The risk that comes with it
More equity buys more expected return and costs more volatility, but the line is bent. Translate the bend into a bad year in dollars, and the client can decide.
- Compute a two-asset portfolio's expected return and volatility
- Turn volatility into a one-in-twenty bad year, in percent and dollars
- Find the highest equity weight that keeps volatility at a target
- Explain why the first bonds remove more risk than return
The intuition
Tell a client their portfolio has 11% volatility and you will get a polite nod. Tell them that in a one-in-twenty bad year they should expect to be down about $124,000 and you will get a decision. The number is the same; the second version is one a person can react to. That translation is most of what this chapter is for.
Behind it is the two-asset formula. Return blends linearly: 60% of the equity return plus 40% of the bond return. Risk does not, because stocks and bonds do not move in lockstep. The variance carries a cross term scaled by the correlation, and when that correlation is below one the portfolio's volatility comes out below the weighted average of the two. That bend is why a little bond in an all-equity portfolio removes a lot of risk and very little return, and why the bad year is not simply 60% of an equity bad year.
E[R] = w × r_eq + (1 − w) × r_bond. σ² = w² σ_eq² + (1 − w)² σ_bond² + 2 w (1 − w) ρ σ_eq σ_bond. Bad year (1 in 20) = E[R] − 1.645 σ. Sharpe = (E[R] − r_f) ÷ σ. Run backwards, the variance formula is a quadratic in w: the higher root is the most equity a volatility target allows.
Why it works
- The conventions here: an equity/bond mix with stated expected returns, annual volatilities and one correlation. Returns are treated as normal for the bad year, so the one-in-twenty outcome is the mean less 1.645 volatilities. Sharpe uses the stated risk-free rate.
- Variances add; volatilities do not. The cross term is what the correlation controls. At a correlation of one the portfolio's volatility is the weighted average; below one it is lower; at zero the cross term vanishes and only the two squared terms remain.
- Equities dominate the risk long before they dominate the money. In a 60/40 mix with the usual volatilities, well over ninety percent of the variance comes from the equity sleeve. "Balanced" describes the capital.
- A bad year is a number, not a feeling. The mean less 1.645 volatilities is the return you would expect to do worse than about once in twenty years. In dollars it is a figure a client can say yes or no to. The caveat is fat tails: real bad years are worse than a normal distribution promises.
- Sharpe is return per unit of risk, not a verdict. A client who needs more return and can bear more risk may rationally accept a lower ratio; it becomes a mistake only when the same return was available more cheaply.
- Run it backwards and the client's limit sets the weight. A volatility the client can live with, put into the variance formula, gives a quadratic in the equity weight. The higher root is the efficient side of the curve.
| Expected return: 60% × 7% + 40% × 4% | 5.80% |
| Variance: (60% × 18%)² + (40% × 6%)² | 116.64 + 5.76 = 122.40 %² |
| Volatility: square root | 11.06% |
| Straight-line volatility, for contrast: 60% × 18% + 40% × 6% | 13.20% |
| Bad year: 5.80% − 1.645 × 11.06% | −12.40%, about −$124,000 |
| Sharpe: (5.80% − 2%) ÷ 11.06% | 0.34 |
With the correlation at zero the cross term is zero. The bend is the gap between 13.20% and 11.06%: two points of volatility the mix gives away for free.
The formulas
Return blends linearly.
Two squared terms and a cross term the correlation scales.
The return you would expect to do worse than about once in twenty years.
Excess return per unit of risk.
A = σ_eq² + σ_bond² − 2cov, B = 2cov − 2σ_bond², C = σ_bond² − σ*².
Worked example
One-in-twenty on the downside is 1.645 standard deviations below the mean. Multiply the volatility, subtract from the expected return, and turn the result into dollars.
See it move
Same client. Change the equity weight, each sleeve's expected return and volatility, the correlation between them, and the size of the portfolio.
- Raise the correlation. Volatility rises and the bad year gets worse; the expected return does not move, because correlation lives only in the cross term.
- Raise the equity weight. The expected return rises and, across the slider's range, so does volatility. The bad year can go either way: watch it.
- Raise the equity volatility. Portfolio volatility rises and the bad year gets worse; the expected return does not move.
- Set the correlation to zero and read the volatility curve. That is the two squared terms alone; move the correlation up and the whole curve lifts.
Run it backwards
The client names the volatility they can live with. Set the variance formula equal to the target squared and solve the quadratic for the equity weight, taking the higher root.
The variance formula is a quadratic in w once the target is fixed: A w² + B w + C = 0 with A, B and C built from the two volatilities and their covariance. The higher root is the efficient side of the curve, the most equity the limit allows.
The follow-up compares that weight with what the client holds: room to add, or a mix already running above the risk they asked for.
Traps
Say it in the interview
“How would you explain this portfolio's risk to a client?”
Check yourself
4 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- σ² = w² σ_eq² + (1 − w)² σ_bond² + 2 w (1 − w) ρ σ_eq σ_bond; the correlation lives in the cross term.
- Bad year = E[R] − 1.645 σ; say it in dollars.
- The first bonds remove risk fastest: the line is bent.
- Volatility target → quadratic in w → take the higher root.