Foundation — The risk-free asset and the Capital Market Line

5.1 What capital market theory adds

Capital market theory builds on Modern Portfolio Theory and develops a model for pricing all risky assets. Capital market theory expands the concepts introduced by Markowitz's Modern Portfolio Theory by introducing a risk-free asset while forming efficient portfolios of risky assets. The availability of risk free asset has significant implications on portfolio choices.

One addition — a risk-free asset — changes everything. Markowitz gave us an efficient frontier of risky portfolios. Add a risk-free asset and, as you will see, the whole curve is replaced by a straight line.

A fundamental question in finance is how the risk of an investment should affect its expected return. The Capital Asset Pricing Model (CAPM) provided the first coherent framework for answering this question. CAPM is a dominant model for the valuation of risky asset and estimation of required rate of return.

It was developed in the early 1960s concurrently by four individuals Jack Treynor (1962), William Sharpe (1964), John Lintner (1965) and Jan Mossin (1966).

Four names and four dates: Treynor 1962, Sharpe 1964, Lintner 1965, Mossin 1966. Note the word concurrently — no single author, unlike Markowitz's 1952 paper.

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5.2 The assumptions, and which may be relaxed

As Capital Market Theory builds on Modern Portfolio Theory (MPT), all the assumptions of MPT hold for capital market theory with some additional assumptions.

The workbook is unusually helpful here: it states each assumption and what happens if it is relaxed. Learn them as pairs.

1. All investors want to target points on the efficient frontier. The exact location on the efficient frontier and, therefore, the specific portfolio selected, will depend on the individual investor's risk-return utility function.

2. Investors can borrow or lend any amount of money at the risk-free rate of return (rf). — And the honest qualification: Practically, it is always possible to lend money at the nominal risk-free rate by buying risk-free securities such as government T-bills. It is not possible to borrow at this risk-free rate. Investors' rate of borrowing in a practical world is higher than the risk free lending rate. However, assuming a higher borrowing rate does not change the general results.

3. All investors have homogeneous expectations i.e. they estimate identical probability distributions for future rates of return. — This assumption can be relaxed. As long as the differences in return expectations are not large, the effects are minor.

4. All investors have the same one-period time horizon such as one-month, six months, or one year. — The model will be developed for a single hypothetical period, and its results could be affected by a different time horizon assumption. A difference in the time horizon would require investors to derive risk measures and risk-free assets that are consistent with their time horizons.

5. All investments are infinitely divisible, which means that it is possible to buy or sell fractional shares of any asset or portfolio. — This assumption allows us to discuss investment alternatives as continuous curves. Changing it would have little impact on the theoretical models.

6. There are no taxes or transaction costs involved in buying or selling assets. — This is a reasonable assumption in many instances. Again, relaxing this assumption modifies the results, but does not change the basic theory.

7. There is no inflation (or inflation is fully anticipated).

8. Capital markets are in equilibrium. This means that we begin with all investments properly priced in line with their risk levels. — This is a very important assumption and can't be relaxed as such.

Assumption 8 is the one that cannot be relaxed. That is a very likely examination point. Every other assumption can be softened with only modest damage; equilibrium pricing cannot, because the entire model is about what prices should be when everything is correctly priced.

And the workbook's balanced verdict:

As already mentioned, some of these assumptions do not hold good in practice. However, Capital Market Theory explains and helps predict market behaviours quite reasonably well.

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5.3 The risk-free asset

A risky asset is defined as one whose future returns are uncertain. A risk free asset is defined as one whose returns are certain. 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 5.1 — the 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 (rf)
• Will lie on the vertical axis of a portfolio graph as it has no risk

That second point answers sample question 2 directly: the correlation between the market portfolio and a risk-free asset is zero. And it follows from the first — a variable with no variation cannot co-vary with anything.

Combining the risk-free asset with a risky portfolio

When a portfolio of risky assets is combined with a risk free asset, the expected return of such a portfolio, like the expected return for a portfolio of two risky assets, is the weighted average of the two returns:

E(Rport) = WRF × (RFR) + (1 − WRF) × E(Ri)

Where WRF is the proportion of wealth invested in the risk free asset, RFR is the Risk Free Rate, and E(Ri) is the expected return on the risky portfolio.

The worked example: with a risk-free return of 5%, a risky asset returning 12%, and 70% in the risk-free asset:

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

And the risk — where the magic happens

Further, if the risk (standard deviation) of the risky asset is 10%. Applying the Markowitz portfolio risk formula from Chapter 4:

As the variance of a risk free asset is zero and its covariance with any risky asset is also zero, the formula will be reduced to:

= (1 − 70%)² × (10%)² = 0.0009

And the portfolio risk will be: = 30% × 10% = 3%

Look at what happened. Chapter 4's formula had three parts — the weighted variances and the covariance term. Here two of the three vanish: the risk-free asset's own variance is zero, and its covariance with the risky asset is zero. What survives is a single term, and taking its square root gives simply the weight in the risky asset multiplied by its standard deviation.

