Time Value of Money *(Part II — skills)*

Concept

A dollar today is worth more than a dollar tomorrow. Today's dollar can be invested to earn a return, and inflation erodes the buying power of a future dollar. This single idea underpins compounding (growing money forward), discounting (valuing future money today), and almost every planning recommendation — from "start saving now" to "why a lump sum beats instalments."

Key rules & facts

• Compounding — you earn returns on the principal *and* on previously accumulated returns. The effect accelerates with time; the last years contribute the most growth.
• Future Value (FV) — what a sum today grows to when compounded forward: FV = PV × (1 + r)^n.
• Present Value (PV) — a future sum's worth today, found by discounting back: PV = FV ÷ (1 + r)^n. A higher rate or a longer horizon → a *lower* PV.
• Annuity — a stream of equal periodic payments. FV of an annuity accumulates regular savings; PV of an annuity values a stream of future income (e.g. a retirement payout).
• Rule of 72 — 72 ÷ annual return% ≈ years to double. At 6% → ~12 years; at 8% → ~9 years. Run it in reverse on inflation to see how fast prices double (3% → ~24 years).
• Inflation & real return — real return ≈ nominal return − inflation. "Safe" cash can quietly lose real value.
• Start early — because compounding rewards time, an early consistent saver frequently ends up ahead of a later, larger saver.

Formula toolkit — which to use when

Worked example — Future Value of a lump sum

Invest $10,000 at 6% for 3 years.

• Year 1: 10,000 × 1.06 = 10,600
• Year 2: 10,600 × 1.06 = 11,236
• Year 3: 11,236 × 1.06 = 11,910.16

So FV ≈ $11,910. Direct formula check: 10,000 × (1.06)^3 = 10,000 × 1.191016 = $11,910.16. Match.

Worked example — Present Value (discounting back)

You need $20,000 in 4 years; you can earn 5%. How much to set aside today?

• (1.05)^4 = 1.05 × 1.05 × 1.05 × 1.05 = 1.21550625
• PV = 20,000 ÷ 1.21550625 = $16,454 (approx.)

Set aside about $16,454 now — the extra ~$3,546 is what compounding supplies.

Worked example — Rule of 72

• At 8%: 72 ÷ 8 = 9 years to double. So $50,000 → ~$100,000 in ~9 years.
• Inflation at 3%: 72 ÷ 3 = 24 years for prices (and your dollar's cost) to double.

Worked example — the cost of ignoring inflation

A "4% return" with 3% inflation gives a real return ≈ 4 − 3 = 1%. On $100,000 that is only ~$1,000 of *real* gain, not $4,000. The nominal figure flatters the result.

Worked example — start-early advantage (FV of an annuity)

Save $3,000/year at 7% for 10 years:

FV = 3,000 × [((1.07)^10 − 1) ÷ 0.07].

• (1.07)^10 ≈ 1.967151
• (1.967151 − 1) ÷ 0.07 = 0.967151 ÷ 0.07 ≈ 13.8164
• FV ≈ 3,000 × 13.8164 ≈ $41,449

Those same deposits started earlier, then left to compound, would grow further with no extra contributions — that is why time beats size.

Exam angle

Mostly reasoning, not heavy math — "which sum is worth more today," a Rule-of-72 doubling estimate, "why start now?", and picking the right lens (grow forward vs discount back). Any arithmetic asked is short and clean.