Why Risk-Free Borrowing and Lending Matter
In the standard CAPM, investors can place wealth in a risk-free asset or borrow at the same risk-free rate to invest more than their own wealth in risky assets. This assumption is not just a financing detail. It allows investors with different risk tolerances to choose different amounts of risk while holding the same optimal mix of risky assets. If borrowing is unavailable or more expensive than lending, that shared-portfolio argument may no longer hold.
The analysis below uses returns measured over one common period. Write \(R_f\) for the risk-free return over that period, \(R_P\) for a risky portfolio’s return, and \(y\) for the fraction of initial wealth allocated to that portfolio. Thus \(y=0.6\) means 60% is invested in the risky portfolio and 40% is lent at the risk-free rate. A weight \(y>1\) means the investor borrows \(y-1\) times initial wealth and invests the borrowed funds in the risky portfolio.
The restriction \(y\geq0\) in this allocation setup rules out short positions in the risky portfolio. “Unrestricted” borrowing and lending here means there is no limit on lending or on borrowing at the common rate; it does not by itself authorize short selling. If short sales are also allowed, the feasible range of risky weights changes and the optimum must be checked against that broader range.
The Mean-Variance Allocation Rule
Suppose the investor evaluates a complete portfolio using mean-variance utility \(U=E[R_C]-\frac{A}{2}\operatorname{Var}(R_C)\), where \(A>0\) is the investor’s risk-aversion coefficient. This objective is a model: it is appropriate when the investor’s preferences are represented by mean and variance, as with quadratic utility, or under standard conditions such as normally distributed returns and exponential utility. It is not a universal description of investor preferences.
Because the risk-free return is constant, it contributes no variance. Therefore \(E[R_C]=R_f+y(E[R_P]-R_f)\) and \(\operatorname{Var}(R_C)=y^2\sigma_P^2\), where \(\sigma_P^2=\operatorname{Var}(R_P)\). Differentiating utility with respect to \(y\) gives the unconstrained stationary point:
Since the second derivative is \(-A\sigma_P^2<0\) for a risky portfolio with positive variance, this stationary point maximizes utility when it is feasible. Under the stated long-only risky-weight condition \(y\geq0\), the optimum is \(\max(0,y^*)\). In particular, a negative unconstrained stationary point does not mean the investor lends a negative amount: the optimum is \(y=0\), all wealth in the risk-free asset. If \(E[R_P]>R_f\), then \(y^*>0\); \(y^*<1\) means lending, \(y^*=1\) means no net risk-free position, and \(y^*>1\) means borrowing.
Worked Example: Choosing a Risky Weight When Borrowing Is Unrestricted
A hypothetical investor has risk aversion \(A=2\). A risky portfolio has expected annual return 14% and annual standard deviation 20%; the annual risk-free rate is 4%. Assume mean-variance utility, the same annual rate for borrowing and lending, and \(y\geq0\). Find the optimal weight, expected complete-portfolio return, and standard deviation.
Method. Use the mean-variance stationary point, then check feasibility and calculate the portfolio moments. The common mistake is to treat a weight greater than 100% as impossible even though borrowing is explicitly allowed.
The market premium for this risky portfolio is \(0.14-0.04=0.10\), and its variance is \((0.20)^2=0.04\). Thus \(A\sigma_P^2=2(0.04)=0.08\), and \(y^*=0.10/0.08=1.25\). This is feasible and exceeds 1, so the investor borrows \(1.25-1=0.25\), or 25% of initial wealth.
The expected return is \(E[R_C]=0.04+1.25(0.14-0.04)=0.04+0.125=0.165\), or 16.5%. The standard deviation is \(\sigma_C=1.25(0.20)=0.25\), or 25%, because the risk-free asset has zero variance. As a check on the optimum, the utility derivative at \(y=1.25\) is \(0.10-2(1.25)(0.04)=0.10-0.10=0\), and the second derivative is \(-0.08<0\).
How the Allocation Assumption Supports CAPM
With a risk-free asset available at one common rate, investors can adjust their overall exposure to risk without changing the mix of risky assets they hold. Under the standard CAPM assumptions—mean-variance preferences (or conditions that make them applicable), common expectations about returns and covariances, competitive markets, and unrestricted borrowing and lending at \(R_f\)—the optimal risky mix is shared. In equilibrium, the aggregate risky holdings must be the market portfolio, denoted by return \(R_M\). The argument below shows why the scale of an individual investor’s position matters when deriving the pricing relation.
Let investor \(i\) hold the market portfolio as the risky mix, with complete-portfolio return \(R_C=R_f+y_i(R_M-R_f)\). If the investor can vary the weight in any risky asset \(j\), the mean-variance first-order condition is \(E[R_j]-R_f=A_i\operatorname{Cov}(R_j,R_C)\). The risk-free asset is the alternative to that risky investment.
