Compound Interest & Depreciation
When you deposit money in a bank or take out a loan, the interest you earn or owe does not always stay the same year after year. Compound interest is the process by which interest is calculated not only on the original principal but also on the interest that has already accumulated. This 'interest on interest' effect causes money to grow exponentially rather than linearly, making it one of the most powerful concepts in personal finance and economics. Compound interest can be applied over different compounding periods — yearly, half-yearly (semi-annually), quarterly, or monthly. The more frequently interest is compounded, the faster the balance grows. Understanding how to calculate these different scenarios is essential for comparing savings accounts, loans, and investment products in the real world. On the flip side, assets like cars, machinery, and electronics lose value over time through a process called depreciation. Annual depreciation follows the same mathematical structure as compound interest but in reverse — value decreases by a fixed percentage each period. Additionally, inflation erodes the real purchasing power of money, so a sum that seems large today may buy far less in the future. Mastering all three concepts — compound growth, depreciation, and inflation — equips you to make informed financial decisions throughout your life.
- The Core Idea: Interest on Interest
- Annual Compounding Formula
- Adjusting for Different Compounding Frequencies
- Annual Depreciation
- Real Value Under Inflation
The Core Idea: Interest on Interest
Simple interest is computed only on the original principal P every period. Compound interest, however, adds earned interest back to the principal at the end of each compounding period, so the next period's interest is larger. After just a few years the difference between simple and compound interest becomes dramatic. For example, ₹10,000 at 10% simple interest for 3 years earns ₹3,000; at 10% compounded annually it earns ₹3,310 — an extra ₹310 purely from compounding.
Annual Compounding Formula
When interest is compounded once per year, the amount A after n years is A = P(1 + r)^n, where r is the annual interest rate as a decimal (e.g., 8% → r = 0.08). The compound interest earned is CI = A − P. Always convert the percentage rate to a decimal before substituting into the formula.
Adjusting for Different Compounding Frequencies
When interest is compounded more than once per year, the annual rate r is divided by the number of compounding periods k per year, and the number of periods becomes n × k. The general formula becomes A = P(1 + r/k)^(nk). For half-yearly compounding k = 2; for quarterly k = 4; for monthly k = 12. A higher k always results in a larger final amount for the same nominal annual rate, because interest is being reinvested more frequently.
Annual Depreciation
Depreciation models the loss in value of an asset over time. If an asset has an initial value P and depreciates at an annual rate of d (as a decimal), its value V after n years is V = P(1 − d)^n. This is structurally identical to compound interest but with subtraction instead of addition. For instance, a machine worth ₹5,00,000 depreciating at 15% per year is worth ₹5,00,000 × (0.85)^n after n years. The amount of value lost is called accumulated depreciation: Dep = P − V.
Real Value Under Inflation
Inflation reduces the purchasing power of money. If the inflation rate is i per year, then the real value (in today's purchasing power) of an amount A received n years in the future is Real Value = A ÷ (1 + i)^n. This is called discounting. Conversely, to find what today's sum P will be worth in terms of future prices, you compute P × (1 + i)^n, which tells you how much you would need in the future to match today's purchasing power. Financial planners combine compound interest and inflation calculations to determine whether an investment actually grows in real terms: the real growth rate ≈ nominal rate − inflation rate (for small rates).
The Key Equation
- A — Final Amount (currency (₹, $, etc.)): Total value including principal and all accumulated compound interest after n years.
- P — Principal (currency): The initial sum of money deposited or invested before any interest is added.
- r — Annual Interest Rate (dimensionless decimal): The nominal yearly interest rate expressed as a decimal; e.g., 12% per annum → r = 0.12.
- k — Compounding Frequency (periods per year): Number of times interest is compounded per year: 1 (annual), 2 (half-yearly), 4 (quarterly), 12 (monthly).
- n — Number of Years (years): The total time duration of the investment or loan in years.
- nk — Total Compounding Periods (periods): The total number of times interest is applied over the full duration of the investment.
Worked Example
Problem. Priya invests ₹25,000 at an annual interest rate of 12% compounded quarterly for 2 years. Separately, her scooter was purchased for ₹80,000 and depreciates at 10% per year. Find (a) the compound amount and compound interest on Priya's investment, and (b) the value of the scooter after 3 years.
Answer. Investment grows to ₹31,669.25 with compound interest of ₹6,669.25; scooter value after 3 years is ₹58,320.
Origins · 1 minJacob Bernoulli and how this was discovered
Jacob Bernoulli was the first mathematician to rigorously study compound interest and its limiting behaviour. While investigating what happens when compounding frequency increases without bound, he discovered in 1683 that the expression (1 + 1/n)^n approaches a constant as n → ∞. This constant, approximately 2.71828, is now called Euler's number e and forms the basis of continuous compounding: A = Pe^(rt). Bernoulli also made foundational contributions to probability theory and the calculus of variations. Bernoulli noticed the mathematical constant e while thinking about a bank problem — not in a pure-math context. He described it as a limit in his work 'Ars Conjectandi', published posthumously in 1713, which also laid the groundwork for the law of large numbers in probability theory.
Related Concepts
- Simple Interest — Simple interest is the foundation on which compound interest is built. Understanding how interest is calculated only on the principal helps highlight why compounding on accumulated interest produces faster growth.
- Ratio and Proportion — Compound interest calculations rely on repeated proportional scaling (multiplying by the growth factor). A strong understanding of ratios and percentage change makes interpreting compound growth and depreciation much more intuitive.
- Exponential Functions — The compound interest formula A = P(1 + r/k)^(nk) is an exponential function of time n. Studying exponential growth and decay unifies compound interest, depreciation, population growth, and radioactive decay under one mathematical framework.
- Percentages, Profit and Loss — Depreciation rates and interest rates are expressed as percentages. Fluency with percentage calculations — including successive percentage changes — is directly applied when computing compound interest over multiple periods.