Interpreting Mean-Variance Weights Above One with Borrowing
Summary
The document explains a two-asset mean-variance allocation in which the risky asset receives a weight greater than one and the risk-free asset receives a negative weight. For an investor with a fixed amount of capital, the interpretation is to invest the available capital in the risky asset and borrow additional funds at the risk-free rate to invest more. The negative risk-free weight represents financing rather than a cash allocation that must be sourced from elsewhere.
The replies clarify the weight ranges: a risky-asset weight of one means all capital is invested in it, while a weight between zero and one divides capital between the risky asset and risk-free holding. A weight above one indicates leverage; a negative risky weight represents shorting the risky asset to hold the risk-free asset. The claim that negative risky weights are irrational is stated for the simplified setup, without assumptions or derivation. Real-world borrowing costs, margin rules, and constraints can change whether the theoretical allocation is feasible.
Key ideas
- A risky-asset weight above one means investing more than the investor’s initial capital in that asset.
- A negative risk-free weight represents borrowing at the risk-free rate to finance the extra investment.
- Weights between zero and one divide capital between the risky asset and risk-free asset.
- The document’s judgment about negative risky-asset weights applies to its simplified setup and does not discuss real-world constraints.
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Full text
# How to use mean-variance weights in practice (when going short is allowed)? # How to use mean-variance weights in practice (when going short is allowed)? I have calculated my optimal portfolio weights following the mean-variance framework where I go $w_1$ in the risky asset and $1-w_1$ in the risk free rate. I get the following result: $w_1$ = 1.5, 1-$w_1$ = -0.5 My question is, how can I interpret this? Obviously I need to short the risk-free rate and go long in the risky asset. However, if I have 100 dollars to use for my portfolio, how do I allocate this amount in practice? Moreover, it should follow that I go long with 150 dollars and short 50, but the question is, where does this money come from? How is it possible that I get money when I go short? One option would be to standardize the sum of the weights. Hence this is not possible as then you never will invest in the risk-free rate. ## Answer by Akavall (score 2) https://quant.stackexchange.com/a/7193 In your set up where you have just two assets, risky asset and risk-free asset, where weight of risky asset is w1, and consequently weight of risk-free asset is 1 - w1: w1 = 1, => You invest all your money in the risky asset. w1 = 0, => You invest all your money in the risk-free asset 0 < w1 < 1, => You invest some of your money in the risky asset and some of it in the risk-free asset w1 > 1, => Your case : you invest all you money (100) in the risky asset, plus you borrow (50) at risk free-rate and invest it in the risky asset. w1 < 0, => implies that you short the risky asset to invest in the risky free rate, this behavior is irrational, because we can find combinations of risky asset and risk-free asset that yield higher mean return at lower variance, thus w1 < 0, should never happen. ## Answer by Bob Jansen (score 1) https://quant.stackexchange.com/a/7191 You borrow the money against the risk free rate.
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