"Life Cycle" Hypothesis of Saving: Aggregate ... - Arabictrader.com

292 Miscellanea
Intuitively, we know that this operation will reduce the risk, but also decrease
the expected return of the portfolio (provided the original portfolio had a positive
excess return). Indeed, if we sell, say, d i % of the portfolio and use the proceeds
to purchase riskless securities, this will reduce the dispersion (sigma) of
the returns of the portfolio by d i % (because d i % of the returns will have been
made constant). It also reduces the excess return of the portfolio by the same d i %.
Similarly, by levering a portfolio we mean increasing the investment in the
portfolio through borrowing. Intuitively, we know that this will increase the risk,
and also increase the expected return of the portfolio (again, assuming a positive
excess return on the original portfolio). If an additional amount, say, d i %, is
invested in the portfolio, and this investment is financed by borrowing, then both
the sigma and the excess return of the portfolio will increase by d i %.
The risk-adjusted return of portfolio i, or RAP(i), is the return of portfolio i,
levered by an amount d i (d i positive or negative), where d i is defined as the leverage
required to make portfolio i risk-equivalent to the market, i.e., to make its
sigma, s(i), match that of the market. The value of d i can be inferred from this
definition:
s()= i ( 1+
d i ) s i = s M
(12.1)
which implies:
d = s s -1
i M i
(12.2)
Taking into account the interest on d i , which is the amount borrowed (if d i is
positive) or lent (if d i is negative), we find: 3
RAP()= i r()= i ( 1+
d ) r - d r
(12.3)
i i i f
Substituting equation (12.2) into equation (12.3), we can rewrite RAP as:
RAP()= i ( s s ) r - [( s s )-1] r = ( s s )( r - r )+ r
M i i M i f M i i f f
(12.4)
Using the definition of e i , we can also rewrite RAP as:
RAP()= i ( s s ) e + r = e()+
i r
where
ei ()= ( s s ) e
M i i f f
M i i
(12.5)
(12.6)
Thus RAP(i) can be computed from total returns, using equation (12.4), or from
excess returns using equation (12.5).
We see from equation (12.5) that e(i) = RAP(i) - r f . That is, RAP(i) and e(i)
differ only by r f , a constant in the sense that it is the same for all portfolios. This