Module 5 — Return measures, risk measures, risk-adjusted return measures (Sharpe, Treynor, Jensen's Alpha), benchmarking, performance attribution
where E = Ending Value, B = Beginning Value, I = Income (dividends/interest)
Portfolio value rises from Rs. 1,00,000 (1 Apr 2018) to Rs. 1,20,000 (31 Mar 2019):
HPR = (1,20,000 − 1,00,000) / 1,00,000 = 20%
If the investor also received Rs. 5,000 as dividend/interest income:
HPR = (5,000 + (1,20,000 − 1,00,000)) / 1,00,000 = 25%
MWRR = the IRR of the cash flows; depends on the timing of cash flows. TWRR = the compound rate of growth, computed by linking sub-period returns; it is unaffected by cash flow timing (essentially the geometric mean return).
| Aspect | MWRR (= IRR) | TWRR |
|---|---|---|
| Sensitive to cash flow timing? | Yes | No |
| Used by | Wealth manager / investor reporting (money flows multiple times) | SEBI-mandated for discretionary PMS managers (preceding 3 years) for uniform comparison |
| Reflects | What the investor actually experienced | The fund manager's underlying skill, isolated from cash-flow timing |
Investor contributes at the start of each year: 2015–19 = Rs. 5,000 / 10,000 / 15,000 / 20,000 / 25,000. Portfolio value grows to Rs. 1,05,920.31 by end-2019.
Solving: IRR (MWRR) = 15.15%
| Year | Value at start of year (Rs.) | Value at end of period (Rs.) | Sub-period return | Wealth Relative (1+r) |
|---|---|---|---|---|
| 2015 | 5,000.00 | 4,750.00 | −5.00% | 0.9500 |
| 2016 | 14,750.00 | 12,508.00 | −15.20% | 0.8480 |
| 2017 | 27,508.00 | 29,736.15 | 8.10% | 1.0810 |
| 2018 | 49,736.15 | 65,030.01 | 30.75% | 1.3075 |
| 2019 | 90,030.01 | 1,05,920.31 | 17.65% | 1.1765 |
Cumulative Wealth Relative = 0.95 × 0.848 × 1.081 × 1.3075 × 1.1765 = 1.3396
TWRR = (1.3396)(1/5) − 1 = 6.021% (annualized; equals geometric return)
Choice 1: −50% (Yr 1), +100% (Yr 2). AMR = (−50%+100%)/2 = 25%; GMR = ((1−0.5)(1+1.0))1/2 − 1 = 0% — Rs. 100,000 returns to Rs. 100,000.
Choice 2: +10% (Yr 1), +10% (Yr 2). AMR = (10%+10%)/2 = 10%; GMR = ((1.1)(1.1))1/2 − 1 = 10% — grows to Rs. 1,21,000.
Despite a lower AMR, Choice 2 leaves the investor better off — GMR is the correct measure for long-run accumulation; AMR is the better single-period forecast.
| Capital Contribution / AUM | 1,00,00,000 |
| Profit @ 20% | 20,00,000 |
| Gross Value of Portfolio | 1,20,00,000 |
| Less: Other Expenses (0.50%) | 60,000 |
| Less: Fixed Mgmt Fee (1.5% of avg of opening/closing) | 1,65,000 |
| Hurdle (Required) Value @ 10% = 1,00,00,000 × 1.10 | 1,10,00,000 |
| Less: Performance Fee (20% of profit over hurdle) | 1,55,000 |
| Less: Exit Load (2%) | 2,32,400 |
| Net Value of Portfolio | 1,13,87,600 |
Gross Return = (1,20,00,000 − 1,00,00,000)/1,00,00,000 = 20%
Net Return = (1,13,87,600 − 1,00,00,000)/1,00,00,000 = 13.88%
Pre-tax return = 5%, capital gains tax = 15%: Post-tax return = 5% × (1 − 0.15) = 4.25%
Rs. 1,00,000 grows to Rs. 1,33,960 over 5 years (returns: −5%, −15.2%, 8.1%, 30.75%, 17.65%):
CAGR = (1,33,960 / 1,00,000)(1/5) − 1 = 6.02%
Returns: 2017 = 15.5%, 2018 = 9.5%, 2019 = −6.9%. Rs. 100 grows: 100 → 115.5 → 126.47 → 117.7459
Compounded Annualised Return = (117.7459/100)(1/3) − 1 = 5.60%
Rs. 100 lacs invested; manager deploys Rs. 75 lacs in equity (return Rs. 7.5 lacs) and keeps Rs. 25 lacs in liquid funds (4% return).
