I have read that some Institutional Investors are utilizing leverage. According to Modern Portfolio Theory, to apply leverage one would:

a) find the tangency portfolio on the efficient frontier from the risk free rate and
b) leverage it up or down to attain the desired level of risk and target return

As a starting point, I assume that the 60:40 Equity:Fixed Income portfolio is the Capital Market portfolio and tangency portfolio, since many Institutional Investors use some version of this allocation currently. I also assume that the risk reward characteristics of all investable assets have not changed. In other words, expected return, volatility and correlation have remained the same.

I further assume that in applying leverage, such investors would borrow at the risk free rate and allocate to the tangency portfolio.

Question 1: Does the tangency portfolio change with the level of interest rates?

The decision to apply leverage was made when the risk free interest rate was considerably lower than currently. As such, this seems to imply that the point of tangency in that environment would be a portfolio that is lower in risk and return than with a higher interest rate. As such, under the current 5% risk free rate, would the current tangency portfolio be something closer to 70:30, Equities:Fixed Income?

Question2: Does it make sense to leverage such a portfolio?

I understand that there are duration, convexity and credit differences between borrowing rates and a fixed income portfolio but if borrowing is considered a negative allocation to Fixed Income, would taking on such leverage essentially be moving the portfolio more toward equities? At the extreme, if one were to apply 130% leverage, and apply it to a 70:30 portfolio, wouldn't one just be allocated to equities? It should at least negate part of the risk free sovereign portion of the fixed income portfolio. Also, does it make sense to apply leverage with an inverted yield curve (borrow at 5% and lend at 4%)?

  • $\begingroup$ As far as applying leverage is concerned, it is the answer MPT gives when you want the best risk-adjusted return but don't like the portfolio drag that comes with diversification. You don't add to the allocation of the risky asset with the highest expected return. You use leverage to raise the return of the "optimal" portfolio. The problem is do we really know what that optimal portfolio will be ex post and would you be comfortable applying 130% leverage -- even if that were feasible. How would that have worked out in 2022? $\endgroup$
    – RRL
    Commented Apr 18 at 20:36

1 Answer 1


The composition of the tangency portfolio in standard mean-variance analysis does depend on the risk-free rate. The degree of that dependence depends on whether or not we hold the expected returns of the risky assets fixed as we vary the risk-free rate or if we assume fixed risk premia.

It is worthwhile to work through the steps leading to the tangency portfolio to understand this dependence.

Consider a portfolio of risky assets where $\mathbf{w}$ is the vector of portfolio weights, $\mathbf{r}$ is the vector of expected returns, and $\mathbf{\Sigma}$ is the covariance matrix. Let $w_0$ denote the weight of the riskless asset with risk-free rate $R$. Setting an excess return target of $\mu - R$, we obtain the minumum variance set by solving the constrained minimization problem: $$\text{min }\, \mathbf{w}'\mathbf{\Sigma}\mathbf{w}\\ \text{subject to }\, (\mathbf{r}-R\mathbf{1})'\mathbf{w} = \mu - R$$ Here $\mathbf{1}$ is a vector with all components equal to $1$ and the superscript $'$ denotes the transpose. Note that no budget constraint is imposed on the weights of the risky assets and the weight of the riskless asset will satisfy $1-w_0= \mathbf{1}'\mathbf{w}$. This accounts for borrowing or lending at the risk-free rate to leverage or deleverage the portfolio of risky assets.

Introducing a Lagrange multiplier $\lambda$, we solve the minimization problem by setting the gradient of the Lagrangian equal to zero:

$$\tag{1}0=\frac{\partial \mathcal{L}}{\partial \mathbf{w}}= \frac{\partial}{\partial \mathbf{w}}\left[\frac{1}{2}\mathbf{w}'\mathbf{\Sigma}\mathbf{w} +\lambda(\mu -R- (\mathbf{r}-R\mathbf{1})'\mathbf{w} \right] = \mathbf{\Sigma}\mathbf{w}-\lambda (\mathbf{r}-R\mathbf{1})$$

The optimal vector of weights that solves (1) is given by

$$\tag{2}\mathbf{w}^*= \lambda \mathbf{\Sigma}^{-1}(\mathbf{r}- R\mathbf{1})$$

Then the return constraint takes the form $\lambda (\mathbf{r}-R\mathbf{1})'\mathbf{\Sigma}^{-1}(\mathbf{r}- R\mathbf{1})= \mu -R$, allowing us to solve for the Lagrange multiplier.

Proceeding, we would find that the standard deviation of return $\sigma$ for the optimal portfolio as a function of the return constraint $\mu$ will lie on a ray in the $\sigma - \mu$ plane. Furthermore, the optimal portfolio set is spanned by two minimum variance portfolios: the riskless asset and the so-called tangency portfolio which contains none of the riskless asset. To find the portfolio weights for the tangency portfolio we take $w_0 = 0$ which implies that $\mathbf{1}'\mathbf{w^*} = 1$. Applying this to (2) we get

$$\mathbf{1}'\mathbf{w}^* = \lambda(\mathbf{1}' \mathbf{\Sigma}^{-1}\mathbf {r}- R\mathbf{1}'\mathbf{\Sigma}^{-1}\mathbf {1})= 1$$

Solving for $\lambda$ and substituting into (2) we get the weights of the risky assets in the tangency portfolio as

$$\mathbf{w}_t= \frac{\mathbf{\Sigma}^{-1}(\mathbf{r}- R\mathbf{1})}{\mathbf{1}' \mathbf{\Sigma}^{-1}\mathbf {r}- R\mathbf{1}'\mathbf{\Sigma}^{-1}\mathbf {1}},$$

showing the dependency of the tangency portfolio on both expected returns and the risk-free rate.

For some numerical results, consider a portfolio comprising the S&P 500 (SPX) and 10-year US Treasury (UST) indices. Over the past century the annual returns were about 10% for SPX and 5% for UST. The historical volatilities were about 15% and 6% for SPX and UST, respectively, and the correlation of returns was 0.1. The average risk-free rate represented by 3-month Treasury bills was about 3%.

With a 3% risk-free rate, the tangency portfolio has weights of 37% for SPX and 63% for UST. If we assume fixed excess returns (risk premia) for stocks and bonds over cash, then those weights will not change regardless of the level of the risk-free rate.

On the other hand, if we hold the expected returns for stocks and bonds fixed at historical averages, the tangency portfolio weights will vary. Assuming a risk-free rate of 1% we get a lower allocation of 26% to stocks and 74% to bonds. The allocation to stocks increases along with the assumed risk-free rate, reaching the 70/30 split when $R \approx $ 4.5%.

With a further increase in the risk-free rate, the tangency portfolio ultimately reaches 100% stocks. It can be shown that the tangency portfolio lies on the mean-variance efficient frontier for portfolios composed solely of the risky assets. As $R$ is increased beyond the expected return of the global minimum-variance portfolio, the point of tangency shifts to the lower branch of the minimum variance set for the risky assets. This can lead to some surprising results.

As this answer is already quite long, such phenomena should be explored with another question.

Finally, it should be emphasized that adhering strictly to the literal interpretation of modern portfolio theory for allocation decisions among a few asset classes is dangerous. The sensitivity to parameter assumptions is particularly pronounced in this example. How one should adjust return expectations for stocks and bonds based on current valuation metrics and in different environments such as an inverted yield curve (where funding cost exceeds the yield of longer duration bonds) is an open question.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.