Solving for the Risky Allocation That Actually Fits You
Once you accept that a portfolio can be reduced to a risky sleeve plus a risk-free sleeve, the remaining question is not philosophical, it is arithmetic: exactly what fraction belongs in each. A single formula answers that question directly, rather than leaving it to an age-based rule of thumb.
The optimization problem behind every allocation
A portfolio built from a single risky asset (or fund) and a single risk-free asset moves along the capital allocation line described elsewhere in this series, with expected return E(r_C) = r_f + y × (E(r_P) - r_f) and standard deviation σ_C = y × σ_P. Every value of y along that line is achievable, but only one value of y is optimal for a given investor, the one that maximizes that investor's utility, using the mean variance utility formula covered earlier in this series: U = E(r_C) - 0.5 × A × σ_C².
Substituting the complete-portfolio formulas into the utility function and solving for the value of y that maximizes U produces a clean closed-form result: y* = (E(r_P) - r_f) / (A × σ_P²). Read left to right, this says the optimal risky weight rises with the size of the risk premium on offer, and falls with both your personal risk aversion and the riskiness of the asset itself. It is the single most useful equation in this entire area of portfolio construction, because it turns three separate, individually estimable inputs, a return assumption, a risk assumption, and a personal risk aversion number, into one specific allocation percentage.
Solving for the optimal risky weight
Suppose a risky portfolio has an expected return of 11% against a risk-free rate of 4%, so the risk premium is 7 percentage points, and its standard deviation is 18% (variance of 0.18² = 0.0324). For an investor with a risk aversion coefficient of A = 4, a fairly typical value for someone with a stable salary and a long horizon:
y* = 0.07 / (4 × 0.0324) = 0.07 / 0.1296 = 0.540, or roughly 54% in the risky portfolio and 46% in the risk-free asset.
Now take the identical risky portfolio and hand it to a less risk-averse investor with A = 2, perhaps someone earlier in their career with substantial future earnings still ahead of them:
y* = 0.07 / (2 × 0.0324) = 0.07 / 0.0648 = 1.080, or roughly 108% in the risky portfolio, meaning this investor's utility is actually maximized by borrowing a small amount, about 8% of their own capital, and investing more than 100% of their wealth in the risky sleeve.
That second result is worth sitting with. It is not a suggestion that everyone with low risk aversion should use leverage; real borrowing costs, covered elsewhere in this series, erode the appeal of any y* above 100%, and few individual investors have practical access to borrowing at anything close to the government's own short-term rate, so the honest, achievable version of this result for most people is simply "hold close to 100% in the risky sleeve," not "borrow to exceed it," and any investor with genuine access to low-cost borrowing should still discount the theoretical result by their actual spread over the risk-free rate before treating a computed y* above 100% as a real target rather than a curiosity of the math, and should still weigh the emotional difficulty of holding a leveraged position through the inevitable bad stretch that any risky asset eventually delivers, a cost the formula itself has no way of pricing in, and one that has ended more than a few leveraged strategies well before the math on paper said it should, forced closures that had nothing to do with the long-run expected return being wrong and everything to do with the investor's inability to sit through the interim drawdown. But it illustrates how directly the formula responds to risk aversion: halving A exactly doubled the optimal risky weight in this example, holding the risk premium and variance fixed, which is the formula behaving exactly as the underlying utility framework predicts.
A third example: what happens when the risk premium shifts
The formula's sensitivity to its inputs is worth seeing in action, because risk premium assumptions are exactly the kind of number that shifts across market cycles even when an investor's own risk aversion has not changed at all. Return to the first investor from above, with A = 4 and a risky portfolio with a standard deviation of 18%, but now suppose the expected risk premium is revised downward from 7 percentage points to 4.5 percentage points, reflecting a more cautious view of forward-looking equity returns after a strong multi-year run has already priced in a great deal of good news.
y* = 0.045 / (4 × 0.0324) = 0.045 / 0.1296 = 0.347, or roughly 35% in the risky portfolio, down sharply from the 54% computed earlier using a 7-point premium assumption, with the investor's actual risk aversion held completely constant. This is a useful check against a common source of confusion: a falling recommended equity allocation does not necessarily mean an investor has become more risk averse, it can equally reflect a legitimate downward revision to the expected reward on offer, and conflating the two leads to misdiagnosing why an allocation is changing.
