Portfolio Management — Practice

Question 1

Security A has an expected return of 12% and standard deviation of 20%; Security B has an expected return of 16% and standard deviation of 30%. A portfolio invests 60% in A and 40% in B. Compute portfolio expected return and portfolio standard deviation under two scenarios: (a) correlation coefficient of +1; (b) correlation coefficient of 0.2. Comment on the diversification benefit.

Accepted Answer

Portfolio expected return (same in both scenarios, since expected return does not depend on correlation) = (0.6 × 12%) + (0.4 × 16%) = 7.2% + 6.4% = 13.6%. (a) With ρ = +1: Portfolio variance = (0.6)²(0.20)² + (0.4)²(0.30)² + 2(0.6)(0.4)(1)(0.20)(0.30) = 0.0144 + 0.0144 + 0.0288 = 0.0576; Portfolio standard deviation = √0.0576 = 24%. Note this equals exactly the weighted average of the individual standard deviations, (0.6×20%)+(0.4×30%) = 12%+12% = 24%, confirming that perfect positive correlation produces zero diversification benefit. (b) With ρ = 0.2: Portfolio variance = 0.0144 + 0.0144 + 2(0.6)(0.4)(0.2)(0.20)(0.30) = 0.0144 + 0.0144 + 0.00576 = 0.03456; Portfolio standard deviation = √0.03456 ≈ 18.59%. This is materially lower than the 24% weighted average and lower than either individual security's own standard deviation reduction would suggest by simple averaging, demonstrating the diversification benefit directly: for the identical 13.6% expected return, the lower correlation of 0.2 produces a portfolio risk of only 18.59%, compared to 24% under perfect correlation, a genuine, quantified risk reduction of 5.41 percentage points achieved at absolutely no cost in expected return, purely because the two securities' returns do not move in lockstep.

Question 2

Using Security A (σ = 20%) and Security B (σ = 30%) from the previous problem, with a correlation coefficient of 0.2, compute the portfolio weight in Security A that minimises overall portfolio variance.

Accepted Answer

The minimum variance weight formula for a two-security portfolio is: w_A = [σ_B² − ρ×σ_A×σ_B] ÷ [σ_A² + σ_B² − 2ρ×σ_A×σ_B]. Substituting: σ_A² = 0.04, σ_B² = 0.09, ρ×σ_A×σ_B = 0.2 × 0.20 × 0.30 = 0.012. w_A = [0.09 − 0.012] ÷ [0.04 + 0.09 − 2(0.012)] = 0.078 ÷ [0.13 − 0.024] = 0.078 ÷ 0.106 = 0.7358, approximately 73.6%. This means the minimum-variance combination allocates approximately 73.6% to Security A and 26.4% to Security B, a materially different weighting from the 60%/40% split used in the previous problem; note that the minimum-variance weight is not simply 100% in the lower-risk Security A, since the diversification benefit from combining a modest allocation to the higher-risk, imperfectly-correlated Security B can still reduce overall portfolio risk below what holding Security A alone would achieve, illustrating precisely why the minimum-variance weighting depends on the correlation coefficient and not merely on ranking the two securities by their individual standard deviations.

Question 3

Portfolio X offers an expected return of 11% with a standard deviation of 15%. Portfolio Y offers an expected return of 11% with a standard deviation of 19%. Explain which portfolio is dominated, and why a rational, risk-averse investor would never choose the dominated portfolio.

Accepted Answer

Portfolio Y is dominated by Portfolio X, since both offer the identical expected return of 11%, but Portfolio Y carries a higher standard deviation (19% versus 15%), meaning Portfolio Y exposes an investor to strictly more risk for exactly the same expected return. A rational, risk-averse investor, by definition, prefers less risk to more risk for the same expected return (this is precisely what risk aversion means), and would therefore never choose Portfolio Y over Portfolio X, since Portfolio X offers everything Portfolio Y offers, in terms of expected return, with strictly less risk; Portfolio Y therefore does not lie on the efficient frontier at all, since a dominated portfolio, by construction, is never the optimal choice for any risk-averse investor regardless of that investor's specific individual degree of risk aversion, distinguishing dominance (a general, universally applicable conclusion) from the separate question of which specific efficient-frontier portfolio a particular investor with a specific risk tolerance would personally prefer.

