Questions
In-depth breakdowns answering specific personal finance questions with first-principles mechanics, verified formulas, and reproducible worked examples.
How to calculate compound interest with monthly contributions and inflation
To calculate compound interest with monthly contributions and inflation, convert the nominal annual rate to a monthly rate using (1 + r)^(1/12) − 1. Iterate the monthly balance recurrence, then deflate the nominal terminal sum by (1 + i)^years, or adjust the rate via the Fisher equation (1 + r)/(1 + i) − 1 if contributions increase with inflation.
Formula for percentage gain needed to recover from a drawdown
The percentage gain required to recover from a portfolio drawdown is g = d / (1 − d), where d is the drawdown as a decimal. Because losses shrink the capital base from which future returns compound, recovery demands an asymmetric percentage increase: a 20% drop needs 25%, while a 50% crash requires a 100% gain.
How is the compound interest formula derived
The compound interest formula is derived by applying an interest rate r repeatedly to an accumulating balance. One period yields A_1 = P(1 + r); n periods produce the recurrence A_n = A_(n−1)(1 + r), resolving to A_n = P(1 + r)^n. Dividing annual rate r across k periods yields P(1 + r/k)^(kn), approaching Pe^(rn) under continuous compounding.
Is there a mathematical proof for dollar-cost averaging
There is a proof that dollar-cost averaging reduces average cost per share below the average market price, but no proof that it beats lump-sum investing. By the AM-HM inequality, investing equal dollar amounts buys more shares at lower prices, yielding a harmonic mean share cost less than or equal to the arithmetic average.
How to find single points of failure in an investment portfolio
To find single points of failure in an investment portfolio, identify any concentrated exposure that can cause catastrophic capital loss. Calculate your top-position weight, Herfindahl-Hirschman concentration index (HHI = Σ w_i²), and effective holding count (N_eff = 1 / HHI). Audit hidden operational bottlenecks across correlated employer equity, single custodians, and localized currency risks.
How to calculate CAGR in Google Sheets and Excel
To calculate Compound Annual Growth Rate in Google Sheets or Excel, use the geometric formula =(End/Start)^(1/Years)-1 or the built-in =RRI(Years, Start, End) function. CAGR determines the constant annual rate that compounds initial capital to its final balance over time, smoothing out periodic volatility into a single annualized metric without requiring interim pricing data.
What is the difference between XIRR and simple return
Simple return measures absolute percentage gain on capital regardless of duration or timing. XIRR calculates the exact annualized internal rate of return for irregular cash flows by solving the daily discounted equation where net present value equals zero. Use XIRR whenever periodic contributions, distributions, or withdrawals occur across calendar dates.
How does inflation affect investment returns
Inflation steadily erodes the future purchasing power of investment returns. Nominal return measures account balance expansion, whereas real return measures the true growth in purchasing capacity. The exact real return is determined by the Fisher relation (1 + r_nom) / (1 + i) − 1, which reveals greater purchasing erosion than the naive subtraction shortcut.
How to track net worth without linking bank accounts
To track net worth privately without sharing financial credentials with third-party aggregators, maintain a monthly snapshot balance sheet. On the last day of each month, record assets across cash, investments, retirement, and real estate, then deduct liabilities. Storing data locally in browser memory or plain text JSON preserves privacy without ongoing security risks.