Foundations · F5 · 9 min
Returns, risk and portfolios
Turn daily swings into yearly risk, estimate a bad day’s loss, and compare funds by their return for each unit of risk.
In the news: When markets get jumpy, or shares and bonds fall together, every desk’s risk numbers jump with them.
The essentials
- Volatility
- The typical size of an asset’s moves: the standard deviation of its returns, usually quoted per year. A share that moves 3% on a typical day is riskier than one that moves 1%.
- Square root of time
- Risk grows with the square root of time, because up and down days partly cancel. A year has about 252 trading days and √252 ≈ 16, so 1% a day is about 16% a year: the .
- Value at risk (VaR)
- The loss a position should exceed only on a set share of days. £10m with 1% daily volatility has a 95% one-day of about £165,000. It says nothing about how big losses beyond it get.
- Correlation and diversification
- runs from −1 to +1. Below +1, a mix swings less than the average of its parts. Correlations change: in 2022, rising interest rates hit shares and bonds together.
- Sharpe ratio
- Return above cash divided by volatility: the return earned for each unit of risk. A 9% return with cash at 4% and 10% volatility gives 0.5; higher is better.
- Beta
- How far a share moves with the market. A beta of 1.3 means about 1.3% for each 1% move in the index, so £10m of the shares behaves like £13m of the index.
Volatility and value at risk
σ_year = σ_day × √252; 95% VaR ≈ 1.65 × σ_day × V
- σ
- volatility: the typical size of moves (standard deviation of returns)
- √252
- about 16: the square root of the trading days in a year
- V
- the value of the position
How it works
1. Returns
Buy a share at £100 and sell it at £110: a 10% return. Returns let you compare investments of different sizes: £10 made on £100 is as good as £1,000 made on £10,000.
2. Volatility: how much it swings
Two shares can both average 8% a year, but one moves about 1% on a typical day and the other 3%. The size of those typical swings is : the standard deviation of returns, usually quoted per year. More volatility means more risk.
3. The square root of time
Up days and down days partly cancel, so risk grows with the square root of time, not with time itself. A year has about 252 trading days, and √252 ≈ 16. So 1% a day is about 16% a year, and 32% a year is about 2% a day: the .
4. Value at risk
Risk managers ask how much they could lose on a bad day, say the worst day in 20. If returns follow the usual bell shape, that is about 1.65 × daily volatility × the position. £10m with 1% daily volatility gives about £165,000. This is the 95% , or VaR.
5. What VaR misses
VaR says how often you should lose more than it, not how much more. It also leans on recent data, which is often calm, so it can look smallest just before trouble. Desks use it to set limits, alongside tests of what extreme moves would cost.
6. Mixing assets
If two assets don’t always move together, holding both is smoother than holding either. How closely they move together is , from −1 to +1. Below +1, a mix swings less than the average of its parts: 60% shares at 18% volatility and 40% bonds at 7%, uncorrelated, swing about 11%, not 13.6%.
7. When the cushion fails
For years, government bonds tended to rise when shares fell, which cushioned a mix of the two. In 2022, with inflation the main worry, interest rates rose fast and hit both, so they fell together. Correlations change, and diversification can fail just when it is needed.
8. Return for each unit of risk
A fund making 10% with wild swings can be worse than one making 7% calmly. The divides the return above cash by volatility: a 9% return with cash at 4% and volatility of 10% gives (9 − 4) ÷ 10 = 0.5. Higher is better.
9. Beta: moving with the market
Some shares move more than the market, some less. measures it: a beta of 1.3 means about 1.3% for each 1% the index moves. So £10m of those shares behaves like £13m of the index, and selling £13m of index futures removes the market risk.
Worked example
You run £50m: £30m of shares (18% volatility a year) and £20m of gilts (7%), with a correlation of −0.2. Cash pays 4% and the mix is expected to return 7%.
- Shares part: 0.6² × 18² = 116.64.
- Bonds part: 0.4² × 7² = 7.84.
- Mixing part: 2 × 0.6 × 0.4 × (−0.2) × 18 × 7 = −12.10.
- Volatility = √(116.64 + 7.84 − 12.10) = √112.38 = 10.6% a year, or 10.6% ÷ 15.87 = 0.668% a day.
- 95% one-day VaR = 1.65 × 0.668% × £50m = about £551,000.
- Sharpe ratio = (7% − 4%) ÷ 10.6% = 0.28.
Volatility about 10.6% a year, a 95% one-day VaR of about £551,000 and a Sharpe ratio of about 0.28.
Check yourself
9 quick questions on the explanation above.
Practice
New numbers every set. Misses and guesses go into your review deck.
In the interview
Answer out loud in under a minute, then compare.
What’s wrong with VaR as a risk measure?
Why might bonds stop hedging equities, and when did that last happen?
How would you compare two funds with different returns and volatility?
Remember
σ_year ≈ σ_day × 16
Risk grows with the square root of time, and diversification works only while correlations stay low.
Sources
Definitions and maths are standard textbook material; dated facts link to the sources above. Worked numbers are illustrative.