Portfolio beta
Also written Beta of a portfolio
The weighted average of the betas of the holdings in a portfolio — the measure of its systematic, non-diversifiable risk used in CAPM, the Treynor ratio and market-neutral strategies.
In plain language
A single share's beta tells you how much it tends to move when the market moves. A portfolio beta answers the same question for the whole fund.
The workbook computes it simply: use portfolio beta as a weighted average beta of the holdings, with weights equal to each holding's share of the portfolio. In its market-neutral example it describes it as the total of the weighted average of the stock-specific beta.
What it measures is systematic risk — in the workbook's words in Chapter 6, the change in the returns earned by the fund portfolio on account of a change in the returns on a broad-based index. The higher the portfolio beta, the more volatile the fund is compared with the market. This risk cannot be diversified away by spreading across industries, sectors and geographies, which is why CAPM pays investors for beta and not for total risk.
For an AIF, portfolio beta is something the manager chooses and manages. A fund holding mainly small-cap and mid-cap shares will have a high beta; a market-neutral Category III AIF works to keep it at or close to zero.
How it works
What the number means (Chapter 5). A beta of 1 moves equally with the market portfolio; 1.50 is more volatile (a 10% rise in the market goes with a 15% change); below 1 is less volatile. The market portfolio's own beta is always 1.
Where portfolio beta is used.
| Use | Workbook section | How beta enters |
|---|---|---|
| Expected return | CAPM, Chapter 5 | E(R) = Rf + β × (Rm − Rf) |
| Risk-adjusted performance | Treynor ratio, Chapter 9 | (Rp − Rf) / β |
| Alpha | Chapter 10 | Alpha = actual return − CAPM expected return |
| Strategy design | Market-neutral and directional strategies, Chapter 11 | Neutral aims for β close to 0; directional does not |
| Beta management | Chapter 6 | Manager keeps systematic risk in line with the strategy in the PPM |
Why portfolios, not single shares. The empirical tests in Chapter 5 found beta not stable for individual stocks but stable for portfolios — more so for larger portfolios (over 50 stocks) and longer periods (over 26 weeks). High-beta portfolios tended to decline toward 1.00 over time and low-beta portfolios to rise toward it. Black, Jensen and Scholes found a positive linear relationship between monthly excess return and portfolio beta.
Beta and bull or bear markets (Chapter 6). High-beta investments are likely to outperform the benchmark in bull markets and underperform in bear markets.
Treynor versus Sharpe. For a well-diversified portfolio the two ratios give nearly identical results because total risk is close to systematic risk; for undiversified portfolios they differ, and the workbook says investors should then prefer the Sharpe ratio.
The formula
Portfolio beta = Σ (weight of holding i × beta of holding i)
For a long-short book, short positions carry a negative sign.
CAPM: E(Rp) = Rf + βp × (Rm − Rf)
Treynor: (Rp − Rf) / βp
Alpha: Actual return − E(Rp)
A worked example
1. The workbook's case, in rupees. An AIF puts ₹15 crore of a ₹100 crore portfolio in stock A (beta 1.2) and ₹85 crore in stock B (beta 1.5). Risk-free rate 6%, market risk premium 10%.
- Portfolio beta = 0.15 × 1.2 + 0.85 × 1.5 = 0.18 + 1.275 = 1.455
- Expected return = 6% + 1.455 × 10% = 20.55%
2. The market-neutral book (Fund PQC, Chapter 11). Large-cap positions:
| Stock | Beta | Position (₹) | Weight | Weighted beta |
|---|---|---|---|---|
| Company B | 0.40 | +5,00,00,000 | 0.25 | +0.10 |
| Company C | 2.00 | +5,00,00,000 | 0.25 | +0.50 |
| Company D | 1.20 | −10,00,00,000 | 0.50 | −0.60 |
| Total | Nil | 0.00 |
Mid-cap: Company E (beta 1.50) long ₹5 crore and Company F (beta 1.50) short ₹5 crore give +0.75 − 0.75 = 0.00. The book is beta-neutral, though each stock still carries unsystematic risk, and the NIFTY50 options with different strikes move the overall beta close to, not exactly, zero.
3. Alpha with a changing beta (Example 10.8). T-bill 5.60%, benchmark 9.25%.
| Best case | Worst case | |
|---|---|---|
| Portfolio beta | 1.3 | 1.5 |
| Expected return | 5.60% + 1.3 × 3.65% = 10.35% | 5.60% + 1.5 × 3.65% = 11.08% |
| Net IRR | 12.77% | 3.38% |
| Alpha | +2.42% | −7.70% |
Why NISM asks about it
Chapter 5 (Introduction to Capital Market Theory) defines beta through CAPM and the security market line, reports the empirical tests on portfolio beta, and its case study computes a weighted portfolio beta. Chapter 6 covers beta management for AIFs; Chapter 9 uses portfolio beta in the Treynor ratio; Chapter 10 uses it to compute alpha; Chapter 11 builds market-neutral and directional strategies around it. Expect a calculation of portfolio beta and CAPM return, a Treynor ratio, and a question on what beta a market-neutral fund targets.
Common exam traps
- Portfolio beta is a simple weighted average of the holdings' betas.
- Beta measures systematic risk only, which diversification cannot remove.
- Market-neutral targets a beta of zero or close to zero; directional strategies do not.
- A zero beta does not mean zero risk — the workbook warns leverage and concentrated positions can still make the fund volatile.
- Treynor divides by beta; Sharpe divides by standard deviation. For undiversified portfolios, prefer Sharpe.
- Beta is more stable for portfolios than for single stocks, and more so with over 50 stocks and over 26 weeks.
