Learning Module 1
Rates and Returns
Key Outcomes Summary & Practice Problems
What you must be able to do
Curriculum Year: 2026
Interpret interest rates as required rates of return, discount rates, or opportunity costs; explain as a sum of a real risk-free rate plus risk premiums.
Calculate and interpret different return measurement approaches over time (HPR, arithmetic, geometric, harmonic) and describe their appropriate uses.
Compare money-weighted and time-weighted rates of return; evaluate portfolio performance using both measures.
Calculate and interpret annualized return measures and continuously compounded returns; describe their appropriate uses.
Calculate and interpret gross/net returns, pre-tax/after-tax returns, real returns, and leveraged returns; describe their appropriate uses.
1 ยท Interest Rates โ Three Interpretations
Required rate of return โ minimum return an investor must receive to accept an investment.
Discount rate โ rate used to bring a future cash flow to present value; terms are near-interchangeable.
Opportunity cost โ the return foregone by choosing one course of action over another.
r = Real risk-free rate + Inflation premium + Default risk premium + Liquidity premium + Maturity premium
Nominal risk-free rate โ Real risk-free rate + Inflation premium
(Exact: (1 + nominal) = (1 + real)(1 + inflation))
2 ยท Return Measures โ Formulas & When to Use
R = (Pโ โ Pโ + Iโ) / Pโ
Multi-period: R = [(1+Rโ)(1+Rโ)โฆ(1+Rโ)] โ 1
Rฬ = (Rโ + Rโ + โฆ + Rโ) / T
Best for: estimating expected return over a single future period.
Rฬ_G = [(1+Rโ)(1+Rโ)โฆ(1+Rโ)]^(1/T) โ 1
Best for: measuring compound growth; historical performance reporting.
Always โค arithmetic mean (equal only when all returns identical).
Xฬ_H = n / ฮฃ(1/Xแตข)
Best for: averaging ratios (e.g., P/E) and cost-averaging strategies.
Always โค geometric โค arithmetic (with non-zero variance).
Arithmetic ร Harmonic = (Geometric)ยฒ
Trimmed mean: removes extreme % from both ends before averaging.
Winsorized mean: replaces extremes with nearest non-extreme value.
1. Value portfolio immediately before each cash flow.
2. Calculate HPR for each sub-period: HPR = (End Value โ Start Value) / Start Value.
3. Link: R_TW = [(1+rโ)(1+rโ)โฆ(1+rโ)]^(1/N) โ 1
4 ยท Annualizing Returns & Continuous Compounding
R_annual = (1 + R_period)^c โ 1
c = number of periods per year (52 weekly, 12 monthly, 4 quarterly, 365 daily)
For >1 year holding period, c = 1/years e.g. 18-month: c = 2/3
r_cc = ln(Pโ/Pโ) = ln(1 + R)
Multi-period CC returns are additive: rโ,T = rโ,โ + rโ,โ + โฆ + r_{T-1,T}
CC return is always < corresponding HPR
PV = FV_N ร (1 + R_s/m)^(โmN)
m = compounding periods/year, R_s = quoted annual rate, N = years
5 ยท Other Return Measures
Gross return โ return before management/admin fees; used to compare manager skill across investors.
Net return โ gross return less all fees; what the investor actually receives. Better for investor decision-making.
Pre-tax nominal โ the default; no adjustment for taxes or inflation.
After-tax nominal โ total return minus taxes on dividends, interest, and realized capital gains.
Real return โ removes inflation effect: (1 + real) = (1 + nominal) / (1 + inflation). Useful across time periods and countries.
After-tax real return โ the true benchmark; what investor earns after taxes and inflation (rarely calculated due to investor-specific tax rates).
Leveraged return โ R_L = R_P + (V_B/V_E)(R_P โ r_D). Leverage amplifies both gains AND losses.