Let us now take one risky bundle (of n assets already held in fixed proportions wi). For n = 3 we might have w1 = 20%, w2 = 25% and w3 = 55%, which makes up our one risky bundle. Now allow investors to borrow or lend at the safe rate of interest r. Because r is fixed over the holding
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period, the variance on the risk-free asset is zero, as is its covariance with our own risky bundle.
Thus, the investor can:
x invest all of her wealth in the one risky bundle and undertake no lending or borrowing;
x invest less than her total wealth in the risky bundle and use the remainder to lend at the risk-free rate;
or
x invest more than her total wealth in the risky bundle by borrowing additional funds at the risk-free rate; in this case she is said to hold a levered portfolio.
The above choices are represented in the transformation line, which is a relationship between expected return and risk for a portfolio consisting of one safe asset plus one risky bundle. The transformation line holds for any portfolio consisting of these two assets and it turns out that the relationship between expected return and risk (measured by the standard deviation of the ‘new’
portfolio) is a straight line.
To derive a particular transformation line, consider the risk-free return r (on, say, a T-bill) and the return on a single ‘bundle’ of risky assets Rq (Table I.A.3.4). Since r = 10% for all scenarios, then Ƴr = 0. The risky-asset bundle q we assume has a mean return ERq = 22.5% and a standard deviationƳq 24.87%. The expected return on this new ‘two-asset’ portfolio is
q
N xr x ER
ER 1 (I.A.3.16)
Table I.A.3.4: Risk-free and risky ‘bundle’
Returns
T-bill (safe) Equity (risky)
Mean r = 10% rq = 22.5%
Std. dev. Ƴr = 0 Ƴq = 24.87%
We can now alter the proportions x held in the risk-free asset and 1 – x held in the risky bundle to obtain a ‘new’ portfolio. The possible combinations of expected return ERN and riskƳN on this ‘new’ portfolio are shown in the last two columns of Table I.A.3.5. This linear opportunity set is shown in Figure I.A.3.6 as the transformation line.
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Table I.A.3.5: ‘New’ portfolio: (transformation line) State Share of Wealth in ‘New’ Portfolio
T-bill Equity ERN ƳN
w1 w2
1 1 0 10% 0%
2 0.5 0. 5 16.25% 12.44%
3 0 1 22.5% 24.87%
4 –0.5 1.5 28.75% 37.31%
Figure I.A.3.6: Transformation line: one riskless asset and one risky ‘bundle’
6 8 10 12 14 16 18 20
0 5 10 15 20 25
Standard deviation
Expected Return(%)
All lending at the risk free rate of 10%
No borrowing/
no lending 0.5 lending +
0.5 in the risky bundle
-0.5 borrowing + 1.5 in the risky bundle
The transformation line has an intercept equal to the risk-free rate (r = 10%). Here the investor puts all her wealth in the safe asset (x = 1). When all of the investor’s own wealth is held in the risky bundle (x = 0) then the ‘new’ portfolio has ERN = 22.5% and ƳN= 24.87%These are of course the expected return and standard deviation of the risky-asset bundle (ERq and Ƴq). This is the ‘no borrow/no lend’ portfolio.
When x = 1, all wealth is invested in the risk-free asset and ERN = r and ƳN = 0.For 0 < x < 1 some wealth is invested in the risk-free asset and the remainder is put in the risky-asset bundle.
When x = 0, all the investor’s wealth is invested in stocks and ERN = ERq. For x < 0 the agent borrows money at the risk-free rate r to invest in the risky asset. For example, when x = –0.5 and initial wealth is $100, the individual borrows $50 (at an interest rate r) and invests $150 in stocks
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(i.e. a levered position). From Table I.A.3.5 we see that this gives ERN = 28.75% and ƳN = 37.31% (see also Figure I.A.3.6).
The transformation line gives us the linear risk–return relationship for any portfolio consisting of a combination of investment in the safe asset and one ‘bundle’ of risky assets. At each point on any transformation line the investor holds the risky assets in the same fixed proportions wi.
All the points (except the intercept) on the transformation line represent fixed proportions wi = 20%, 25% and 55% (say) in our one risky bundle (of n = 3 risky assets): ‘alpha’, ‘beta’ and
‘gamma’. The only ‘quantity’ that varies along the transformation line is the proportion held in the one risky bundle of assets relative to that held in the one safe asset. The investor can borrow or lend and be anywhere along the transformation line. For example, the (0.5, 0.5) point (Figure I.A.3.6) represents 50% in the safe asset and 50% in the single bundle of risky securities. Hence, an investor with $100 would hold $50 in the risk-free asset and $50 in the one risky bundle made up of 0.2 × $50 = $10 in alpha, 0.25 × $50 = $12.50 in beta and 0.55 × $50 = $27.50 in the gamma securities.
Since r is known and fixed over the holding period, the standard deviation of the ‘new’ portfolio depends only on the standard deviation of the one risky bundle and this is why the opportunity set in this case is a straight line. For any portfolio consisting of two assets, one of which is a single risky bundle and the other is a safe asset, the relationship between the expected return on this new portfolio ERN and its standard deviation ƳN is linear with intercept r. When a portfolio consists only of n risky assets then, as we have seen, the efficient frontier in (ERp,Ƴp) space is curved. This should not be unduly confusing since the portfolios considered in the two cases are very different.
If we choose a single risky bundle with Ƴk = 30% (and ERk = 15%) then we can draw the corresponding transformation line L (Figure I.A.3.7). Similarly for our original single risky bundle Vq = 24.87% (ERq = 22.5%) we have a higher transformation line LȨ. Hence, each single risky bundle (each with different but fixed weights wi) has its own transformation line.
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Figure I.A.3.7: Transformation lines
0 5 10 15 20 25 30 35
0 5 10 15 20 25 30 35 40
Standard deviation
Expected return (%)
L L’
Vk= 30 Vk= 25
Figure I.A.3.8: Efficient frontier and CML
0 5 10 15 20 25 30
0 5 10 15 20 25
Standard deviation
Expected return
D M
C
A CML
The PRM Handbook – I.A.3 Capital Allocation