Lending portfolio

Buying a government treasury bill is a way of lending money. You hand over cash and get a certain return back.

So when an investor splits her money between a risk-free asset and a portfolio of risky assets, she is lending part of it and investing the rest. The combination is a lending portfolio.

Every such mix sits on the Capital Market Line, somewhere between the risk-free rate and the market portfolio M. The more she lends, the lower the risk and the lower the return.

The opposite choice is to borrow at the risk-free rate and put more than her own wealth into M. That gives a borrowing, or leveraged, portfolio, which sits to the right of M.

So the line has two halves. Lending is the left half.

The risk-free asset (Box 16.1, section 16.4). A risk-free asset is one whose returns are certain, so its standard deviation is zero. Its correlation with all other risky assets is zero and so is its covariance with them. It provides the risk-free rate of return, and it lies on the vertical axis of a portfolio graph because it has no risk. Practically, the workbook notes, it is always possible to lend at the nominal risk-free rate by buying risk-free securities such as government treasury bills — but it is not possible to borrow at that rate.

The return of a lending portfolio. It is the plain weighted average of the two returns:

The workbook's worked example: the risk-free asset returns 5%, the risky portfolio 12%, and 70% of wealth goes into the risk-free asset.

E(Rport) = (70% x 5%) + (30% x 12%) = 7.10%

The risk of a lending portfolio. Because the variance of the risk-free asset is zero and its covariance with any risky asset is zero, the Markowitz two-asset formula collapses to a single term. With the risky asset's standard deviation at 10%:

(1 − 70%)² x (10%)² = 0.0009, so portfolio risk = 30% x 10% = 3%

Risk is therefore in linear proportion to the weight in the risky asset. Table 16.1 sets out 11 such combinations, and because both risk and return are linear the graph of these portfolios is a straight line joining the risk-free asset to the risky portfolio.

On the CML (section 16.5). All the possible combinations on the line drawn from the risk-free return on the y-axis to the market portfolio M are equivalent to lending at the risk-free rate — the investor's entire wealth is invested, split between the two. The workbook's CML figures, with M returning 20% and the risk-free asset 5%:

Illustrative figures, following the workbook's method. Mrs Iyengar has Rs 60,00,000 and will not borrow. The market portfolio M is expected to return 16% with a standard deviation of 18%. Treasury bills yield 6%.

She chooses to lend 40% and put 60% into M.

• Amount lent: Rs 24,00,000 in treasury bills
• Amount in M: Rs 36,00,000
• Expected return = (0.40 x 6%) + (0.60 x 16%) = 2.4% + 9.6% = 12.0%
• Portfolio risk = 0.60 x 18% = 10.8%

Expected gain in a year: Rs 7,20,000.

Her sister wants less risk and lends 75%:

• Expected return = (0.75 x 6%) + (0.25 x 16%) = 8.5%
• Portfolio risk = 0.25 x 18% = 4.5%

On the same Rs 60,00,000 that is Rs 5,10,000 expected — Rs 2,10,000 less, in exchange for cutting risk by well over half.

Neither of them can earn more than M's 16% while they only lend. That ceiling is the defining feature of the lending half of the line: the highest return available without borrowing is the market portfolio itself, at 100% in M and nothing lent.

Chapter 16 (Introduction to Capital Market Theory). Section 16.4 introduces the risk-free asset with Box 16.1 and works the 70% / 5% / 12% return and the 3% risk example. Section 16.5 extends the CML past M and labels everything to the left of M as lending.

Expect a computation of the return or risk of a risk-free-plus-risky mix, a question on the characteristics of a risk-free asset (zero standard deviation, zero correlation with risky assets, plots on the vertical axis), and a question on which side of M a lending portfolio plots.

• Lending means buying the risk-free asset, not making a loan to a borrower. Investing in treasury bills is the lending in this model.
• Risk is linear in the risky weight, and the risk-free asset contributes nothing to it. So portfolio risk is simply the weight in the risky asset times its standard deviation — no covariance term survives.
• 100% in M is the boundary case. It is the highest-returning lending portfolio, and the point at which lending becomes zero. Beyond it you are borrowing.
• The borrowed weight is subtracted, not added. That is a borrowing portfolio, and mixing up the sign is the classic error.
• Lending at the risk-free rate is realistic; borrowing at it is not. The workbook says so plainly — a real investor's borrowing rate is above the risk-free lending rate, though assuming otherwise does not change the model's general conclusions.
• Zero correlation with risky assets is a property of the risk-free asset, not an assumption about the market. Do not confuse it with perfect negative correlation, which is a hedge, not a risk-free asset.