Nguyễn Thảo Nghiêm Túc
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![➡️ Quantitative Finance from Scratch, part 44: Arbitrage Pricing Theory (APT). Stephen Ross published it in 1976 as an alternative to the CAPM. The CAPM explains expected returns with one risk, the market. The APT allows several risks, called factors, for example surprises in economic growth, inflation or interest rates. Every stock gets its own beta for each factor, plus swings of its own that have nothing to do with any factor. A factor model describes how a return moves when a factor changes. The APT asks a different question: what return should an investor expect for carrying that risk? In a factor model each stock can have its own alpha. The APT says this alpha should be zero, because two stocks with the same exposures carry the same risk and should earn the same expected return. The formula: E[R_i] = r_f + β_i1 λ_1 + β_i2 λ_2 + ... + β_iK λ_K, with r_f the risk free rate, β the exposure of the stock to one factor and λ the premium the market pays per unit of that factor's risk. If two stocks have the same exposures and one is expected to earn more, you buy that one and sell the other. Every factor cancels out, and with many such pairs their own swings average away. Traders who exploit such gaps push prices back, so expected returns line up on the formula, at least approximately. The theory leaves open which factors to use; researchers pick them from data, like Chen, Roll and Ross (1986) with economic factors, and the Fama-French model follows the same logic. The code (apt_reel.py, 15 lines of plain Python without packages) is general on purpose: the expected return from the risk free rate, the betas and the premiums, the alpha as the part the betas leave unexplained, and the arbitrage rule for two stocks with the same betas. Honest notes: the plane, the stocks and the long-short pairs in the video come from a simulation (seed 44), not from market data. Assumed: 100 stocks, two factors, betas between 0 and 1.5, risk free rate 2 %, premiums 4 % and 2 % per year, one stock with an alpha of 3 %, monthly factor surprises with 4.5 % and 3 % volatility, own swings 6 % per month, 64 pairs over 120 months. In the simulation the hedged position has exactly zero factor exposure and zero net cost, and the leftover swings shrink roughly with the square root of the number of pairs. Real arbitrage has costs, limits and estimation error in the betas. Sources: Ross, S. A. (1976), The Arbitrage Theory of Capital Asset Pricing, Journal of Economic Theory 13(3), 341-360. Chen, N.-F., Roll, R. and Ross, S. A. (1986), Economic Forces and the Stock Market, Journal of Business 59(3), 383-403. #quantfinance #apt #arbitragepricingtheory #factormodels #capm Education only, not financial advice.](/video/cover/7705772526953106720.webp)
