19.2. THE SINGLE-COUNTRYCAPM 713
Figure 19.1:Combinations of risky stock portfoliosand asset 06 x 6 E( ̃rp)sd( ̃-rp)0 r 0
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sd( ̃rs)rs)1 ........................................E( ̃ rDoItYourself problem 19.2
Rework the example by changing the initial investment in asset 1 from 400 to 800,
maintaining the other risky positions and adjusting the risk-free one. Check that
the weighted excess-return formula still gives the right answer.
19.2.2 The Tangency Solution: Graphical Discussion
Consider the feasible combinations of expected return and standard deviation. The
simplest case is one with a risk-free asset, subscripted “0”, and a risky stock denoted
ass. We invest a fractionxinto the risky stock portfolio with return ̃rswhile 1−x
is invested risk-free. The portfolio return is
̃rp=xr ̃s+ (1−x)r 0 =r 0 +x( ̃rs−r 0 )⇒
E( ̃rp) =r 0 +xE( ̃rs−r 0 ),sd( ̃rp) =|x|sd( ̃rs)(19.8)
So for non-negative values ofx, both expected return and standard deviation are
linear functions ofx. This will imply that all (E, sd) combinations achievable with
the risk-free asset and the risky portfolio are found on a halfline. To show this
we use a trick that is often applied in thermometers, where heat is measured on
two scales, say Celsius and Fahrenheit, that are linearly related. Same here:xand
E( ̃rp) are linearly related, so we can show them as two scales on one axis, as we do
in Figure 19.1. To link thexand Epscales we calibrate them using any two known
corresponding points: x= 1 means E( ̃rp) = E( ̃rs) whilex= 0 means E( ̃rp) =r 0.
All this gives us the double-scaled axis shown in Figure 19.1. If sd( ̃rp) is linear inx+,
then looking at the other scale of the axis we must conclude it is linear in E( ̃rp)>r 0
too. The sd values for the calibration points are 0 and sd( ̃rs), respectively, and all
risk-return combinations for intermediate or higher values ofxor E( ̃rp) are on one