Time Value of Money
FoundationA dollar today is worth more than a dollar tomorrow: it can earn interest, and inflation erodes purchasing power. Simple interest is paid on principal only, while compound interest rolls interest into principal — FV = PV(1+r)ⁿ; discounting works back from future amounts to present value. Nominal returns must be adjusted for inflation to show real growth.
- Money has time value: early money earns interest sooner, dodges inflation and cuts risk
- Simple interest = principal × rate × years; compound interest rolls on — FV = PV(1+r)ⁿ
- Discounting reverses compounding: PV = FV÷(1+r)ⁿ; invest only if NPV is positive
文體小知識：商業知識說明文——講解金錢的時間值，逐步演算簡單利息與複息、現值與貼現、淨現值及名義與實際回報率。
§Text Passage
1The time value of money means a given sum is worth more today than in the future. Three reasons: money can be invested to earn interest from the moment it is received; inflation erodes purchasing power year by year; and there is uncertainty over whether a future payment will arrive. Personal finance and business investment decisions both rest on this concept.
2Calculating time value starts with two kinds of interest. Simple interest is charged on the principal only: interest equals principal times the annual rate times the number of years. Compound interest rolls interest into principal — each period's interest is added to the principal and itself earns interest next period, so growth accelerates. The longer the term and the higher the rate, the wider the gap between compound and simple interest: the compound-interest effect.
3Future value tells how much today's money will become after a number of years: future value equals present value times one plus the rate to the power of the number of periods. Discounting is the reverse — converting a future amount into today's present value using the same formula backwards. Amounts at different points in time must first be converted to the same date, or surface figures mislead. Summing the present values of cash flows and subtracting the initial outlay gives the net present value, used to judge whether an investment is worthwhile.
4When comparing returns, distinguish nominal from real rates of return: the nominal rate is the stated rate, while the real rate deducts inflation and reflects true growth in purchasing power. At a four per cent deposit rate with inflation at two and a half per cent, purchasing power grows by only about one and a half per cent. Judge wealth by real returns, or a high nominal rate may distract from the erosion of rising prices.
1Summary 重點摘要
A dollar today is worth more than a dollar tomorrow: it can earn interest, and inflation erodes purchasing power. Simple interest is paid on principal only, while compound interest rolls interest into principal — FV = PV(1+r)ⁿ; discounting works back from future amounts to present value. Nominal returns must be adjusted for inflation to show real growth.
2Vocabulary cards Vocabulary
The concept that a sum of money today is worth more than the same amount in the future.
The value today of a future sum after discounting.
The total value of today's principal after growing at compound interest for a number of periods.
Interest charged on the principal only: interest = principal × annual rate × years.
Interest earned on both principal and accumulated interest, so growth accelerates.
Converting a future amount into present value at a given rate — compound interest in reverse.
Total present value of cash flows minus the initial outlay; a positive figure suggests a worthwhile investment.
The stated rate of return before adjusting for inflation.
The return after deducting inflation, reflecting real growth in purchasing power.
A sustained rise in prices that erodes the purchasing power of money.
3Step by step Steps
- Step 1: Why today's money is worth moreThe time value of money is the cornerstone of personal finance: the same sum is worth more received today than in three years. Three reasons — money in hand can be deposited or invested to earn interest immediately; inflation means the same amount buys less later; and future payment carries the uncertainty of default. For example, $10,000 received today and put into a three per cent time deposit becomes $10,300 in a year; receive it three years later and that $300 of interest is lost. Early money starts compounding early — the starting point of all financial calculations.
- Step 2: Calculating simple interestSimple interest is charged on the principal only, so each year earns the same amount: interest = principal × annual rate × years, and the maturity value = principal × (1 + annual rate × years). For example, $20,000 in a simple-interest deposit at three per cent for three years earns 20,000×3% = $600 each year, $1,800 in total, returning $21,800 at maturity. Keep units consistent: annual rates go with years, monthly rates with months — mixing them wrecks the result.
- Step 3: Calculating compound interestCompound interest adds each period's interest to the principal, so the next period earns interest on interest: future value FV = PV×(1+r)ⁿ, where r is the periodic rate and n the number of periods. For example, $20,000 at three per cent compounded yearly for three years: year one 20,000×1.03 = $20,600; year two 20,600×1.03 = $21,218; year three 21,218×1.03 = $21,854.54. Total interest is $1,854.54 — $54.54 more than simple interest. In one step: 20,000×1.03³ = $21,854.54.