Ignoring cash drag: Return = 7.5/75 = 10%
Adjusting for cash drag: (10% × 75 lacs) + (4% × 25 lacs) = Rs. 8.5 lacs ⇒ 8.5% on Rs. 100 lacs — the figure that must actually be reported.
Rf = risk-free return; β = market beta; Rm = return on market portfolio. Beta return rewards bearing market (systematic) risk; Alpha return rewards bearing non-market (unsystematic) risk via manager skill.
Portfolio return = 25%, Market return = 15%, Beta = 1.5, T-bill (risk-free) yield = 5%
Risk-free return = 5%
Beta return = 1.5 × (15% − 5%) = 15%
Jensen's Alpha = 25% − {5% + 1.5 × (15% − 5%)} = 25% − 20% = 5%
(Note: simple "excess return over benchmark" = 25% − 15% = 10% — some professionals call this "alpha," distinguishing it from Jensen's alpha.)
| Security | Return | Weight |
|---|---|---|
| A | 15% | 30% |
| B | 10% | 20% |
| C | 12% | 20% |
| D | 18% | 30% |
Portfolio Return = (15%×30%)+(10%×20%)+(12%×20%)+(18%×30%) = 14.30%
| Risk Type | Description / Measure |
|---|---|
| Total risk | Variability in expected return — measured by Variance / Standard Deviation |
| Downside risk | Losses or worse-than-expected outcomes — measured by Semi-variance/Semi-SD or Target semi-variance |
| Market risk | Arises from market-wide price fluctuations affecting all investments; cannot be diversified away (can be hedged); measured by Beta |
| Tracking error | Standard deviation of the difference between portfolio and benchmark total returns; lower = closer to benchmark; calculated against the Total Returns Index |
| Systematic risk | Risk from common factors — interest rates, exchange rates, commodity prices; cannot be diversified away (can be hedged); measured by Beta |
| Unsystematic risk | Sector/company-specific; CAN be diversified away; alpha return rewards bearing this risk |
| Liquidity risk | Uncertainty from difficulty of converting an asset to cash near its economic worth (e.g., T-bills = low; art/illiquid assets = high) |
| Credit risk | Risk that a borrower fails to repay on time — arises in debt instruments |
Stock A: β = 1.2 (60% weight); Stock B: β = 1.1 (40% weight)
βp = 0.60 × 1.2 + 0.40 × 1.1 = 1.16
β > 1 ⇒ more volatile than benchmark; β < 1 ⇒ less volatile than benchmark.
Annualized Return Rp = 10.50%, Risk-free Rate Rf = 5.50%, SD σp = 6.50%
Sharpe Ratio = (10.50% − 5.50%) / 6.50% = 0.7692
Interpretation: 0.7692 percentage points of excess return generated for each percentage point of standard deviation (total risk).
Limitation: Ignores the diversification potential of the portfolio (uses total risk, not just systematic risk).
Using the same Rp = 10.50%, Rf = 5.50%, and Beta = 1:
Treynor Ratio = (10.50% − 5.50%) / 1 = 0.05
Interpretation: 0.05 percentage points of excess return per unit of systematic risk.
| Aspect | Sharpe Ratio | Treynor Ratio |
|---|---|---|
| Risk measure used | Standard deviation (total risk) | Beta (systematic risk only) |
| Best suited for | Investors who have NOT achieved adequate diversification (evaluating standalone/mutually exclusive portfolios) | Investors with well-diversified wealth, where unsystematic risk is minimal |
| Ranking vs each other | Identical for a fully-diversified portfolio (total risk = systematic risk); for a poorly-diversified portfolio, Treynor ranking can exceed Sharpe ranking because Treynor ignores unsystematic risk | |
Adjusts excess return for downside risk only — appeals to investors who define risk as "chance of losing money" rather than overall variability. Higher Sortino = superior performance.