What real-world glide paths reveal
Target-date retirement funds are, in effect, a pre-built approximation of this formula applied across a lifetime, even though most providers do not publish the underlying A they assume. A fund targeting a retirement date three decades out typically holds a y above 90%, consistent with either a low assumed risk aversion, a high assumed risk premium, or usually some blend of both, on the theory that decades of future contributions and recovery time reduce the effective cost of a temporary decline. As the target date approaches, published glide paths taper y down, often to somewhere between 30% and 50% at the retirement date itself, which is broadly consistent with the same investor's assumed A staying roughly constant while their effective time horizon, and therefore their tolerance for a multi-year drawdown, shrinks.
Where target-date providers differ meaningfully from each other, sometimes by 15 to 20 percentage points of equity exposure at the same target date, is a useful reminder that reasonable professionals disagree about the right inputs to this formula, particularly the long-run risk premium assumption and how quickly risk aversion should be treated as rising near retirement. There is no single "correct" glide path, only a range of defensible ones built on different assumptions.
Applying this to your own money
Applying the formula to your own situation starts with honest inputs, not precise-looking ones. A long-run equity risk premium assumption in the 4% to 6% range over Treasury bills is broadly consistent with long-run historical averages and a reasonable starting point absent a stronger view of your own. Your own volatility estimate should reflect the actual risky sleeve you plan to hold, a diversified global stock fund typically shows lower long-run volatility than a concentrated domestic growth fund. Your own A, per the earlier discussion in this series, is best estimated from how you actually behaved during a real past drawdown, not from a hopeful guess.
For a high earner with substantial, stable future income, a physician, senior engineer, or partner at a professional firm mid-career, the honest input for A is often lower than it feels in the moment, because a large share of that person's total wealth is really their future earning power, a bond-like, steady asset that is not sitting in the brokerage account at all. Ignoring that human capital and setting A based purely on emotional comfort with the visible portfolio balance tends to understate true risk capacity, sometimes substantially, for exactly this population.
There is a mirror-image case worth naming as well: a self-employed consultant or a commission-based sales professional with real income volatility from year to year has less stable human capital than the salaried employee example above, even at an identical account balance and an identical stated comfort with market swings. For this investor, the honest input is a higher effective A, or equivalently a larger allocation to the risk-free sleeve than the formula would suggest using return and volatility assumptions alone, because a bad year in the market arriving alongside a bad year in the business is a real, correlated risk that the two-asset formula on its own does not capture.
Actionable breakdown
- Gather honest inputs before computing anything.
- Use a long-run risk premium estimate, not a recent hot streak.
- Use the volatility of the actual fund you will hold, not a generic figure.
- Use an A estimated from past behavior, not aspiration.
- Compute y* and compare it to your current allocation.
- A large gap between y* and your actual holdings is worth investigating.
- Do not force an immediate jump to y*; move deliberately if the gap is large.
- Account for human capital explicitly if your income is stable and substantial.
- Treat a secure salary as a bond-like asset when setting true risk capacity.
- Revisit the inputs periodically, not the output percentage in isolation.
- Re-run the formula when rates, your wealth, or your career stability change.
Common pitfalls
The most common pitfall is treating the formula's output as more precise than its inputs deserve. A risk premium assumption that is off by even one or two percentage points, entirely plausible given how uncertain long-run forecasts are, moves y* meaningfully, so the right use of this formula is as a directional guide and a discipline check, not a number to follow to the decimal point.
A second pitfall is computing y* once and never revisiting it, even as risk-free rates, wealth, and career circumstances change materially over time. The formula is a snapshot, not a permanent setting.
A third pitfall, common among high earners with volatile but ultimately generous compensation, business owners, commissioned salespeople, is applying a standard risk aversion estimate without adjusting for genuinely lower income stability, which understates true risk capacity constraints and can push y* higher than someone in that position can actually sustain through a bad year for their business or industry. A fourth pitfall is applying a single portfolio-wide A across every goal an investor holds simultaneously, a retirement account, a child's education fund, a near-term home purchase, when each of these goals really carries its own effective time horizon and therefore arguably its own optimal y*, rather than one blended figure across the entire household balance sheet.
The bottom line
Your optimal split between a risky portfolio and a risk-free asset is a specific, computable number driven by the risk premium, the asset's variance, and your own risk aversion, not a round percentage borrowed from your age.
Related reading: building an asset allocation, funds and ETFs, quantifying your own risk aversion, the capital allocation line, translating this into a real world allocation.