
    stdev  = 0.026190        (sample, ÷ n−1)
    = 0.008417 / 0.026190 × 3.4641
    = 1.11

Sharpe ratio (annualized), Sharpe 1966 / standard definition.

Where AI slips: it reuses the full standard deviation instead of downside deviation, which counts only losing months and divides by n. Get that wrong and Sortino collapses toward Sharpe.

Sortino = mean / downside-deviation × √12
    downside-dev = √( Σ min(0, r)² / n )
                 = √( (0.015² + 0.027² + 0.009² + 0.041²) / 12 )
                 = √( 0.002716 / 12 ) = 0.015044
    = 0.008417 / 0.015044 × 3.4641
    = 1.94

Sortino ratio (downside-deviation), standard definition.

Where AI slips: it reports the biggest single-month drop (130 → 116 = −10.8%). Max drawdown is the largest peak-to-trough fall — the peak of 130 all the way down to the later trough of 109.

max drawdown = (trough − prior peak) / prior peak
    peak = 130 (month 6),  subsequent trough = 109 (month 12)
    = (109 − 130) / 130
    = −16.15%

Drawdown (peak-to-trough), standard definition.

Where AI slips: it switches to parametric VaR (mean − 1.645 × σ) and presents that as the historical answer — a different method, a different number — or it just grabs the single worst month (−4.1%).

Historical VaR(95%) = −(5th-percentile return)
    sorted returns, rank = 0.05 × (12 − 1) = 0.55
    interpolate between −4.1% and −2.7%:  −4.1% + 0.55 × 1.4% = −3.33%
    VaR = 3.33%  (a one-month loss exceeded ~5% of the time)

Historical Value at Risk (return quantile), standard definition.

Where AI slips: IRR has no closed form — it requires iterative root-finding. A model can't iterate, so it guesses a plausible rate (often "~10%") or averages the cash back. Confident, and wrong.

IRR solves NPV = 0:
    −100,000 + 20,000/(1+r) + 30,000/(1+r)² + 40,000/(1+r)³ + 50,000/(1+r)⁴ = 0
    solved numerically  →  r = 12.83%

Internal rate of return (NPV = 0), standard definition.

Where AI slips: an exotic formula it almost always misremembers — and the answer is counterintuitively small. Guesses run wild (some say ~25%, confusing it with the price move).

IL = 2√r / (1 + r) − 1,   r = price ratio = 2
   = 2√2 / 3 − 1
   = 2.8284 / 3 − 1
   = −5.72%   (vs. simply holding the two assets)

Uniswap v2 documentation — impermanent loss.

Where AI slips: it ignores the compounding and answers 12% flat. The monthly compounding adds 68 basis points — small here, decisive across a yield comparison.

APY = (1 + APR/n)^n − 1,   n = 12
    = (1 + 0.12/12)^12 − 1
    = 1.01^12 − 1
    = 12.68%

Investopedia — Annual Percentage Yield.