Rb = benchmark return; Stdev(p−b) = standard deviation of (portfolio − benchmark) differences = tracking error (active risk). Numerator = "active return" (manager skill vs. benchmark). Helps determine whether observed alpha is due to skill or chance.
Adjusts the portfolio's risk (via levering/de-levering with T-bills) to match the market portfolio's risk, then compares the resulting return with the market return.
Portfolio return = 35%, σportfolio = 42%; Market return = 28%, σmarket = 30%; T-bill rate = 6%
Weight in Portfolio (to match market vol) = 30/42 = 0.714; Weight in T-bills = 1 − 0.714 = 0.286
r* = (0.714 × 0.35) + (0.286 × 0.06) = 26.7%
Since 26.7% < market's 28% (by 1.3%), the managed portfolio underperformed on a risk-adjusted basis.
GIPS defines a benchmark as "an independent rate of return (or hurdle rate) forming an objective test of effective implementation of investment strategy."
Customized benchmarks are used when market indices don't fit a manager's style — advantage: a valid fit; disadvantage: high construction/maintenance cost.
Common benchmarking errors: choosing a non-representative benchmark, an un-investable/hard-to-replicate benchmark, one with frequently-changing components, or one lacking available (Total Return Index) data.
Managers' Universe / Peer Group Analysis: ranks portfolios with similar investment characteristics and risk-return profiles on a risk-adjusted return measure — the median portfolio serves as the yardstick.
Attribution dissects portfolio return into: (a) return driven by the benchmark and (b) the differential return — then determines whether the differential was driven by manager skill or random factors (luck).
Differential return from being overweight outperforming sectors or underweight underperforming sectors/categories vs. the benchmark. The "allocation effect" quantifies this.
Differential return from picking winning securities within a sector/category, or avoiding poor performers, relative to the benchmark.
Market timing = anticipating market moves and shifting exposure accordingly; selectivity = picking securities that generate extra returns. Most studies show timing rarely adds value; results on selectivity are mixed.
An absolute measure: annualised fund return minus risk-free yield, further reduced by (standardised expected market premium × total portfolio risk). Shows the excess return earned that could not have been earned by simply holding the market portfolio.
Indian investor invests Rs. 50 lacs in a US equity fund at ₹70/$. Fund earns 15%; INR appreciates to ₹65/$.
Capital in USD = 50,00,000 / 70 = $71,428.57
Value after 15% gain = 71,428.57 × 1.15 = $82,142.86
Value in INR = 82,142.86 × 65 = Rs. 53,39,285.62
Rupee return = (53,39,285.62 − 50,00,000)/50,00,000 ≈ 6.79% (not 15%) — the gap is explained by INR's appreciation from Rs. 70/$ to Rs. 65/$.
Note: in most cases INR depreciates against major currencies, which then ADDS to the rupee-return (e.g., part of gold's INR returns reflect INR depreciation, not just LBMA gold price moves).
1. A portfolio has an annualized return of 10.50%, a risk-free rate of 5.50%, and an annualized standard deviation of 6.50%. What is its Sharpe Ratio?
2. Using the same portfolio (Rp = 10.50%, Rf = 5.50%) but with Beta = 1, what is the Treynor Ratio?
3. A portfolio earned 25% return when the market returned 15%, the portfolio's beta is 1.5, and the risk-free rate is 5%. What is the Jensen's Alpha?
4. Which risk-adjusted return measure is most appropriate for an investor whose overall wealth is already well-diversified (so unsystematic risk is minimal)?
5. An investment grows from Rs. 1,00,000 to Rs. 1,33,960 over 5 years. What is its CAGR (approximately)?