Start with one share. Then follow the money through returns, portfolios, and trading costs.
1.1 / Stocks and returns
What do you own when you buy a stock?
A share is a small unit of ownership in a company. A stock price is the price of one share. If you own 10 shares and each is worth $50 in the market, your shares have a market value of $500.
A higher share price does not, by itself, mean a larger company or a better investment. Companies have different numbers of shares. Value can also mean an estimate of what a share is worth, based on the business and its future. That estimate may differ from its current market price.
You can gain or lose money when the price changes. A company may also pay a dividend: cash paid to its shareholders. Dividend payments are not guaranteed.
Price return = (P1 − P0) / P0
Total return = (P1 − P0 + d) / P0
P0 is the starting price; P1 is the ending price; d is the cash dividend per share during the period. These simple formulas assume one share is held throughout, with no trading costs or share splits. The dividend is kept as cash.
START$50one share
→
END$52 + $1share value + cash dividend
The share price rose by $2. The dividend adds $1. Together, the gain is $3 per share.
A portfolio is the collection of assets you hold. Here it contains Stock A, Stock B, and cash. Its value is the sum of all three parts.
V = C + qAPA + qBPB
V is portfolio value; C is cash; q is the number of shares; P is price per share. The labels A and B identify the two stocks.
A weight is the fraction of the portfolio invested in one asset. For A, wA = qAPA / V. The cash weight is wC = C / V. With no borrowing or short selling, the weights are non-negative and add to 1.
THE STARTING $1,000100% of the account
A · 50%B · 30%Cash · 20%
Stock A $500Stock B $300Cash $200
A weight tells us where the money is. Cash is part of the portfolio, too.
The same result comes from the starting weights and each asset's return:
rp is the portfolio return. Each r is that asset's return over the same period. This equation uses starting weights and assumes no trades during the period.
Diversification means spreading money across different investments. It can reduce the effect of one company doing badly. But many stocks can fall together. Correlation measures how returns move together, from −1 to +1. Lower correlation can make a mix more stable; it does not remove all risk. Two fictional stocks are enough to explain the math, not to show a well-diversified real portfolio.
A return in the next period acts on the money left after the previous period. This is compounding.
V2 = V0(1 + r1)(1 + r2)
V0 is starting portfolio value; V2 is value after two periods; r1 and r2 are the two portfolio returns. No money is added or withdrawn.
Up 10%, then down 10%: the ending value is 1% below the start. The drop from the peak is 10%. The path is illustrative.
Risk concerns uncertain outcomes, including losses. Volatility describes how much returns vary. Drawdown describes a fall from a previous peak. These describe different parts of the same path.
Trading fees are direct charges. One basis point (bp) is 0.01 percentage point. A fee of 10 bps is 0.10%: buying $500 of A costs $0.50 in fees. Selling it for $520 costs another $0.52 at the same rate.
The bid is the best displayed buying price; the ask is the best displayed selling price. Their difference is the spread. If A's bid is $49.95 and ask is $50.05, buying 10 shares at the ask and immediately selling at the unchanged bid loses $1, even with no commission.
Slippage is the difference between an expected trading price and the actual price received. Prices can move, and a large order may fill at several prices. Fees, spreads, and slippage are different costs; a simulation may include only some of them.