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.
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.
Practice Questions
- Shares bought for £1 800 are sold for £2 430. Calculate the ROI.
- An investment of £5 000 grows at 6% per year. Find its value after 15 years.
- A property bought for £220 000 is sold for £275 000 after 4 years. Find the annual capital gain and the ROI.
- Two investments each start at £10 000. Investment A grows at 5% per year; Investment B at 9% per year. How much more is Investment B worth after 10 years?
- Using the Rule of 72, estimate how long it takes to double an investment growing at 7% per year.