Graphical Presentation of Portfolio Risk Return of Two Securities
This sub‑topic explains how to graphically represent the risk‑return characteristics of a portfolio that contains two securities. You will learn to plot expected returns against portfolio risk, understand the shape of the combination line, and identify the efficient frontier. Mastery is essential for NISM Series X‑A questions that test your ability to interpret and construct risk‑return graphs.
Learning Objectives
- 1Define the axes and key points on a risk‑return graph for two assets.
- 2Calculate portfolio expected return and standard deviation using weights, variances and correlation.
- 3Interpret how changes in correlation affect the shape of the combination line.
- 4Identify the minimum‑variance portfolio and the efficient frontier on the graph.
Understanding the Risk‑Return Plane
The risk‑return plane is a two‑dimensional chart where the horizontal axis (X‑axis) measures portfolio risk, usually expressed as the standard deviation (σ) of returns, and the vertical axis (Y‑axis) measures the expected portfolio return (E(R)).
Each individual security is plotted as a single point using its own σ and E(R). When you combine two securities, every possible weight allocation creates a new point, and the collection of these points forms a curve (or a straight line when correlation equals one). This visual tool helps advisers quickly assess whether a portfolio offers a favourable trade‑off between risk and return.
For the NISM exam, you must be able to recognise the axes, locate the individual securities, and describe what the curve represents. Questions often ask you to identify the minimum‑variance point or to state which part of the curve is the efficient frontier.
- Risk is measured by standard deviation, not variance.
- Return is the expected (mean) return, not a single historical observation.
Students sometimes plot variance on the X‑axis. The NISM syllabus requires standard deviation (the square‑root of variance). Using variance will give a wrong shape and lead to loss of marks.
Expected Return of a Two‑Asset Portfolio
Where:
w_{1}= Weight of security 1 in the portfolio (decimal, sum of weights = 1)w_{2}= Weight of security 2 in the portfolio (decimal)E(R_{1})= Expected return of security 1 (percentage per annum)E(R_{2})= Expected return of security 2 (percentage per annum)Worked Example
Given w_{1}=0.60, w_{2}=0.40, E(R_{1})=12\%, E(R_{2})=8\%: Step 1: E(R_{p}) = (0.60 \times 12) + (0.40 \times 8) Step 2: E(R_{p}) = 7.2 + 3.2 = 10.4\% Verification: (0.60*12)+(0.40*8) = 10.4.
The expected return formula is linear because return is a first‑order moment. As you increase the weight of the higher‑return security, the portfolio return moves proportionally toward that security’s return.
Remember that the weights must always add up to one. If the weights do not sum to one, the result will be overstated or understated, which is a common mistake in exam calculations.
In NISM questions, you may be given the returns of two mutual funds and asked to compute the portfolio return for a specific allocation. Plug the numbers directly into the formula; no need for logarithms or compounding adjustments because the formula already reflects expected (average) returns.
- Weight of 0% means the portfolio consists entirely of the other security.
- Weight of 100% places the portfolio at the point of the chosen security on the graph.
Portfolio Risk (Standard Deviation) for Two Assets
Where:
w_{1}= Weight of security 1 (decimal)w_{2}= Weight of security 2 (decimal)\sigma_{1}= Standard deviation of security 1 (decimal, e.g., 0.15 for 15%)\sigma_{2}= Standard deviation of security 2 (decimal)\rho_{12}= Correlation coefficient between the returns of security 1 and security 2Worked Example
Given w_{1}=0.50, w_{2}=0.50, \sigma_{1}=0.15, \sigma_{2}=0.08, \rho_{12}=0.30: Step 1: Variance = (0.5^{2}\times0.15^{2}) + (0.5^{2}\times0.08^{2}) + 2\times0.5\times0.5\times0.15\times0.08\times0.30 Step 2: Variance = 0.005625 + 0.001600 + 0.001800 = 0.009025 Step 3: \sigma_{p}=\sqrt{0.009025}=0.0950 (or 9.5\%) Verification: sqrt(0.009025) = 0.0950.
The portfolio risk formula incorporates three elements: the individual risks of each security, the proportion of each security in the portfolio, and the correlation between the securities. The correlation term (\rho_{12}) can either increase or decrease overall risk, which is why diversification works.
If the two securities are perfectly positively correlated (\rho_{12}=1), the cross‑term adds maximally and the portfolio risk becomes a straight line between the two points on the graph. When the correlation is less than one, especially when it is low or negative, the cross‑term reduces total variance, causing the curve to bow inward.
For the exam, you may be asked to compute portfolio risk for a given weight and correlation, or to infer the effect of changing correlation on the shape of the risk‑return curve. Always convert percentages to decimals before substituting, and double‑check that the correlation lies between -1 and +1.
- Zero correlation does not mean zero risk; it only removes the covariance contribution.
- Negative correlation can produce a portfolio risk lower than either individual security.
Many candidates set \rho_{12}=0 by default. While this simplifies the calculation, the syllabus expects you to use the given correlation. Forgetting it leads to an over‑estimated risk and loss of marks.
Plotting the Combination Line
To draw the combination line, select a series of weight allocations (e.g., 0%, 25%, 50%, 75%, 100% in security A) and compute the corresponding expected return and standard deviation using the formulas above. Plot each (σ, E(R)) pair on the risk‑return plane and join the points.
If the correlation between the two securities is exactly +1, the plotted points lie on a straight line connecting the two individual security points. For any correlation less than +1, the line bows toward the origin, creating a curved shape known as the "efficient frontier" for two assets.
Exam questions may present a partially drawn graph and ask you to identify the missing point, or they may give you a set of weights and ask you to calculate the coordinates of the point you would plot.
- Higher correlation → flatter, more linear curve.
