Unit study guide · NISM Series XXI-B
Introduction to Modern Portfolio Theory
6 questions in this unit
What this unit covers
- What characterises a portfolio on the mean-variance efficient frontier?
- A security returns −5%, 10% or 20% with probabilities 20%, 50% and 30%. What is its expected return?
- A portfolio allocates 60% to an asset expecting 12% and 40% to an asset expecting 7%. What is expected portfolio return?
- Two assets have return standard deviations of 20% and 10% and correlation 0.30. What is their covariance using decimal returns?
- An equally weighted two-asset portfolio has standard deviations of 20% and 10% with zero correlation. What is its standard deviation, approximately?
- Two assets have 30% and 10% standard deviations and perfect negative correlation. What first-asset weight produces zero variance under these exact assumptions?
Key ideas
From the practice bank · 1
What characterises a portfolio on the mean-variance efficient frontier?
An efficient-frontier portfolio offers the highest available expected return at its risk level. An alternative with equal risk and a higher expected return would dominate it. Remember: Efficiency is a comparison of expected risk and return, not a profit guarantee.
Hint Compare portfolios at the same risk or the same expected return. Common mistake: Confusing an efficient portfolio with a risk-free or always-winning portfolio.
XXI-B-089From the practice bank · 2
A security returns −5%, 10% or 20% with probabilities 20%, 50% and 30%. What is its expected return?
Expected return is 10.00% because it is the probability-weighted average of outcomes. Working: 0.20 × (−5%) + 0.50 × 10% + 0.30 × 20% = −1% + 5% + 6% = 10%. Remember: Expected return uses probabilities and can differ from every simple unweighted average.
Hint Give each possible return its stated probability weight. Common mistake: Ignoring outcome probabilities or dropping the sign of a possible loss.
XXI-B-090From the practice bank · 3
A portfolio allocates 60% to an asset expecting 12% and 40% to an asset expecting 7%. What is expected portfolio return?
Expected portfolio return is 10.00% because each asset contributes its weighted expected return. Working: 0.60 × 12% + 0.40 × 7% = 7.20% + 2.80% = 10.00%. Remember: Expected portfolio return is a weighted average of expected asset returns.
Hint Combine expected returns using the amounts exposed to each asset. Common mistake: Using an unweighted average despite unequal portfolio allocations.
XXI-B-091From the practice bank · 4
Two assets have return standard deviations of 20% and 10% and correlation 0.30. What is their covariance using decimal returns?
Covariance is 0.006 when returns are expressed as decimals. Covariance = correlation × standard deviation of the first asset × standard deviation of the second. Working: 0.30 × 0.20 × 0.10 = 0.006. Remember: Correlation and covariance are related but not interchangeable.
Hint Correlation is unitless, while covariance also reflects both assets’ volatility. Common mistake: Using percentage numbers as if they were decimal returns.
XXI-B-092From the practice bank · 5
An equally weighted two-asset portfolio has standard deviations of 20% and 10% with zero correlation. What is its standard deviation, approximately?
Portfolio standard deviation is approximately 11.18% because zero correlation reduces combined volatility. Variance = 0.50² × 0.20² + 0.50² × 0.10² + 2 × 0.50 × 0.50 × 0.20 × 0.10 × 0 = 0.0125. Standard deviation = √0.0125 = 0.111803, or 11.18%. Remember: Portfolio risk depends on covariance, not just average individual volatility.
Hint Combine variances and co-movement before taking a square root. Common mistake: Calculating a weighted average of standard deviations as portfolio risk.
XXI-B-093From the practice bank · 6
Two assets have 30% and 10% standard deviations and perfect negative correlation. What first-asset weight produces zero variance under these exact assumptions?
A 25% weight in the first asset cancels the two volatility contributions under perfect negative correlation. Set 0.30w = 0.10(1 − w). Then 0.40w = 0.10 and w = 0.25. The remaining 75% is in the second asset. This exact cancellation depends on the stated model assumptions. Remember: With perfect negative correlation, unequal volatilities need unequal balancing weights.
Hint Make the two weighted volatility contributions cancel. Common mistake: Assuming equal money weights always eliminate risk when correlation is minus one.
XXI-B-094