That is why the relationship is a straight line. The workbook says so: the risk of a portfolio having risk-free asset with risky assets is in linear proportion to the risk of the risky asset. Since risk and the return of such portfolios are linear combinations, a graph of such portfolios will be a straight line combining risky asset and risk free asset.

Table 5.1 — the eleven combinations

Notice the perfect linearity in the last column: the portfolio risk is always exactly the weight in the risky asset × 10%. That never happens when two risky assets are combined — there the curve bulges to the left.

The risk-return relationship holds for every combination of the two components: one risk-free asset and the other risky assets. Investors would obviously like to maximize their expected compensation for bearing risk (i.e. they would like to maximize the risk premium they receive).

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Finding the Capital Market Line

The concept of efficient frontier is described as an umbrella shaped curve comprising of investment opportunities providing highest levels of returns for given levels of risk. Depending upon the risk appetite and investment objectives, the investor can choose to be on any point on the efficient frontier.

Now the crucial argument:

by assuming that the efficient frontier... is constructed considering all the possible traded risky securities in the investment, and also by introducing a risk free asset in the universe of securities, the investor can now allocate his/her wealth between the risk free asset and the risky assets.

It can also be observed from the above figure, that the portfolios combining risk free asset with risky asset portfolio 'A' lying on efficient frontier are inferior to the combinations offered by portfolios combining risk free asset with risky asset portfolio 'B'. Similarly, portfolios combining risk free asset with risky asset portfolio 'B' are inferior to the combinations offered by portfolios combining risk free asset with risky asset portfolio 'M'.

Hence, every rational investor would choose a combination of risk free asset with risky portfolio 'M' depending upon their risk appetite since these combinations are superior to any other combinations. Thus 'M' is the most desirable risky portfolio.

Follow the logic. Each risky portfolio on the frontier, when paired with the risk-free asset, produces its own straight line. Some lines lie above others. Keep rotating the line upward until it can rotate no further — it is then tangent to the frontier at a single point, M. Every combination on that tangent line beats every combination on every other line.

The two definitions that follow:

This combination of risky portfolios at point 'M' on the efficient frontier is the most optimal portfolio of all possible combinations of risky assets, when the existing risk free rate is plotting as the intercept in the graph. It is also called the Market Portfolio.

The line connecting risk free asset and tangent on efficient frontier at Portfolio 'M' (Market Portfolio) is called Capital Market Line (CML). In the presence of risk free asset, this is the efficient frontier. All the other risky portfolios lying below CML are no longer desirable/efficient.

In the presence of a risk-free asset, the CML replaces the efficient frontier. That is the single most important sentence in the chapter. Markowitz's curve was the best available; introduce risk-free borrowing and lending and a straight line does better everywhere except at the one point where they touch.

As can be observed, a portfolio combination of risk free asset and portfolio M at 'C' is preferable to portfolio 'A'.

The CML in equations

In summary, the Capital Market Line helps investors to choose an optimal combination of the market portfolio and the risk free asset. So if the investors can decide how much of her wealth to invest the market portfolio... according to her risk appetite, then the remaining wealth can be invested in the risk free asset.

E(Rp) = ((1 – Wmp) × Rf) + (Wmp × Rm)

where Wmp is the proportion of wealth invested in the market portfolio, Rf is the risk free rate, Rm is the return on market portfolio.

Rearranged — and this is the intuition behind CAPM:

E(Rp) = Rf + Wmp (Rm − Rf)

Every investor could expect a minimum of Risk Free Rate of return and then a proportion of the Excess Return that the market portfolio generates over and above the risk free rate. That is a proportion of the market risk premium.

And the risk side:

The risk of the above portfolio where Wmp is invested in the market portfolio and the remaining in the risk free asset will be simply Wmp × σmp.

This can be written as σp = Wmp × σmp, and rearranging, Wmp = σp / σmp. That is the proportion of investment should be the ratio of risks of the desired portfolio and the market portfolio. This relationship forms the basis of CAPM model.

Hold on to that last substitution, because section 5.6 uses it. If Wmp = σp / σmp, then substituting into E(Rp) = Rf + Wmp (Rm − Rf) gives E(Rp) = Rf + (Rm − Rf) × σp / σmp — which is one step away from CAPM.