The risk-free asset and the capital market line

Add one more asset, a Treasury bill paying 3.0% for certain, and the question “which efficient portfolio should we hold?” gets a surprising answer: the same risky portfolio for everyone. A cautious pension fund and an adventurous hedge fund should hold global equities, emerging markets and bonds in the same proportions; they differ only in how much of their money sits in T-bills, or is borrowed.

Mixing with a risk-free asset

A risk-free assetAn asset whose return over the holding period is known in advance: no variance, and no covariance with anything. In practice, a short-dated government bill in the investor's own currency. pays a known return $r_f$: no variance and no covariance with anything. Put a share $y$ of your money in a risky portfolio P and $1 - y$ in the risk-free asset. The two-asset formula of L3.1.01: Portfolio return and risk loses two of its three terms, and both expected return and volatility scale with $y$. Plotted, the mixes form a straight line from $(0, r_f)$ through P: the capital allocation lineThe straight line of all mixes of the risk-free asset with one risky portfolio P, in the risk-return picture. Its slope is P's Sharpe ratio. (CAL).

The slope is P's Sharpe ratioExcess return over the risk-free rate divided by volatility: (μ − r_f)/σ. The slope of a capital allocation line., met in L2.1.03: Risk and return: the extra expected return for each extra point of volatility. Below P on the line ($y < 1$) you are lendingA mix with part of the money in the risk-free asset (y < 1): less risk and less expected return than the tangency portfolio.; above it ($y > 1$), you are borrowingA mix that borrows at the risk-free rate (or more) to hold more than 100% in the risky portfolio (y > 1): leverage. to hold more than 100% of P.

The steepest line

Every risky portfolio has its own line. The group's current mix (5.45% at 9.50%) has a Sharpe ratio of 0.258; government bonds alone, 0.083; global equities, 0.250. A steeper line gives more return for every level of risk, so rotate the line upwards until it just touches the efficient frontier. The point it touches is the tangency portfolioThe risky portfolio with the highest Sharpe ratio, where the steepest capital allocation line touches the efficient frontier. T: the risky portfolio with the highest Sharpe ratio, here 0.288.

The calculus is in the formal version below; the answer is a formula you have nearly met before. In L3.1.07: Mean-variance optimisation the frontier was built from $\Sigma^{-1}\mu$ and $\Sigma^{-1}\mathbf 1$. The tangency portfolio uses the excess returns instead:

For global equities, emerging-market equities, government bonds, with the inverse from L3.1.07: Mean-variance optimisation:

Check the shortcut: $\sqrt{(\mu - r_f\mathbf 1)'z} = 0.2873$, the same Sharpe ratio. For all five asset classes, T holds 29.4% global equities, 18.3% emerging-market equities, 41.8% government bonds, 13.1% corporate bonds, −2.6% property: 5.51% expected return at 8.72% volatility. Check the three-asset weights in the calculator

Two-fund separation

Once T is found, the investor's appetite for risk decides only one number: how much to put in T. This is two-fund separationWith a risk-free asset, the choice of risky portfolio (the tangency portfolio, the same for everyone) is separate from the choice of how much risk to take (how much to lend or borrow). Tobin, 1958.: every mean-variance investor holds the same two funds, the risk-free asset and T. With a volatility target $\sigma^*$, hold $y = \sigma^*/\sigma_T$ in T:

The cautious investor lends 54% of its money at 3.0%: no portfolio of risky assets alone has volatility as low as 4% (the minimum is 5.27%). The adventurous one borrows 38% and earns 6.46%, more than the 6.39% of the best risky-only portfolio with the same risk. Except at T itself, the line lies above the curve. The two-fund theorem of L3.1.07: Mean-variance optimisation needed two risky funds; with a risk-free asset, one of the two funds is risk-free.

When borrowing costs more

Few investors can borrow at the Treasury bill rate. If lending earns 3.0% but borrowing costs 3.5%, there are two tangency portfolios: T for lending and a second one, T_b, for the borrowing rate, with 7.42% expected return at 16.16% volatility. The best set of portfolios is now in three pieces: the lending line up to T, the risky frontier from T to T_b, and a flatter borrowing line beyond. At 12% volatility the best portfolio is on the frontier itself (6.39%). At 18%, borrow 11% at 3.5% to earn 7.87%, against 8.19% if borrowing cost only 3.0%. And if the borrowing rate reaches the minimum-variance portfolio's return (3.92%), borrowing is never worth it.