Question 4

Explain the separation theorem in portfolio theory, and why two investors with very different personal risk tolerances should, under this theory, hold the identical portfolio of risky assets despite their differing risk preferences.

Accepted Answer

The separation theorem holds that the investment decision (which specific portfolio of risky assets to hold) can be separated from the financing decision (how much of an investor's total wealth to allocate between that risky portfolio and the risk-free asset), and that once a risk-free asset is available, every rational investor should hold the identical tangency, or market, portfolio of risky assets, the single specific combination of risky assets where a straight line drawn from the risk-free rate is tangent to the efficient frontier of risky assets, since this combination dominates every other risky-asset combination once mixed with the risk-free asset in appropriate proportions. Two investors with very different personal risk tolerances should, under this theory, hold this identical risky-asset portfolio, differing from each other only in how much of their total wealth each allocates to this shared risky portfolio versus the risk-free asset: a highly risk-averse investor holds a larger proportion of wealth in the risk-free asset and a smaller proportion in the shared risky portfolio, moving toward the lower-risk, lower-return end of the capital market line, while a less risk-averse investor holds a larger proportion, or even, in principle, borrows at the risk-free rate to hold more than 100% of wealth in the shared risky portfolio, moving further out along the capital market line toward higher risk and higher expected return; the composition of the risky portfolio itself, however, remains identical for both investors, since any deviation from the tangency portfolio's specific composition would move an investor onto a less efficient combination that a more appropriately weighted mix of the shared tangency portfolio and the risk-free asset would strictly dominate.

Question 5

A bond portfolio has a modified duration of 6.5 years. If market yields are expected to rise by 0.5 percentage points, estimate the approximate percentage change in the portfolio's value, and explain why this is described as an approximation rather than an exact figure.

Accepted Answer

Approximate percentage price change = −Modified duration × Change in yield = −6.5 × 0.005 (0.5 percentage points expressed as a decimal) = −0.0325, or approximately −3.25%, meaning the portfolio's value is expected to fall by approximately 3.25% given the anticipated 0.5 percentage point rise in yields. This is described as an approximation, not an exact figure, because duration measures only the linear, first-order sensitivity of a bond's price to a change in yield, effectively approximating the genuinely curved (convex) relationship between bond prices and yields with a straight line at the current yield level; for small changes in yield, this linear approximation is reasonably accurate, but for larger yield changes, the true price change deviates from the duration-based estimate because of convexity, the degree to which the actual price-yield relationship curves away from the straight-line duration approximation, meaning a more precise estimate for larger yield movements would need to incorporate a convexity adjustment on top of the basic duration-based approximation used here.

Question 6

A pension fund has a single, known liability of ₹50 crore due in 8 years, and wishes to immunise a bond portfolio against interest rate risk to ensure it can meet this liability regardless of future rate movements. Explain the core principle of immunisation, and why simply investing in bonds maturing exactly in 8 years does not, on its own, fully achieve this goal if those bonds pay periodic coupons.

Accepted Answer

The core principle of immunisation is to construct a bond portfolio whose duration exactly matches the duration of the future liability it is meant to fund, here 8 years, so that a change in interest rates affects the portfolio's value and the reinvestment income it generates on its own coupon payments in genuinely offsetting directions: a rise in interest rates reduces the bond portfolio's market value but increases the reinvestment return earned on coupons received before the liability date, while a fall in interest rates increases the portfolio's market value but reduces reinvestment income, and matching duration ensures these two offsetting effects approximately cancel, leaving the portfolio's value at the liability date approximately protected regardless of which direction rates move. Simply investing in bonds maturing exactly in 8 years does not, on its own, achieve this goal if those bonds pay periodic coupons, because a coupon-paying bond's duration, a weighted average of the times to each of its cash flows including the periodic coupons received well before maturity, is actually shorter than its stated maturity of 8 years; a portfolio of 8-year-maturity coupon bonds therefore has a duration meaningfully less than 8 years, mismatched against the 8-year liability duration, and genuine immunisation requires selecting bonds (or a combination of bonds and other instruments) whose duration, not merely whose maturity, matches the liability's own duration of 8 years.