Since \(R_f\) is constant, \(\operatorname{Cov}(R_j,R_C)=y_i\operatorname{Cov}(R_j,R_M)\). Therefore \(E[R_j]-R_f=A_i y_i\operatorname{Cov}(R_j,R_M)\). The factor \(y_i\) cannot be dropped: it records how much of the investor’s total wealth is exposed to the market portfolio.
Setting \(j=M\) gives \(E[R_M]-R_f=A_i y_i\operatorname{Var}(R_M)\). The coefficient \(A_i y_i\) is the same in the condition for asset \(j\) and for the market portfolio for this investor.
When \(\operatorname{Var}(R_M)>0\), divide the asset condition by the market condition. This gives the CAPM relation \(E[R_j]-R_f=\frac{\operatorname{Cov}(R_j,R_M)}{\operatorname{Var}(R_M)}(E[R_M]-R_f)\). With \(\beta_j=\frac{\operatorname{Cov}(R_j,R_M)}{\operatorname{Var}(R_M)}\), it is \(E[R_j]=R_f+\beta_j(E[R_M]-R_f)\).
This ratio argument is the key justification: for an investor holding the market risky mix at positive scale, both first-order conditions contain the same \(A_i y_i\), so that investor’s risk-aversion coefficient and portfolio scale cancel. The argument does not equate an individual’s risk aversion with an aggregate coefficient. It also requires a nonzero market variance; if the market portfolio has zero variance, its beta-based pricing relation is not defined in this form.
Unrestricted borrowing and lending support the shared risky mix because an investor can choose the desired scale \(y_i\) at the same rate. Investors can differ in risk aversion and therefore in \(y_i\), while their risky holdings remain proportional to the same portfolio. This does not mean borrowing is required: some investors lend, and others may choose \(y_i=1\).
When Borrowing Costs More Than Lending
If the borrowing rate \(R_b\) exceeds the lending rate \(R_f\), the complete-portfolio return is piecewise. For \(0\leq y\leq1\), the investor lends the uninvested fraction, so \(E[R_C]=R_f+y(E[R_P]-R_f)\). For \(y>1\), the investor borrows \(y-1\) at \(R_b\), so \(E[R_C]=yE[R_P]-(y-1)R_b=R_b+y(E[R_P]-R_b)\). In both regions, assuming a single risky portfolio is scaled, \(\operatorname{Var}(R_C)=y^2\sigma_P^2\).
Utility is continuous at \(y=1\), but its slope may change there. The derivative on the lending side is \(E[R_P]-R_f-Ay\sigma_P^2\); on the borrowing side it is \(E[R_P]-R_b-Ay\sigma_P^2\). The higher borrowing rate makes the right-hand derivative smaller. If the left-hand derivative at \(y=1\) is nonnegative and the right-hand derivative is nonpositive, utility is maximized at the kink: the investor uses the risky portfolio without borrowing. A borrowing rate premium therefore can make an otherwise attractive leveraged allocation suboptimal.
Worked Example: Finding the Optimum at a Borrowing-Rate Kink
A hypothetical investor has \(A=2\). A risky fund has expected annual return 13% and annual standard deviation 20%. The investor can lend at 3% but must borrow at 8%. Assume mean-variance utility and \(y\geq0\). Find the optimal risky weight and the return and standard deviation there.
Method. Check the one-sided utility slopes at \(y=1\), then verify that neither region has a feasible stationary point that improves on the kink. The common mistake is to use the lending rate in the borrowing region.
The variance is \((0.20)^2=0.04\), so \(A\sigma_P^2=2(0.04)=0.08\). The derivative just to the left of 1 is \(0.13-0.03-0.08=0.02>0\). Utility is still increasing as the investor approaches \(y=1\) from below. The derivative just to the right is \(0.13-0.08-0.08=-0.03<0\). Utility decreases after entering the borrowing region.
Thus the optimum is \(y=1\), with no net borrowing or lending. For confirmation, the lending-region stationary point is \((0.13-0.03)/0.08=1.25\), outside the region \(y\leq1\); the borrowing-region stationary point is \((0.13-0.08)/0.08=0.625\), outside the region \(y\geq1\). At the optimum, expected return is \(0.13\), or 13%, and standard deviation is \(1(0.20)=0.20\), or 20%.
Borrowing Constraints and the Limits of the Prediction
A hard borrowing limit, such as \(y\leq1\), can bind even when an investor would prefer a larger risky position under the unconstrained model. The investor’s actual allocation is then determined by the feasible set, not by the unconstrained first-order condition. A higher borrowing rate can have a related effect by reducing the incentive to choose \(y>1\), as the kink example shows.
