Portfolio Choice with Different Lending and Borrowing Rates

Summary

The document explains how unequal lending and borrowing rates change the usual capital market line. Starting from the efficient frontier of risky assets, it describes a line from the lending rate that touches the frontier at one portfolio, followed by a section along the frontier, and then a borrowing line that touches at another portfolio. The transition between lending and borrowing therefore involves a risky portfolio region rather than one tangency portfolio serving both rates.

To find a portfolio at a target standard deviation, the proposed procedure is to draw the efficient frontier, construct both tangents and the intervening frontier segment, then locate the target risk on the resulting piecewise capital market line. The example question supplies four risky securities and a target volatility, but the answer provides no numerical portfolio weights or calculation steps. Applying it requires estimating the frontier from the return and covariance inputs and matching the interest rate units to the data frequency.

Key ideas

• Different lending and borrowing rates produce distinct tangency portfolios on the efficient frontier.
• The capital market line has a lending segment, a curved frontier segment, and a borrowing segment.
• The portfolio for a target standard deviation is found by locating that risk level on the full piecewise opportunity set.
• The document gives a graphical method but does not calculate weights for the example securities.

Tags

# optimal portfolio with different lending and borrowing rates


# optimal portfolio with different lending and borrowing rates












I have 4 risky securities (have returns and var-cov matrix for monthly data), and I can lend at 1% per annum, but borrow at 5% per annum. If i wish to obtain the s.d. of 5%, what is the optimal portfolio? How would I go about solving this? I tried finding tangency portfolios for both rates, but not sure how to proceed from there.

## Answer by Alex C (score 1)

https://quant.stackexchange.com/a/31747

In this case the Capital Market Line is composed of three sections: (1) a straight line segment from the 1% point on the y axis that is tangent to the frontier at a point T. A (curved) section of the frontier from T to another point S on the frontier. S is determined as the point of tangency of a line from 5% on the y axis to the frontier. And (3) a ray (half line) that starts at S and is part (the rightmost part) of the tangent mentioned earlier.

So just draw the frontier, the two tangents, the above mentioned three part Capital Market Line (straight, curved, then straight again) and find the point on the CML having a s.d. of 5%.

This figure may help (though unfortunately the points T and S are not clearly labeled as such):

http://www.d42.com/_/rsrc/1272735029294/portfolio/theory/the-efficient-frontier-with-risk-free-lending-and-borrowing/figure_3.7.jpg

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.