Question 7

Portfolio manager P's fund has a return of 15%, standard deviation of 22%, and beta of 1.1. The risk-free rate is 6%. This fund represents an individual investor's entire investable wealth. Compute the appropriate performance evaluation ratio for this investor's use, and explain why the alternative ratio would be less appropriate here.

Accepted Answer

Since this fund represents the investor's entire investable wealth, the Sharpe ratio is the appropriate performance evaluation measure, because the investor's total risk, standard deviation, genuinely matters to them in this case, with no further diversification available elsewhere in their overall holdings to eliminate the fund's own unsystematic risk. Sharpe ratio = (Portfolio return − Risk-free rate) ÷ Portfolio standard deviation = (15% − 6%) ÷ 22% = 9% ÷ 22% = 0.409. The Treynor ratio, by contrast, would be less appropriate here: Treynor ratio = (Portfolio return − Risk-free rate) ÷ Beta = (15% − 6%) ÷ 1.1 = 9% ÷ 1.1 = 8.18%, but this measure is designed for evaluating one portfolio held as part of a broader, well-diversified overall portfolio, where only the portfolio's systematic risk (beta) is genuinely relevant to the investor because any unsystematic risk specific to this one portfolio is assumed to be already diversified away at the level of the investor's total combined holdings; since this fund is the investor's entire wealth, with no other holdings to diversify away its unsystematic risk, using Treynor here would ignore genuinely relevant total risk that this investor cannot diversify away elsewhere, making the Sharpe ratio, based on total risk, the theoretically correct and more informative choice in this specific scenario.

Question 8

A pension fund's strategic asset allocation policy targets 60% equity and 40% debt. Its portfolio manager, believing equities are currently overvalued based on short-term market conditions, temporarily reduces the equity allocation to 50% and increases debt to 50%, intending to revert to the 60/40 policy once conditions normalise. Classify this action and explain how it differs from rebalancing.

Accepted Answer

This is a tactical asset allocation decision, a deliberate, temporary deviation from the fund's long-term strategic policy mix, based on the manager's active, short-to-medium-term view that equities are currently overvalued relative to their longer-run expected return; it represents a genuine, active bet layered on top of the underlying strategic policy, intended to be reversed once the manager's specific view about current market conditions changes or plays out. This differs from rebalancing, which is a mechanical, policy-driven action to bring an actual portfolio mix that has drifted away from its strategic target weights, purely due to differential market performance of the asset classes over time, back into line with those target weights; rebalancing is not based on any active view about current relative valuation, but purely on restoring the fund's long-term policy mix after market movements have caused it to drift, whereas the tactical shift described here is explicitly and deliberately based on the manager's own active market view, a fundamentally different motivation even though both actions happen to involve changing the equity-debt mix.

Question 9

An investor purchases the debt of a company currently undergoing bankruptcy proceedings, at a substantial discount to face value, expecting a partial but uncertain recovery once the restructuring concludes. Identify two distinct risks this investment carries beyond ordinary market price risk, and explain why these risks are difficult to hedge using conventional instruments.

Accepted Answer

First, default and recovery risk: the actual amount ultimately recovered through the restructuring or liquidation process is inherently uncertain until the process concludes, depending on factors such as the company's realisable asset values, the claims of other creditor classes with potentially competing or senior priority, and the specific terms of whatever restructuring plan is eventually approved, none of which can be reliably estimated with the precision ordinary credit analysis of a performing company's debt would allow. Second, legal and process risk: the outcome depends heavily on the specific bankruptcy or restructuring jurisdiction's legal process, its typical timeline (which can extend for years), and how the specific court or restructuring authority resolves disputes among competing creditor classes, introducing genuine legal and procedural uncertainty distinct from ordinary market or credit risk. These risks are difficult to hedge using conventional instruments because standard hedging tools such as interest rate derivatives or credit default swaps are typically designed and priced around performing, ongoing-concern credit risk with reasonably well-modelled default probabilities and recovery assumptions, whereas a company already in bankruptcy presents a fundamentally different, more idiosyncratic and process-dependent risk profile that conventional derivative instruments are not well suited to price or hedge efficiently, leaving direct, specialised distressed-debt expertise as the primary tool for managing this specific risk rather than conventional hedging.

