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Intro to portfolio theory

Jamal Munshi, Sonoma State Univesity, 1992
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returns generating assets a1, a2, a3, .... are projected to yield uncertain returns y1, y2, y3, ..... the uncertainty in returns is modeled with a gaussian distribution with expected value = k1, k2, k3, .... and standard deviation = s1, s2, s3, ...
the question

if we combine assets a1, a2, a3, ... with relative dollar amounts invested in each asset given by the weight vector w1, w2, w3, ...and form a portfolio p then what will the returns distribution of p look like? i.e., what are values of kp and sp?

computation of kp and sp

kp will be simply the average of the ki
kp = sum(wi*ki)
spp = sumi(sumj(wi*wj*sij))
sp=sqrt(sumi(sumj(wi*wj*sij)))

here spp is the variance of portfolio returns and sp is the standard deviation. sij is the covariance between a1 and aj returns. when i=j then the term reduces to sii, the variance of asset i returns.

plots of kp, sp against wi
if we form many portfolios by changing the w vector and plot kp and sp against w then what will these plots look like?. for kp the answer is easy. note that the expression for kp is linear in w. so the plot will be LINEAR. it will be a straignt line.
but what about sp?
this expression is not linear but quadratic. to see what these curves will look like let us first define
rij=sij/(si*sj)
where rij is the correlation coefficient between a1 returns and aj returns and rewrite the spp equation as
spp = sumi(sumj(wi*wj*s1*sj*rij))
for two assets 1 and 2 this equation simplifies to
spp = w1*w1*s11 + w2*w2*s22 + 2*w1*w2*s1*s2*r12
consider the following cases for the two asset equation
  • when the correlation coefficient is -1
    • we get spp = spp = w1*w1*s11 + w2*w2*s22 - 2*w1*w2*s1*s2
    • which can be written as spp = (s1w1-s2w2)(s1w2-s2w2).
    • which means that sp = sqrt(spp) = s1w1-s2w2.
    • he relationship between sp and w1 is LINEAR.
  • when the correlation coefficient is +1
    • we get spp = spp = w1*w1*s11 + w2*w2*s22 + 2*w1*w2*s1*s2
    • which can be written as spp = (s1w1+s2w2)(s1w2+s2w2).
    • which means that sp = sqrt(spp) = s1w1+s2w2.
    • he relationship between sp and w1 is LINEAR.
  • when the correlation coefficient is between -1 and +1 the relationship between sp and w1 is sp = sqrt(w1*w1*s11 + w2*w2*s22 - 2*w1*w2*s1*s2) which is quadratic or CURVED
  • these relationships are shown in the left panel of the graph below. this curvature is the key to understanding portolio theory.
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when one of the assets is riskless
  • set s22 to zero and s2 to zero and get
  • sp = sqrt(w1*w1*s11) and taking the square root we find that
  • sp = w1s1
  • LINEAR
the right panel of the figure above
  • the curve represents portfolios formed from risky assets with imprefect rij
  • the line represents portfolios formed with one of the risky portolios and a riskless asset
  • this line dominates all other lines that could be drawn
  • this line is the basis of the capital asset pricing model which is the subject of another lecture