Worked Example: Applying a Maximum Risky-Weight Rule
A hypothetical investment policy permits an investor to invest at most 100% of wealth in a risky portfolio, so \(0\leq y\leq1\). The investor has \(A=2\); the portfolio has expected annual return 12% and annual standard deviation 15%; and the risk-free rate is 4%. Assume the same rate applies to borrowing and lending, but the policy limit remains in force. Find the constrained allocation and its moments.
Method. First calculate the unconstrained stationary point, then compare it with the allowed interval. The common mistake is to report an infeasible leveraged weight as the investor’s actual choice.
The premium is \(0.12-0.04=0.08\). The variance is \((0.15)^2=0.0225\), giving \(A\sigma_P^2=2(0.0225)=0.045\). Therefore \(y^*=0.08/0.045=1.777\ldots\), which exceeds the permitted maximum. Utility is increasing throughout the allowed interval, so the constrained optimum is \(y=1\).
The expected return at that allocation is \(0.04+1(0.12-0.04)=0.12\), or 12%. Its standard deviation is \(1(0.15)=0.15\), or 15%. For a comparison with the unconstrained model, that model would give expected return \(0.04+(1.777\ldots)(0.08)=0.1822\ldots\) and standard deviation \((1.777\ldots)(0.15)=0.2667\ldots\), or about 18.22% and 26.67% when rounded to two decimals. Those figures are not feasible under the policy.
When borrowing constraints bind for some investors, they cannot necessarily scale the same risky portfolio to their preferred level. If borrowing rates also vary across investors, their preferred risky mixes or positions may differ further. The standard shared-portfolio derivation then does not follow simply from the assumptions used above. This weakens the derivation’s prediction; it does not establish that every security must depart from the CAPM relation in every constrained market.
Common Exam Traps
- Using a negative risky weight despite a long-only restriction. If \(y^*<0\) but the feasible set is \(y\geq0\), the optimum is \(y=0\), not a negative allocation.
- Applying the lending rate to borrowed funds. For \(y>1\), the return calculation must use the borrowing rate on \(y-1\) of initial wealth.
- Dropping the scale factor in the CAPM derivation. An investor holding the market portfolio at scale \(y_i\) has \(\operatorname{Cov}(R_j,R_C)=y_i\operatorname{Cov}(R_j,R_M)\), not merely \(\operatorname{Cov}(R_j,R_M)\).
- Claiming a constrained investor must choose the unconstrained optimum. Calculate the stationary point, then check whether it is feasible. A binding limit can make the boundary optimal.
Check Your Understanding
For each problem, use annual returns and mean-variance utility where specified. Check both the calculated weight and the stated borrowing or lending conditions.
- An investor with \(A=2\) can borrow and lend at 3%. A risky portfolio has expected return 11% and standard deviation 20%. Find the optimal risky weight, expected complete-portfolio return, and standard deviation. (Answer: \(y=1\); expected return \(=0.11\); standard deviation \(=0.20\).)
- An investor with \(A=2\) can borrow and lend at 4%. A portfolio has expected return 10% and standard deviation 15%. Find the unrestricted optimal weight and the resulting expected return and standard deviation. (Answer: \(y=0.06/[2(0.15^2)]=4/3\); expected return \(=0.12\); standard deviation \(=0.20\).)
- An investor with \(A=1.5\) faces a maximum risky weight of 1. The risk-free rate is 5%, and the risky portfolio has expected return 12% and standard deviation 20%. Find the constrained optimum and its moments. (Answer: unconstrained \(y^*=0.07/[1.5(0.20^2)]=1.1667\), so constrained \(y=1\); expected return \(=0.12\); standard deviation \(=0.20\).)
- An investor with \(A=2\) lends at 4% and borrows at 10%. A risky portfolio has expected return 14% and standard deviation 20%. Check the one-sided utility slopes at \(y=1\) and identify the optimum. (Answer: left slope \(=0.14-0.04-2(0.20^2)=0.02\); right slope \(=0.14-0.10-2(0.20^2)=-0.04\); optimum \(y=1\), with no borrowing.)
- An investor with \(A=3\) may choose only \(y\geq0\), and can borrow and lend at 5%. The risky portfolio has expected return 3% and standard deviation 10%. Find the optimal allocation and the complete-portfolio moments. (Answer: unconstrained \(y^*=(0.03-0.05)/[3(0.10^2)]=-2/3\); constrained optimum \(y=0\); expected return \(=0.05\); standard deviation \(=0\).)