Capital Market Line
Also written CML
The straight line from the risk-free rate tangent to the efficient frontier at the market portfolio — with a risk-free asset available, it is the efficient frontier, priced in standard deviation.
In plain language
Once investors can hold a risk-free asset, they no longer need to pick a point on the curved efficient frontier. They can hold one risky portfolio and adjust their risk by mixing it with lending or borrowing at the risk-free rate.
The best risky portfolio to use is the one where a line drawn from the risk-free rate just touches the frontier. The workbook calls that point M, the market portfolio. The line itself is the Capital Market Line.
Every point on the line is better than the curve beneath it. So, in the presence of a risk-free asset, the CML is the efficient frontier, and all other risky portfolios below it are no longer desirable or efficient.
Left of M, the investor is lending — part of the wealth is in the risk-free asset. At M, everything is in the market portfolio. Right of M, the investor is borrowing at the risk-free rate to put more than 100% of wealth into M.
How it works
The workbook rewrites the expected return of a risk-free-plus-market mix:
E(Rp) = ((1 − Wmp) × Rf) + (Wmp × Rm)
= Rf + Wmp × (Rm − Rf)
and the risk:
σp = Wmp × σmp so Wmp = σp ÷ σmp
Substituting gives the equation of the CML:
E(Rp) = Rf + (Rm − Rf) × σp ÷ σmp
Every investor expects at least the risk-free rate, plus a proportion of the market risk premium equal to the proportion of market risk taken.
Extending the line (section 3.12.2). With the market portfolio returning 20% and the risk-free asset 5%:
| Position | Calculation | Return |
|---|---|---|
| All in M | (1 × 20%) + (0 × 5%) | 20% |
| Half lending | (0.5 × 20%) + (0.5 × 5%) | 12.5% |
| Borrow 50% | (1.5 × 20%) − (0.5 × 5%) | 27.5% |
The debt is subtracted because it is a repayment, not a receipt. Lending combinations lie left of M; borrowing combinations lie right of M. The workbook calls this "the crux of capital market theory".
The CML measures risk with total risk (standard deviation) and applies to efficient portfolios. The Security Market Line measures it with beta and applies to individual securities too.
The formula
E(Rp) = Rf + [(Rm − Rf) ÷ σm] × σp
Slope of the CML = (Rm − Rf) ÷ σm (reward per unit of total risk)
A worked example
The workbook gives the 20% and 5% returns; the 16% market standard deviation and the −10% scenario are illustrative.
An investor has Rs 40 lakh. The market portfolio is expected to return 20% with standard deviation 16%; the risk-free rate is 5%.
Borrowing position — 150% in M. She borrows Rs 20 lakh at 5% and invests Rs 60 lakh in M.
Expected gain on M = Rs 60 lakh × 20% = Rs 12 lakh
Interest on loan = Rs 20 lakh × 5% = Rs 1 lakh
Net gain = Rs 11 lakh → 11 ÷ 40 = 27.5%
Risk = 1.5 × 16% = 24%
That matches the workbook's 27.5%. Now check it sits on the line:
Slope = (20 − 5) ÷ 16 = 0.9375
E(Rp) = 5% + 0.9375 × 24% = 5% + 22.5% = 27.5% ✓
What leverage does in a bad year. If M returns −10% instead:
Levered: (Rs 60 lakh × −10%) − Rs 1 lakh = −Rs 7 lakh → −17.5%
Unlevered: Rs 40 lakh × −10% = −Rs 4 lakh → −10%
The line promises more expected return to the right of M, and delivers exactly proportionate extra risk. The workbook's Exhibit 3.5 is labelled "the effect of leverage" for this reason.
Why NISM asks about it
Chapter 3, section 3.12, builds the CML from the risk-free asset and the tangency portfolio; section 3.12.2 extends it with borrowing (the 20% / 12.5% / 27.5% example); section 3.13 turns the CML equation into CAPM. Expect the definition, "left of M is lending, right of M is borrowing", and a question separating the CML (standard deviation, efficient portfolios) from the SML (beta, any security).
Common exam traps
- CML uses standard deviation; SML uses beta. This is the single most tested distinction in the chapter.
- Left of M = lending (holding the risk-free asset). Right of M = borrowing. Not the other way round.
- With a risk-free asset, the CML is the efficient frontier. The old curve is no longer efficient except at M.
- The debt is subtracted in the borrowing calculation: (1.5 × 20%) − (0.5 × 5%), not plus.
- Workbook wording slip: section 3.12.2 twice says combinations lie left or right of M "on the SML" and cites "Exhibit 3.6 SML" while describing the CML of Exhibit 3.5. The concept being taught there is the CML; answer CML questions with standard deviation on the x-axis.
- The line is straight because the risk-free asset has zero variance and zero covariance with M.
Check yourself
1.The correlation coefficient between the return on the market portfolio and the return on a risk-free asset is:
- a)+1
- b)-1
- c)Between 0 and +1
- d)Zero
Show the answer
Answer: (d) Zero
Zero. Since the return of a risk-free asset is CERTAIN, its STANDARD DEVIATION IS ZERO. Also, the COVARIANCE between the return of the risk-free asset and the risky asset IS ZERO TOO.
Box 3.2 lists four characteristics of a risk-free asset:
- An asset where its returns yield ZERO STANDARD DEVIATION
- ZERO CORRELATION with all other risky assets
- Provides the RISK-FREE RATE OF RETURN
- Will lie on the VERTICAL AXIS of a portfolio graph, as it has no risk
The intuition is simple: correlation measures how two things move together. A risk-free asset does not move at all - its return is certain - so it cannot move with anything. There is nothing to correlate.
And this fact has a large consequence. When a risk-free asset is combined with a risky portfolio, every term in the Markowitz risk formula involving the risk-free asset vanishes, leaving portfolio risk as simply the weight in the risky asset multiplied by the risk of the risky asset. Because that relationship is linear, such portfolios plot on a straight line - which is what makes the Capital Market Line straight, where Markowitz's frontier was curved.
Where this is taught
Free preparation for NISM Series XXI-BRelated terms
- CAPMA model that prices the return an investor should demand from a share: the risk-free rate plus beta times the market risk premium.
- LeverageControl of a large contract value for a small upfront outlay — premium for an option buyer, margin for a futures position — which multiplies percentage gains and percentage losses by the same factor.
- 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.
- Market PortfolioThe tangency portfolio M of all risky assets worldwide in market proportions — the one risky portfolio every rational investor holds once a risk-free asset exists; indices serve as its proxy.
- Risk-free AssetAn asset whose return is certain: zero standard deviation, zero correlation with every risky asset, earning the risk-free rate and plotting on the vertical axis of a risk-return graph.
- 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.
- 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.
- Capital Market TheoryThe extension of Modern Portfolio Theory that adds a risk-free asset and builds a model for pricing all risky assets, giving the Capital Market Line, the market portfolio and CAPM.
- Homogeneous expectationsThe Capital Market Theory assumption that all investors estimate identical probability distributions for future returns — one the workbook says can be relaxed with minor effects if differences are small.
- Infinite divisibilityThe Capital Market Theory assumption that any asset or portfolio can be bought or sold in fractions — which is what lets the theory draw its frontiers and lines as continuous curves.