Capital market line

Capital market line with efficiency curve. The special point M is the market portfolio for market risk .

The capital market line ( KML , English Capital Market Line , CML ) is a relationship from capital market theory . It is the expected (in the sense of claimed) risk / return combination of efficient portfolios in equilibrium . All risk-return combinations that the investor is willing to accept are on the capital market line.

The capital market line is a component of the Capital Asset Pricing Model , which is a further development of the portfolio theory . It tries to find a balance between several market participants, whereas portfolio theory only looks at a representative individual. The central equation of the CAPM, which explains the expected returns by the respective beta factor , is the security line .

Assumptions

The concept of the capital market line is based on the following assumptions:

They are risk averse investors with possibly different preferences.
Investors only choose efficient portfolios.
Investors have the same expectations based on the same information.
The investors invest over the same planning horizon .
There is a risk-free investment and debt option, whereby the debit interest corresponds to the credit interest.

An equilibrium can be derived based on these assumptions.

Derivation of the balance

In contrast to the market portfolio, not all stocks have to be included in the individual portfolio . Every investor holds a share of the tangential portfolio.

For market clearing to occur, every share must be included in the tangential portfolio. This is the case when the total demand and the number of shares in circulation are exactly the same.

The budget condition is ${\ displaystyle x_ {0} ^ {k} + \ alpha ^ {k} + \ Sigma _ {i = 1} ^ {m} x_ {i} = 1 ~ with ~ \ Sigma _ {i = 1} ^ { n} x_ {i} = 1 \}$

${\ displaystyle \ alpha ^ {k}}$ : Share of the tangential portfolio (risky part)
${\ displaystyle x_ {0} ^ {k}}$ : risk-free portion of the portfolio
${\ displaystyle x_ {i}}$ : Share in the tangential portfolio, m identical for each investor k

Market Clearance Condition

${\ displaystyle \ Sigma _ {k = 1} ^ {K} x_ {0} ^ {k} ~ A_ {k} = 0}$

${\ displaystyle A_ {k}}$ : Investor's budget
${\ displaystyle x_ {0} ^ {k}}$ : risk-free portion of the portfolio

Budget condition

${\ displaystyle \ Sigma _ {k = 1} ^ {K} A_ {k} = 0 = \ Sigma _ {k = 1} ^ {K} \ alpha ^ {k} A_ {k} = 0}$

${\ displaystyle \ Sigma _ {k = 1} ^ {K} A_ {k}}$ : Sum of the budget
${\ displaystyle \ Sigma _ {k = 1} ^ {K} \ alpha ^ {k} A_ {k} = 0}$ : Investment in the tangential portfolio

Equation of the capital market line

${\ displaystyle \ mu _ {x} = r + {\ frac {\ mu _ {m} -r} {\ sigma _ {M}}} \ cdot \ sigma _ {x}}$

The greatest possible diversification is the purchase of the market portfolio. This well-diversified portfolio offers the best risk / return ratio
Then the risk can only be reduced by adding a risk-free investment
The slope is a quotient of the market's excess return and its risk. So there is a normalization.

Result

In equilibrium, the market portfolio is the same as the tangential portfolio . This is the only empirically testable statement of the CAPM.
Every investor holds a risky stock portfolio, the structure of which is consistent with. ${\ displaystyle x ^ {T}}$
The Tobin separation applies , i.e. H. the decision about the risky part and the decision about its structure are separate.
All efficient portfolios are completely positively correlated because they are composed of the risk-free interest rate and the market portfolio.