Risk-free Asset

A risky asset is one whose future returns are uncertain. A risk-free asset is one whose returns are certain.

Because the return is certain, it does not vary, and everything else follows from that. Its standard deviation is zero. It cannot move together with anything, so its covariance and correlation with every risky asset are zero. On a graph with risk along the bottom, it sits right on the vertical axis.

The workbook's practical example is government T-bills: investors can always lend at the nominal risk-free rate by buying them.

The risk-free asset is the ingredient that turns Modern Portfolio Theory into Capital Market Theory. Once investors can mix a risky portfolio with something certain, the shape of every good choice changes from a curve to a straight line.

Box 3.2 — characteristics of a risk-free asset:

• returns yield zero standard deviation;
• zero correlation with all other risky assets;
• provides the risk-free rate of return (rf);
• lies on the vertical axis of a portfolio graph, as it has no risk.

Mixing it with a risky asset. Expected return is the ordinary weighted average. Risk is where the zero does its work: because the risk-free asset's variance is zero and its covariance with the risky asset is zero, the Markowitz formula collapses to one term.

The workbook's numbers: risk-free rate 5%, risky asset expected return 12% and standard deviation 10%, with 70% in the risk-free asset:

E(Rport) = (70% × 5%) + (30% × 12%) = 7.10%
σ²port  = (1 − 70%)² × (10%)² = 0.0009
σport   = 30% × 10% = 3%

Table 3.5 runs all eleven combinations from 0% to 100% risky. Return climbs by 0.70 points and risk by 1 point for every 10% moved into the risky asset — risk rises in linear proportion. That straight line is what becomes the Capital Market Line.

One of the capital market theory assumptions is that investors can borrow or lend any amount at the risk-free rate. The workbook concedes that lending at it is always possible but borrowing is not; assuming a higher borrowing rate does not change the general results.

E(Rport) = W(RF) × RFR + (1 − W(RF)) × E(Ri)

σ(port)  = (1 − W(RF)) × σi

where W(RF) is the proportion of wealth in the risk-free asset.

An investor has Rs 1 crore and uses the workbook's figures: T-bills at 5%, a risky equity portfolio expected to return 12% with a standard deviation of 10%.

Check the 70:30 row:

(0.70 × 5%) + (0.30 × 12%) = 3.5% + 3.6% = 7.1%  → Rs 7.1 lakh
Risk = 0.30 × 10% = 3%

Each extra Rs 10 lakh moved into the risky portfolio adds Rs 70,000 of expected gain and one percentage point of risk. There is no diversification bonus and no penalty — the relationship is a straight line, because the T-bill contributes nothing to risk and does not co-move with the equities.

That is why the investor can dial risk precisely: a 3% risk budget means 30% risky and 70% T-bills, whatever the risky portfolio is.

Chapter 3, section 3.12 and Box 3.2, set out the four characteristics and the 70:30 worked example; Table 3.5 gives all eleven combinations; section 3.11 carries the borrow-and-lend assumption. Sample question 5 asks for the correlation between the market portfolio and a risk-free asset — zero. Chapter 1 builds the risk-free rate itself from the real rate and inflation.

• Correlation with any risky asset is zero, not −1. The risk-free asset does not hedge; it simply does not move.
• Portfolio risk is linear in the weight of the risky asset when combined with a risk-free asset: 30% risky means 30% of its risk.
• Lending at the risk-free rate is realistic; borrowing at it is not. The workbook says so, and says the general results survive a higher borrowing rate.
• Zero standard deviation does not mean zero return. It earns rf.
• It sits on the vertical (return) axis, not the horizontal one.