Question 10

Name three distinct triggers that would appropriately prompt an investor to revise an existing portfolio's holdings, beyond simple periodic rebalancing to a fixed target allocation.

Accepted Answer

First, a genuine change in the investor's own objectives or circumstances, such as approaching retirement (typically prompting a shift toward a more conservative allocation), a change in income needs, or a change in risk tolerance, all of which change what the appropriate portfolio composition genuinely should be for this specific investor going forward. Second, a change in an individual security's own risk-return characteristics, such as a company's credit rating being downgraded, its competitive position materially deteriorating, or new information emerging that changes the analyst's own fundamental valuation of a holding, warranting a reassessment of whether that specific security remains an appropriate holding regardless of the overall portfolio's asset-class weights. Third, a genuine, sustained change in the broader market or economic environment, such as a structural shift in interest rate regimes or a fundamental change in the correlation relationships between asset classes the original portfolio construction assumed, which could warrant revisiting the portfolio's overall strategy and asset allocation approach itself, not merely rebalancing back to an allocation that may no longer be appropriate given how the underlying environment has genuinely changed.

Question 11

An investor holds a portfolio entirely of domestic equities with an average correlation of 0.75 among the holdings. A financial adviser suggests adding foreign equities with an expected correlation of only 0.35 to the existing domestic holdings. Explain why this suggestion is grounded in the same portfolio theory principle developed earlier in this chapter, and what specific benefit the lower correlation is expected to provide.

Accepted Answer

This suggestion is grounded directly in the correlation-driven diversification principle this chapter opened with: portfolio risk depends not merely on the average risk of the individual holdings but critically on the correlation between them, and a lower correlation between asset groups produces a greater risk-reduction benefit when those groups are combined into a single portfolio than a higher correlation would. The investor's existing domestic-only portfolio, with an average correlation of 0.75 among holdings, already benefits from some diversification (since 0.75 is below the perfect-correlation value of 1), but adding foreign equities with a materially lower expected correlation of 0.35 to the domestic holdings is expected to provide additional, meaningful diversification benefit beyond what further domestic diversification alone could achieve, since foreign equities are exposed to different economic cycles, currency movements, political and regulatory environments, and industry compositions than domestic equities, making their returns less likely to move in lockstep with the existing domestic holdings; this lower correlation, following exactly the same mathematical logic developed for the two-security portfolio variance formula, is expected to reduce the combined portfolio's overall risk for a given level of expected return more effectively than continuing to diversify only within the domestic market, where the average correlation among holdings is already higher.

Question 12

Explain the primary rationale for including private equity or hedge fund allocations within an institutional investor's broader portfolio, and one genuine trade-off this allocation carries relative to traditional public equity and debt holdings.

Accepted Answer

The primary rationale is diversification: private equity and hedge fund strategies have historically exhibited lower correlation with traditional public equity and debt markets than traditional asset classes exhibit with each other, since these alternative strategies often pursue return sources, such as private company operational improvement, arbitrage strategies, or specialised market-neutral positioning, that are structurally less tied to the broad market movements driving traditional equity and debt returns; including such allocations can therefore reduce overall portfolio risk for a given level of expected return, following the same correlation-driven logic this chapter's opening principle establishes, applied here to an entirely different, less traditional asset class. A genuine trade-off this allocation carries is materially reduced liquidity: private equity investments typically lock up capital for extended periods, often many years, with no ability to redeem or sell on demand the way a publicly traded security can be, and many hedge fund structures similarly impose lock-up periods or redemption notice requirements restricting an investor's ability to access their capital quickly; an institutional investor allocating to these strategies must therefore weigh the genuine diversification and potentially enhanced return benefit against this reduced liquidity, ensuring the illiquid allocation is sized appropriately relative to the investor's own genuine need for liquidity and ability to commit capital for the required holding period.

