Perfect Negative Correlation
Also written Correlation of -1
A correlation coefficient of −1: two assets' returns move in opposite directions, delivering the complete benefit of diversification and making a zero-risk combination possible.
In plain language
At a correlation of −1, the workbook says, the returns of two assets move in opposite directions. When one rises, the other falls, in a perfectly linear way.
This is the best case for diversification. The workbook: when the correlation between two assets is −1, the complete benefits of diversification are realised. The negative correlation term completely offsets the combined variance terms, so a unique combination of such assets could yield a zero-risk portfolio.
In real life, the workbook adds, perfectly negatively correlated assets are hard to find, except for insurance products or put options on the portfolio (Chapter 17).
How it works
Portfolio variance for two assets (Chapter 15, section 15.7.1):
σp² = wA²σA² + wB²σB² + 2·wA·wB·ρAB·σA·σB
Put ρ = −1 and the expression becomes:
σp² = (wA·σA − wB·σB)² → σp = |wA·σA − wB·σB|
The zero-risk mix. Risk is zero when wA·σA = wB·σB. With wA + wB = 1:
wA = σB ÷ (σA + σB) wB = σA ÷ (σA + σB)
This weight rule is simple algebra on the workbook's formula; the workbook itself states only that a unique combination can give zero risk.
Graphically (section 15.8, Exhibit 15.2). The workbook plots two-security portfolios for ρ = +1.00, +0.50, 0.00, −0.50 and −1.00. The lower the correlation, the further the curve bulges left. At −1 the plot reaches the vertical (zero-risk) axis.
Chapter 17 (section 17.3.2). A fund manager diversifies with less correlated assets, sectors and countries. Mathematically the entire portfolio risk could be reduced to zero; the intent is to find perfectly negatively correlated assets, but in real life they are hard to get except for insurance products or put options.
A worked example
Illustrative figures. Two assets with perfect negative correlation.
| Asset X | Asset Y | |
|---|---|---|
| Expected return | 12% | 8% |
| Standard deviation | 18% | 12% |
Zero-risk weights: wX = 12 ÷ (18 + 12) = 12 ÷ 30 = 0.40 wY = 18 ÷ 30 = 0.60
Check the risk: σp = |0.40 × 18 − 0.60 × 12| = |7.2 − 7.2| = 0%
Portfolio return: 0.40 × 12 + 0.60 × 8 = 4.8 + 4.8 = 9.6%
On ₹50 lakh: ₹20 lakh in X, ₹30 lakh in Y, earning an expected ₹4.8 lakh a year with (in theory) no volatility.
A 50:50 mix instead: σp = |0.5 × 18 − 0.5 × 12| = |9 − 6| = 3%, with return 0.5 × 12 + 0.5 × 8 = 10%. Compare that 3% with the 15% weighted average (0.5 × 18 + 0.5 × 12) that the same mix would carry at ρ = +1.
Why NISM asks about it
Chapter 15 (Introduction to Modern Portfolio Theory), sections 15.7–15.8, defines perfect negative correlation and states that it gives the complete benefit of diversification and a possible zero-risk portfolio. Chapter 9 (section 9.2) introduces the −1 to +1 range, and Chapter 17 (section 17.3.2) notes that such assets are rare in practice except for insurance and put options. Expect conceptual questions and simple two-asset risk calculations.
Common exam traps
- −1 gives the complete benefit of diversification; +1 gives none.
- Zero risk needs the right weights, not just any mix. A 50:50 split of assets with different risks still carries risk.
- Opposite direction, not zero relationship. A correlation of 0 means no linear relationship; −1 is a perfect inverse relationship.
- Return is unaffected by correlation. Only risk changes.
- Rare in practice. The workbook's real-world examples are insurance products and put options on the portfolio.
Where this is taught
Free preparation for NISM Series XXI-BRelated terms
- DiversificationSpreading an exposure across holdings that do not move together, so that total risk falls by more than total return does — minimising risk per unit of return.
- Put optionA contract giving its buyer the right, but never the obligation, to sell the underlying at a fixed strike price — insurance against a fall, bought for a premium.
- CorrelationA measure of the strength and direction of the relationship between two variables, running from -1 to +1, and the single factor that determines how much risk diversification actually removes.
- CovarianceA measure of the degree to which the returns of two assets move together relative to their own averages — the building block of portfolio risk, standardised into the correlation coefficient.
- Efficient frontierThe umbrella-shaped curve of portfolios offering the highest expected return for each level of risk, or the lowest risk for each level of return — any portfolio below it is dominated.
- Perfect Positive CorrelationA correlation coefficient of +1: two assets' returns move together in a completely linear way, so combining them gives no diversification benefit at all.
- Lending portfolioAny mix of the risk-free asset and the market portfolio that uses only the investor's own money — it plots on the Capital Market Line between the risk-free rate and M, and lending means buying the risk-free asset.