- Check the workbook's low-beta sentence in section 5.6. It says a stock with beta 0.80 would fall 15% when the market falls 10%. By its own definition (beta times the change in the market) the fall is 8%; the 15% belongs to its beta 1.50 example.
- The CAPM expected return rose in the worst case of Example 10.8 because beta rose — the workbook calls this a limitation of CAPM.
Check yourself
1.A fund holds 15 per cent in a stock of beta 1.2 and 85 per cent in a stock of beta 1.5. With a risk-free rate of 6 per cent and a market risk premium of 10 per cent, the expected portfolio return is:
- a)20.55 per cent
- b)21 per cent
- c)18 per cent
- d)20 per cent
Show the answer
Answer: (a) 20.55 per cent
This is the workbook's own case study, whose explanation reads: Use Portfolio beta as a weighted average beta (0.151.2) + (0.851.5) = 1.455. Then using CAPM approach the expected return is 6% + (1.455 * 10%) = 20.55%. The step candidates miss is the first — portfolio beta is a simple weighted average of component betas. That works because beta captures only systematic risk, which cannot be diversified away, so unlike portfolio standard deviation there is no covariance term to add.
2.A Category III fund taking long positions in small and mid-cap securities will have a high portfolio beta. What follows?
- a)The risk can be diversified away by spreading across industries and geographies
- b)The risk cannot be diversified in that way, and high beta investments are likely to outperform the benchmark during bull markets but underperform during bear markets
- c)The fund will always outperform a large-cap fund
- d)Beta is irrelevant for AIFs
Show the answer
Answer: (b) The risk cannot be diversified in that way, and high beta investments are likely to outperform the benchmark during bull markets but underperform during bear markets
if Fund ABC takes long positions primarily in small-cap and mid-cap securities, the overall portfolio Beta will be high, as compared to Fund XYZ which takes long positions in large-cap securities. This is due to the inherent risky nature of the underlying shares in which Fund ABC is investing. This risk cannot be diversified, by investing across industries, sectors and geographies. And: High beta investments are likely to outperform the benchmark during bull markets, but are also likely to underperform during bear markets. This is Chapter 5's systematic risk restated — beta is precisely what diversification cannot remove.
3.How does a short-bias strategy differ from a long-short strategy?
- a)A short-bias fund takes only short positions
- b)A short-bias fund takes both long and short positions but always maintains a NET SHORT exposure, whereas a long-short fund may maintain either a net long or a net short exposure
- c)A short-bias fund is market-neutral
- d)A short-bias fund may not use derivatives
Show the answer
Answer: (b) A short-bias fund takes both long and short positions but always maintains a NET SHORT exposure, whereas a long-short fund may maintain either a net long or a net short exposure
Short-bias Strategy is a type of Directional Strategy, wherein the investment manager takes both long and short positions in selected stocks or a broad-based market index, but maintains a net short exposure to the broad market. And the comparison: in a Long-short Strategy, the investment manager may maintain a net long or a net short exposure, as compared to the net short exposure to be maintained in a Short-bias Strategy. In a typical "130/30" long-short strategy, the investment manager maintains a net long position, up to 100 percent of the total investable funds. Option (a) describes something closer to a Dedicated-Short posture; the workbook's contrasting term is the Dedicated-Long Strategy, where the manager takes exclusively long positions. And remember that a Directional Strategy is an opposite of Market-Neutral Strategy, as the investment manager will not aim at having a Portfolio Beta of zero.
Where this is taught
Free preparation for NISM Series V-DRelated terms
- BetaHow sharply a share moves relative to the market index — beta 1 moves with the index, above 1 amplifies it, below 1 dampens it. The standard measure of systematic risk.
- CAPMA model that prices the return an investor should demand from a share: the risk-free rate plus beta times the market risk premium.
- Sharpe ratioReturn earned above the risk-free rate divided by standard deviation — how much reward an investment produced for each unit of total risk its holder had to live with.
- Systematic riskThe part of an investment's risk that comes from economy-wide forces moving every asset at once — it cannot be diversified away, and it is the only risk the market pays you to carry.
- Treynor ratioRisk premium per unit of market risk — the return a scheme earned above the risk-free rate, divided by its beta rather than by its standard deviation.
- Unsystematic riskThe part of an investment's risk that belongs to one company or one issuer — a strike, a fraud, a downgrade — and which diversification can remove, unlike market-wide systematic risk.
- AlphaThe return a fund earned above what its beta and the benchmark say it should have earned — the slice of performance left over once the market has been given credit for its share.
- Directional StrategyA Category III AIF strategy that takes a net long or net short position on stocks or an index, betting on the market's direction; short-bias and dedicated-long are its variants.
- Market-Neutral StrategyA Category III AIF strategy taking equal long and short exposures within a sector, industry or market-cap segment so that portfolio beta is zero or close to zero.
- Security Market LineThe graph of CAPM: expected return plotted against beta. In equilibrium every asset lies on it — a security plotting above is undervalued, one plotting below is overvalued.
- Regression of beta toward the meanAn empirical finding in the tests of CAPM: high-beta portfolios tend to drift down towards 1.00 over time, and low-beta portfolios tend to drift up towards 1.00.
- Expected Portfolio ReturnThe weighted average of the expected returns of the holdings in a portfolio, each weight being that asset's share of portfolio value — the return half of Modern Portfolio Theory.
- Variance of a SecurityThe probability-weighted average of the squared gaps between a security's possible returns and its expected return — the forward-looking measure of a single asset's risk in Modern Portfolio Theory.
- Beta ManagementThe manager's job of keeping a fund's systematic risk in line with the strategy stated in its private placement memorandum, rather than letting the portfolio beta drift with the market.
- Cross hedgeHedging an exposure with futures on a different but closely associated asset, because no futures contract exists on the asset itself — for example, hedging a stock portfolio with index futures.