Why Mean-Variance Portfolio Weights Need Not Sum to One
Summary
This exchange explains that mean-variance optimization does not inherently require portfolio weights to sum to one. That condition is an added modeling choice, commonly used to represent allocating all available capital to long-only positions, or to require short positions to be financed by longs. When those assumptions do not fit, other constraints—or no fixed sum constraint—may be appropriate.
The answer uses currency positions and household balance sheets to illustrate why an investor’s total exposures can differ from one unit of capital. It also distinguishes a portfolio’s mathematical optimum from practical restrictions imposed by the investor. The discussion does not provide a crisis-specific optimization method or establish that all assets falling implies every position should be short; the right constraints depend on capital, leverage, shorting, and financing assumptions.
Key ideas
- Mean-variance optimization does not impose a universal sum of weights constraint.
- A weights sum of one can represent fully investing a fixed capital base in long-only assets.
- Long-short portfolios may use a sum constraint to link shorts to long financing.
- Portfolio constraints should reflect the investor’s capital, leverage, and financing rules.
- Falling asset prices alone do not determine the appropriate portfolio weights.
Tags
Full text
# What if all the weights are negative in mean-variance optimization during a crisis? # What if all the weights are negative in mean-variance optimization during a crisis? Usually the constraint is that all weights sum up to 1. But in a crisis when all assets are falling in prices, intuitively, all the weights should be negative in the optimization. But it contradicts with the constraint. Should I adjust the constraint to something like the sum is greater than -x and smaller than 1? Thanks in advance! ## Answer by demully (score 2) https://quant.stackexchange.com/a/50876 Mean-Variance (MV) optimal simply gives you MV optimal. This is bounded between negative infinity to positive infinity. Any constraints on this you night wish to apply to this are at your supplemental discretion. Sum(W) = +1 is simply the traditional constraint to enforce a long-only portfolio; or applied to a long-short where shorts have to be financed by longs. Imagine I was in positioning cash to different currencies: looking at EURUSD (risky), USDJPY (conservative), GBPUSD (risky) and AUDUSD (risky). I am long-only and unlevered, allocating my cash between these five currencies. I would then be negative across the board, summing to -1, across the board if I was risk-seeking! First-time buyers are habitually 3-500% long of their home, their biggest asset on their household balance sheet. Before they even start looking at and adding in their investment portfolio, their weights do not equal 1. They massively exceed that (imaginary) constraint. So why the constraint of sum(W) = 1? This Wizard of Oz applies if you have 1 capital to allocate between competing - and unlevered, no shorting - demands on your capital. Absent those conditions, sum(W) is bounded by plus or minus infinity ;-)
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.