- Lower or negative correlation → more pronounced curvature, greater risk reduction.
Risk‑Return Curve for Varying Weights in Asset A
Efficient Frontier for Two Securities
The efficient frontier is the segment of the combination curve that offers the highest expected return for a given level of risk, or equivalently, the lowest risk for a given expected return. For two assets, the frontier starts at the minimum‑variance point and extends to the asset with the higher return‑to‑risk ratio.
The minimum‑variance portfolio (MVP) can be derived analytically using the formula: \(w_{1}^{*}=\frac{\sigma_{2}^{2}-\sigma_{1}\sigma_{2}\rho_{12}}{\sigma_{1}^{2}+\sigma_{2}^{2}-2\sigma_{1}\sigma_{2}\rho_{12}}\). The weight of the second asset is simply \(w_{2}^{*}=1-w_{1}^{*}\). This point is crucial because any portfolio to the left of the MVP is unattainable, and any portfolio to the right is sub‑optimal.
In the NISM exam, you may be asked to identify the MVP weight, to state which part of the curve is the efficient frontier, or to choose the portfolio that lies on the frontier among several options.
- The MVP does not necessarily have the lowest risk; it has the lowest risk among all possible combinations.
- When correlation is negative, the MVP can have a risk lower than both individual securities.
Sample portfolio points for different weightings in Asset A (Equity) and Asset B (Debt)
| Weight in Equity (%) | Expected Return (%) | Risk (Std Dev %) |
|---|---|---|
| 0 | 7.00 | 5.00 |
| 25 | 8.25 | 6.41 |
| 50 | 9.50 | 9.81 |
| 75 | 10.75 | 13.80 |
| 100 | 12.00 | 18.00 |
Scenario
Ramesh, an Indian retail investor, wants to allocate his ₹1,00,000 between an equity mutual fund (expected return 12% p.a., σ = 18%) and a debt fund (expected return 7% p.a., σ = 5%). The correlation between the two funds is 0.2. He wants the portfolio with the lowest possible risk.
Solution
Step 1: Compute the MVP weight for the equity fund using \(w_{1}^{*}=\frac{\sigma_{2}^{2}-\sigma_{1}\sigma_{2}\rho}{\sigma_{1}^{2}+\sigma_{2}^{2}-2\sigma_{1}\sigma_{2}\rho}\). Here, \(\sigma_{1}=0.18\), \(\sigma_{2}=0.05\), \(\rho=0.20\).\n\nCalculate \(\sigma_{1}^{2}=0.0324\), \(\sigma_{2}^{2}=0.0025\), \(\sigma_{1}\sigma_{2}\rho=0.0018\).\n\nNumerator = 0.0025 - 0.0018 = 0.0007.\nDenominator = 0.0324 + 0.0025 - 2\times0.0018 = 0.0349 - 0.0036 = 0.0313.\n\nWeight in equity (w1*) = 0.0007 / 0.0313 ≈ 0.0224 (2.24%).\nWeight in debt (w2*) = 1 - 0.0224 = 0.9776 (97.76%).\n\nStep 2: Portfolio expected return = 0.0224\times12 + 0.9776\times7 = 0.2688 + 6.8432 = 7.112% (≈7.11%).\n\nStep 3: Portfolio variance = w1*^2\sigma_{1}^{2}+w2*^2\sigma_{2}^{2}+2w1*w2\sigma_{1}\sigma_{2}\rho\n= (0.0224^2\times0.0324) + (0.9776^2\times0.0025) + 2\times0.0224\times0.9776\times0.18\times0.05\times0.2\n≈ 0.0000163 + 0.002389 + 0.0000788 = 0.002484.\nPortfolio standard deviation = sqrt(0.002484) ≈ 0.0498 or 4.98%.\n\nThus, the minimum‑variance portfolio consists of about 2.2% equity and 97.8% debt, delivering an expected return of ~7.11% with a risk of ~4.98%.
Conclusion
The example shows that even a small allocation to a higher‑return equity fund can raise expected return while keeping risk close to the debt fund’s level. NISM questions often test this calculation and the interpretation of the MVP on the risk‑return graph.
⭐Exam Takeaways
- Risk‑return graph uses standard deviation on the X‑axis and expected return on the Y‑axis; plot each security as a point.
- Portfolio expected return is a weighted linear combination: E(Rp)=w1E(R1)+w2E(R2).
- Portfolio risk incorporates correlation: σp = sqrt(w1^2σ1^2 + w2^2σ2^2 + 2w1w2σ1σ2ρ).
- When correlation = +1 the combination line is straight; lower correlation creates a curved efficient frontier.
- The minimum‑variance portfolio weight is w1* = (σ2^2 – σ1σ2ρ) / (σ1^2 + σ2^2 – 2σ1σ2ρ).
- The efficient frontier starts at the MVP and extends to the asset with the higher return‑to‑risk ratio.
- Always convert percentages to decimals before substituting into formulas and verify that weights sum to 1.
Practice Questions
8 questions on Graphical Presentation of Portfolio Risk Return of Two Securities
What is plotted on the horizontal axis of the risk‑return plane?
Which formula correctly computes the expected return of a two‑asset portfolio?
If the correlation between two securities equals +1, how does the combination line appear on the risk‑return graph?
Using w1=0.60, w2=0.40, E(R1)=12% and E(R2)=8%, what is the portfolio expected return?
For σ1=0.15, σ2=0.08, ρ12=0.30 and equal weights, what is the portfolio standard deviation?
Applying the MVP weight formula with σ1=0.18, σ2=0.05 and ρ=0.20, what is the weight of the equity fund?
Which portion of the two‑asset combination curve is defined as the efficient frontier?
What common exam trap does the material warn about regarding the risk axis?