Question 13

Fund A has a return of 14%, standard deviation of 18%; Fund B has a return of 17%, standard deviation of 25%. The risk-free rate is 6%. An investor is choosing one fund to hold as their entire investment portfolio. Using the Sharpe ratio, determine which fund offers superior risk-adjusted performance, even though Fund B has the higher raw return.

Accepted Answer

Sharpe ratio for Fund A = (14% − 6%) ÷ 18% = 8% ÷ 18% = 0.444. Sharpe ratio for Fund B = (17% − 6%) ÷ 25% = 11% ÷ 25% = 0.440. Fund A has the marginally higher Sharpe ratio, 0.444 versus 0.440, meaning Fund A offers a very slightly superior risk-adjusted return per unit of total risk taken, despite Fund B's higher raw, unadjusted return of 17% versus Fund A's 14%. Since this investor is using the fund as their entire investment portfolio, meaning total risk (standard deviation) is the genuinely relevant risk measure rather than only systematic risk, the Sharpe ratio is the appropriate comparison tool, and it reveals that Fund B's higher raw return comes at a proportionately even higher level of total risk (25% standard deviation versus 18%), such that on a purely risk-adjusted basis, the two funds are very close, with Fund A very marginally ahead; this illustrates precisely why raw, unadjusted return comparisons between funds of differing risk levels can be misleading, and why a risk-adjusted performance measure is the more appropriate basis for choosing between them.

Question 14

Explain, conceptually, why utility sector stocks and technology growth stocks might be expected to exhibit relatively low correlation, and how a portfolio manager might use this expectation in constructing a diversified equity portfolio.

Accepted Answer

Utility sector stocks are typically characterised by stable, regulated revenue streams, modest but predictable growth, and returns that are comparatively insensitive to the broader business cycle and to interest rate-driven growth expectations, since utility demand tends to remain relatively stable regardless of economic conditions, and their valuations are often more directly tied to bond-like considerations such as dividend yield and interest rate levels than to growth expectations. Technology growth stocks, by contrast, are typically characterised by returns highly sensitive to changing growth expectations, innovation cycles, and investor risk appetite, with valuations often heavily dependent on distant future growth assumptions that can shift dramatically based on changing sentiment or economic conditions; these two return drivers, stable regulated demand and bond-like interest rate sensitivity for utilities, versus growth expectations and risk appetite for technology, are sufficiently different in nature that the two sectors' returns would be expected to exhibit relatively low correlation with each other. A portfolio manager constructing a diversified equity portfolio might deliberately combine holdings across such sectors with differing, low-correlation return drivers, rather than concentrating holdings within sectors that share similar underlying return drivers and would therefore be expected to move together more closely, achieving genuine diversification benefit within the equity allocation itself, not merely across broader asset classes such as equity versus debt.

Question 15

A portfolio consists of three stocks: Stock P (weight 40%, beta 1.4), Stock Q (weight 35%, beta 0.9), Stock R (weight 25%, beta 0.6). Compute the portfolio's overall beta, and explain what this figure indicates about the portfolio's systematic risk relative to the market.

Accepted Answer

Portfolio beta = weighted average of individual betas = (0.40 × 1.4) + (0.35 × 0.9) + (0.25 × 0.6) = 0.56 + 0.315 + 0.15 = 1.025. A portfolio beta of 1.025, very close to 1, indicates that this portfolio's systematic risk is very close to that of the market portfolio as a whole, meaning the portfolio's returns are expected to move, on average, in almost exactly the same proportion as the overall market's returns; a 10% rise in the broad market would be expected to produce approximately a 10.25% rise in this portfolio's value, and equally, a market decline would be expected to produce an almost proportionate decline in this portfolio, reflecting that the portfolio, despite its individual stocks having quite different betas ranging from 0.6 to 1.4, has been combined in weights that happen to produce an overall systematic risk profile very similar to the market average, illustrating that portfolio beta, exactly like portfolio expected return, is computed as a straightforward weighted average of the individual components' betas, unlike portfolio variance, which requires the full covariance-based formula this chapter developed earlier.

Practice — Portfolio Management

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