M6 β CM-EIP: Securities, CIS (EIP) and Foreign Exchange
Return & Time Value Calculations
Chapter 2 of CM-EIP turns risk and return into numbers. You need to measure a return (holding-period, annualised, arithmetic and geometric), move money through time (compound interest, future and present value, effective rates, annuities, perpetuities, loan and sinking-fund payments, NPV and IRR), strip out inflation with the Fisher relationship, and describe risk with expected return, standard deviation and the coefficient of variation. Calculation items are quick marks if you pick the right formula and match the rate to the period, and costly if you fall for the standard slips: dividing a cumulative return by the years, using an ordinary annuity when payments start today, or quoting variance as risk.
9 sectionsΒ·~5 min read
βChecked against the IBF CM-EIP syllabus chapter 2 (checked 13 Sep 2026); MAS MoneySense 'Effects of compounding interest' and 'What is investing?' pages, checked 13 Sep 2026; standard time-value-of-money and statistics formulas, all worked figures recomputed. Unofficial prep, not endorsed by MAS or IBF.
βAnnualised return over n years = (ending value Γ· beginning value)^(1/n) β 1. Never divide a cumulative return by n.
βGeometric mean β€ arithmetic mean; they are equal only when every period's return is the same. +100% then β50% gives arithmetic 25%, geometric 0%.
βSimple interest = P Γ r Γ n. Compound FV = PV Γ (1 + r)^n; PV = FV Γ· (1 + r)^n.
βRule of 72: years to double β 72 Γ· rate %. Most accurate near 8%; at 24% it says 3 years against an exact 3.22.
βNPV = PV of inflows β cost; accept if NPV > 0. IRR is the discount rate at which NPV = 0. A higher discount rate lowers PV and NPV.
βExact Fisher: real = (1 + nominal) Γ· (1 + inflation) β 1. 8% nominal and 3% inflation give 4.85% real, not 5%.
βExpected return = Ξ£ probability Γ return. Variance = Ξ£ probability Γ (return β expected)Β²; standard deviation = βvariance, in percent.
βCoefficient of variation = standard deviation Γ· expected return: lower means less risk per unit of return. Unreliable when expected return is near zero or negative.
βFees compound: a 1% p.a. fee on a 6% gross return leaves about 75% of the gross ending wealth after 30 years.
Measuring return: holding-period, annualised, arithmetic and geometric
β’The exam gives you prices, income and a time span and asks for a single return figure. Total return always has two parts: income (dividends, distributions, coupons) and capital gain or loss. MAS MoneySense defines return as the gain or loss made on an investment.
β’Arithmetic mean = sum of period returns Γ· number of periods.
β’Geometric mean = [(1 + r1)(1 + r2)β¦(1 + rn)]^(1/n) β 1. It is the rate the investor actually compounded at.
β’Worked example: you buy at $5.00, receive $0.25 of dividends and sell at $5.40. HPR = (0.40 + 0.25) Γ· 5.00 = 13.0%. A holding that grows from $20,000 to $26,000 in 5 years has an annualised return of 1.30^(1/5) β 1 = 5.39%, not 30% Γ· 5 = 6%.
β’Why geometric β€ arithmetic: a loss needs a bigger percentage gain to recover. A share that doubles (+100%) then halves (β50%) has an arithmetic mean of 25% but a geometric mean of 0%, and the investor is back where they started. Roughly, geometric β arithmetic β variance Γ· 2, so the gap widens with volatility.
β’Trap: dividing by the selling price instead of the purchase price, leaving out income, or dividing a cumulative return by the number of years.
β’Takeaway: past compound growth is the geometric mean; the arithmetic mean overstates it whenever returns vary.
Simple vs compound interest, future value and present value
β’Simple interest is paid on the original principal only. Compound interest is paid on the principal and on interest already credited; MAS MoneySense calls it interest earned on top of interest already earned.
β’Simple interest = P Γ r Γ n; maturity value = P Γ (1 + r Γ n).
β’Future value of a lump sum: FV = PV Γ (1 + r)^n.
β’Present value of a lump sum: PV = FV Γ· (1 + r)^n.
β’Changing rates: chain the growth factors, e.g. 1.02 Γ 1.03 Γ 1.04, rather than compounding the average rate.
β’Solving for time: n = ln(FV Γ· PV) Γ· ln(1 + r). Solving for rate: r = (FV Γ· PV)^(1/n) β 1.
β’Worked example (hypothetical 3% rate): $10,000 compounded annually for 10 years grows to 10,000 Γ 1.03^10 = $13,439.16. Simple interest gives $13,000.00. The gap is zero after year 1 and widens every year after that. To have $50,000 in 8 years at 5%, you need 50,000 Γ· 1.05^8 = $33,841.97 today.
β’Discount rate effect: PV falls when the discount rate rises or the wait lengthens, and the longer the wait, the larger the percentage fall. $10,000 due in 10 years is worth $6,755.64 at 4% but $5,583.95 at 6%, a 17.3% drop.
β’Trap: using simple interest when the question says compounded, rounding a fractional number of years down when the target must be reached, or compounding forwards when the question asks for a present value.
β’Takeaway: FV multiplies by (1 + r)^n; PV divides by it. Higher rates and longer waits shrink PV.
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