The market portfolio and the capital market line

Now suppose every investor uses mean and variance, agrees on the inputs, and can lend and borrow at $r_f$. Then every investor holds the same T. Add up all their holdings and you get T, scaled up; but together investors must hold every risky asset there is. So in equilibrium T must be the market portfolioAll risky assets, each held in proportion to its total market value. M, every asset in proportion to its market value. Its capital allocation line has a name of its own, the capital market lineThe capital allocation line through the market portfolio: E[R] = r_f + (E[R_M] − r_f)/σ_M × σ. It holds for efficient portfolios. (CML):

Its slope, the market's Sharpe ratio, is the market price of risk. The CML prices efficient portfolios only: single assets and inefficient mixes, such as the group's current one, lie below it. What return a single asset should offer is the question of L3.1.09: CAPM, beta and the security market line.

Risky assets with expected returns μ and covariance matrix Σ; a risk-free asset with return $r_f < A/C$ (the minimum-variance portfolio's return, 3.92% here).

The same answer another way. With a risk-free asset, the budget constraint disappears from the risky weights: whatever is not invested is lent. Minimise $\tfrac12 w'\Sigma w$ subject only to a target excess return $w'(\mu - r_f\mathbf 1) = e$; the Lagrangian of L3.1.07: Mean-variance optimisation gives $w = \lambda\,\Sigma^{-1}(\mu - r_f\mathbf 1)$. Every optimal risky holding points in the same direction, and only its size $\lambda$ depends on the target. Rescaled to add up to one, it is $w_T$.

How T moves with the rate. T depends on $r_f$. As the rate rises towards $A/C$, the tangency point slides up the frontier and leverage in the risky weights grows; at $A/C$ itself, $A - r_f C = 0$ and there is no tangency portfolio on the efficient half.

Start with lending at 3.0% and a 10% volatility target: you borrow a little to hold more than 100% in T. Raise the cost of borrowing and watch the line kink and the second tangency point move up. Then lower the volatility target until you are lending, and switch to long-only to see how little the ban on short sales changes T here.

All figures use the module's illustrative inputs. In practice the group would weigh liabilities, liquidity and capital, not only volatility.

Regulation has to decide what counts as risk-free, and it treats a government's own debt in its own currency generously. Under the bank capital rules (in the EU, Article 114 of the Capital Requirements Regulation), such exposures can carry a 0% credit risk weight. In the EU, Solvency II's standard formula exempts a member state's central government debt in its own currency from the spread and concentration risk charges an insurer such as RiskAtoms Insurance must hold, but not from interest-rate risk: free of default risk is not free of price risk. In September 2022 long-dated gilt yields rose so fast that the Bank of England announced temporary purchases of long-dated gilts on 28 September to restore orderly markets (Bank of England, 2022). Insurers also value their liabilities with a prescribed risk-free rate curve, published by EIOPA in the EU and by the PRA in the UK. More on this in M4.6: Solvency II and insurance regulation.

• Mixing a risky portfolio with a risk-free asset gives a straight line, the capital allocation line, whose slope is the portfolio's Sharpe ratio.
• The best line touches the frontier at the tangency portfolio, w_T ∝ Σ⁻¹(μ − r_f1), the risky portfolio with the highest Sharpe ratio.
• Two-fund separation: everyone holds the same risky portfolio; appetite for risk only sets how much is lent or borrowed.
• If borrowing costs more than lending, the best set kinks: a lending line, part of the frontier, then a flatter borrowing line.
• In equilibrium the tangency portfolio is the market portfolio, and its line is the capital market line: the price of risk for efficient portfolios.

Risk-free asset

An asset whose return over the holding period is known in advance: no variance, and no covariance with anything. In practice, a short-dated government bill in the investor's own currency.

Capital allocation line

The straight line of all mixes of the risk-free asset with one risky portfolio P, in the risk-return picture. Its slope is P's Sharpe ratio.

Sharpe ratio

Excess return over the risk-free rate divided by volatility: (μ − r_f)/σ. The slope of a capital allocation line.

Tangency portfolio

The risky portfolio with the highest Sharpe ratio, where the steepest capital allocation line touches the efficient frontier.

Two-fund separation

With a risk-free asset, the choice of risky portfolio (the tangency portfolio, the same for everyone) is separate from the choice of how much risk to take (how much to lend or borrow). Tobin, 1958.

Lending portfolio

A mix with part of the money in the risk-free asset (y < 1): less risk and less expected return than the tangency portfolio.

Borrowing portfolio

A mix that borrows at the risk-free rate (or more) to hold more than 100% in the risky portfolio (y > 1): leverage.

Market portfolio

All risky assets, each held in proportion to its total market value.

Capital market line

The capital allocation line through the market portfolio: E[R] = r_f + (E[R_M] − r_f)/σ_M × σ. It holds for efficient portfolios.