‘Kinked’ Capital Allocation Line

What is meant by ‘Kinked’ Capital Allocation Line (CAL)?

CAL is ‘kinked’ when investment in a complete portfolio is leverage and when the borrowing rate is greater than risk free rate.

What is CAL?

CAL is a straight line in investment opportunity set with risky assets and a risk free asset in the expected return-standard deviation graph.

What is expected return, risk premium and the standard deviation of a complete portfolio?

                                                                                                  

The expected return of a complete portfolio is denoted as

                            E(r_c) = r_f + y[E(r_p) - r_f]

Example: 
Let's say the proportion of investment budget in risky asset is y= .5 and that of risk free asset is (1-y).                        

The standard deviation of risk free asset (r_f ) = 0 and that of risky assets = 12%.

The expected return of risky asset  E(r_p) = 8%

§   The expected return of risk free asset E(r_f) = 4%

The expected return of the complete portfolio is:

                             E(r_c) =  4% + .5 (8%-4%)

                                      =  6%

The risk premium of a complete portfolio is 2%  (i.e., 2% greater return  for additional risk)

The standard deviation of a complete portfolio is 6% (= .5 x 12%) (i.e., risk has reduced by 50%).

What is the slope of CAL?

The slope of CAL (S) is the incremental return per incremental risk.  It is called reward to volatility (Sharpe) ratio.

                           

                           S = [E(r_p) –r_f ] / σ_p

                                                 

                                   = 4% /12%   = .33

Let's see what would be the slope of CAL when the portfolio has borrowed at risk free rate

Now, the proportion of investment in risky assets ‘y’ would be greater than one.

Let’s say an investor has investment budget of $2,000,000 and has borrowed $1,000,000 and plans to invest all the funds in risky asset.

                                    

                              Y = 3,000,000/2,000,000

                              Y = 1.5 and

                         (1-y) = -.5  (i.e., the proportion of risk free asset)

                                                                                  

 Other things remaining same, 

 Levered  E(r_c) = 4% + 1.5 (8%-4%) = 10%

Standard deviation of the portfolio (levered) = 1.5 x 12% = 18%

Slope of CAL (levered)    S = [E(r_c)-r_f] / σ_c

                                           = [10%-4%]/18% 

                                           = .33                                    

                                                

As long as risk free rate is equal to borrowing rate the slope of CAL will remain unaffected and CAL is straight.

Kinked CAL

Not everyone can borrow at risk free rate, particularly non-government investors. Default risk of borrowers induce lenders to charge higher rate of interest. Suppose the borrowing rate is 5% the slope of CAL now is:  

                          S = [E(r_p) – r_f] / σ_p

                             = [8% -5%]   / 12%                   

                             = .25

The reward-to-volatility ratio or the slope of CAL gets reduced from .33 to .25 and hence CAL is slightly flattened or kinked.