- Step 4: The compound effect grows with timeThe gap between compound and simple interest widens rapidly as years and rates rise: the longer the term, the more layers of compounding and the steeper the curve. For example, $20,000 at three per cent for ten years earns $6,000 in simple interest, reaching $26,000; compounded, it grows to 20,000×1.03¹⁰ ≈ $26,878.33 — a gap of about $878. At five per cent, ten years of compounding reaches $32,577.89. Start saving early precisely to give compounding time to work: starting young beats starting ten years later by a wide margin.
- Step 5: Present value and discountingDiscounting is compound interest in reverse: converting a future amount into today's present value, PV = FV÷(1+r)ⁿ. For example, you will receive $21,854.54 in three years and discount at three per cent: PV = 21,854.54÷1.03³ = 21,854.54÷1.092727 = $20,000 — so $20,000 today equals $21,854.54 in three years. Always discount amounts occurring at different dates to the same point before comparing, or surface figures mislead and a smaller cash flow can look larger.
- Step 6: Judging investments with NPVNet present value discounts each future cash flow, sums them, and subtracts the initial outlay: a positive result means the project is worth considering at that rate of return; a negative one means it destroys value. For example, a cha chaan teng considers a second-hand blender costing $9,500 that should save $2,000 a year for three years, discounted at five per cent: PV = 2,000÷1.05 + 2,000÷1.05² + 2,000÷1.05³ ≈ 1,904.76 + 1,814.06 + 1,727.68 = $5,446.5. NPV = 5,446.5 − 9,500 = −$4,053.5 — negative, so not worth buying unless further benefits lie beyond.
- Step 7: Nominal versus real returnsWhen comparing returns, distinguish nominal from real rates: the nominal rate is the stated rate, while the real rate deducts inflation — roughly the nominal rate minus the inflation rate. For example, a time deposit pays four per cent while inflation runs at two and a half per cent: the real return is about 4% − 2.5% = 1.5%, so purchasing power grows only about one and a half per cent; if inflation hits five per cent, the real return turns negative and savings shrink in real terms. Plan with real returns, or a flashy nominal rate will mask the erosion of rising prices.
5Worked example Example
Calculation question: Yan receives a scholarship of $30,000 that she will not need for a while. The bank offers two plans: Plan A, a three-year time deposit at three per cent per annum compounded yearly; Plan B, a simple-interest deposit at three and a half per cent per annum for three years. (1) Find Plan A's future value. (2) Find Plan B's future value. (3) Compare the two and explain the source of the difference.
- (1) Plan A (compound): year one 30,000×1.03 = $30,900; year two 30,900×1.03 = $31,827; year three 31,827×1.03 = $32,781.81; i.e. FV = 30,000×1.03³ = $32,781.81, interest $2,781.81
- (2) Plan B (simple): interest each year = 30,000×3.5% = $1,050; three years total 1,050×3 = $3,150; future value = 30,000 + 3,150 = $33,150
- (3) Compare: Plan B exceeds Plan A by 33,150 − 32,781.81 = $368.19 — the difference comes from the rate, not the compounding itself
- Decompose: in simple interest, three-and-a-half per cent beats three per cent by 30,000×0.5%×3 = $450; compounding (at the same three per cent) adds only $81.81; 450 − 81.81 = $368.19 — the two parts sum exactly to the gap
- Answer: (1) Plan A compounds: FV = PV(1+r)ⁿ = 30,000×1.03³ = 30,000×1.092727 = $32,781.81, of which interest is $2,781.81. (2) Plan B uses simple interest: $1,050 per year (30,000×3.5%), $3,150 over three years, giving $33,150. (3) Plan B is higher by $368.19. The difference comes from the rate, not the method: at half a point more, Plan B's simple interest gains 30,000×0.5%×3 = $450 over three years, while Plan A's compounding advantage over the same-rate simple interest is only $81.81; 450 − 81.81 = $368.19. The lesson: compare deposits by working out the maturity value from both the rate and the compounding method — never assume the word 'compound' alone means a better deal.
6Common mistakes Mistakes
7Quick check Quick check
Q1Why is a dollar today worth more than a dollar in the future?
Q2Calculate: $10,000 in a simple-interest deposit at 2% p.a. for 3 years — how much interest?
Q3Calculate: $5,000 at 4% p.a. compounded yearly for 2 years — the future value?
Q4A nominal return of 5% with inflation at 3% — the approximate real return?
Q5What does a negative net present value mean for a project?
8Recap Recap
- Money has time value: early money earns interest sooner, dodges inflation and cuts risk
- Simple interest = principal × rate × years; compound interest rolls on — FV = PV(1+r)ⁿ
- Discounting reverses compounding: PV = FV÷(1+r)ⁿ; invest only if NPV is positive
- Real return = nominal return − inflation — the true growth of purchasing power
- The earlier saving starts, the more time the compound effect has to work