Investments - The Mathematics of Growing Wealth
An investment is putting your money to work so that it grows over time. Unlike leaving money in a current account (where it earns little), investing means accepting some level of risk in exchange for the potential of higher returns. Understanding the mathematics of investments – returns, growth, and risk – gives you the tools to make informed decisions.
The modern stock market traces back to 1602, when the Dutch East India Company (VOC) became the first company in history to sell tradable shares to the public, raising funds for its trading voyages through what became the Amsterdam Stock Exchange. The mathematical principle behind diversification, meanwhile, is much more recent: the American economist Harry Markowitz formalised it in his 1952 paper “Portfolio Selection”, proving mathematically that combining assets whose returns don't move in lockstep can reduce a portfolio's overall risk without necessarily reducing its expected return. The work later earned Markowitz the Nobel Memorial Prize in Economic Sciences in 1990 and remains the theoretical foundation of nearly all professional portfolio management today.
Why Invest?
Simply saving money in a low-interest account may not keep pace with inflation – meaning your money actually loses purchasing power over time. Investing aims to grow your money at a rate that beats inflation, building real wealth over the long term.
Key Investment Terms
| Term | Meaning |
|---|---|
| Principal | The initial amount invested |
| Return | The gain made on an investment |
| Return on Investment (ROI) | (Gain / Cost) × 100 |
| Capital gain | Profit from selling an asset for more than its purchase price |
| Dividend | A share of company profits paid to shareholders |
| Portfolio | The collection of all investments held by an individual |
| Diversification | Spreading investments across different assets to reduce risk |
Return on Investment (ROI)
ROI = [(Final value − Initial value) ÷ Initial value] × 100
ROI tells you the percentage gain (or loss) on your investment.
Annualised ROI accounts for the time period: if total ROI is 40% over 5 years, annual ROI ≈ 40 / 5 = 8% per year (approximate simple method).
Compound Growth of Investments
When investment returns are reinvested, the growth compounds:
A = P(1 + r)t
Where r is the annual growth rate and t is years. This is the same formula as compound interest and it shows why long-term investing is so powerful.
Worked Examples
Gain = £3 150 − £2 400 = £750.
ROI = (750 / 2 400) × 100 = 31.25%.
Annual return ≈ 31.25 / 2 = 15.6% per year (approximate).
A = 10 000 × (1.07)20 = 10 000 × 3.8697 = £38 697.
The investment nearly quadruples in 20 years without adding any extra money.
Gain = £240 000 − £180 000 = £60 000.
ROI = (60 000 / 180 000) × 100 = 33.3%.
Annual gain = £60 000 / 5 = £12 000 per year.
Risk vs Return
| Investment Type | Typical Return | Risk Level |
|---|---|---|
| Cash savings account | Low (1–5%) | Very low |
| Government bonds | Low to medium (2–6%) | Low |
| Property | Medium (4–8%) | Medium |
| Index funds / ETFs | Medium to high (6–10%) | Medium |
| Individual shares | Variable (can be negative) | High |
| Cryptocurrency | Highly variable | Very high |
Diversification
Spreading investments across different asset types, sectors, and countries reduces the risk that any single loss destroys your portfolio. If one investment falls 50%, a diversified portfolio may only fall a few percent because other holdings absorb the shock.
Key Takeaways
- ROI = [(Final − Initial) / Initial] × 100.
- Compound growth: A = P(1 + r)t. Time is the investor's greatest advantage.
- Higher potential returns always come with higher risk.
- Diversification spreads risk across multiple investments.