Systematic and Unsystematic Risk, and What Spreading Does

Systematic risk is the part of a portfolio's variability that comes from the market itself, and adding holdings does not remove it. Unsystematic risk is specific to one holding, and spreading across holdings that do not move together does reduce it. Diversification therefore has a floor: it works only to the extent holdings differ, and it stops at the market exposure underneath them.

Almost every conversation about diversification skips the second half of that. Somebody counts the holdings, the count sounds large, and the room relaxes. Volatility, variance and correlation are settled under their own headings and are applied here rather than explained again.

The running example is the Anantara Multi-Asset Portfolio, an invented Rs 500 crore discretionary mandate run by Faiz Ahmad Ansari for a charitable endowment whose committee is chaired by Rukmini Deshpande. Its policy weights, set in advance: equity 60.0 per cent at Rs 300 crore, fixed income 30.0 per cent at Rs 150 crore, cash 10.0 per cent at Rs 50 crore. Every figure below belongs to one stated twelve month period.

One thing before the arithmetic. Volatility is symmetric: deviations are squared, so a rise and a fall of the same size count exactly alike. Volatility measures the width of the spread, and never the chance of losing money.

Its neighbours carry the rest: portfolio return and risk settles that the covariance term is the only place spreading enters the arithmetic, and capital market expectations that correlation is the least stable input. The split between what spreading removes and what it cannot decides how much a committee's limits could ever achieve.

What is systematic risk, and why does adding holdings not remove it?

Systematic riskThe part of an investment's variability that comes from conditions affecting the whole market at once, such as interest rates, growth or currency. It is also called market risk or undiversifiable risk. is the part of a holding's variability that comes from conditions affecting everything at once. Rates move. Growth expectations move. The currency moves. Choosing a different business avoids none of it. The other business stands in the same weather.

The everyday version: twelve shops outside one office block, twelve genuinely different businesses. The block closes for a month and all twelve lose the same customers on the same morning. The variety along the street was never carrying the risk. The office block was.

Add another holding that also moves with the market and you add more of exactly what the spreading was meant to remove. Systematic risk therefore survives any number of holdings. Two hundred such names are not two hundred small exposures cancelling one another. The set is one large market exposure with two hundred labels on it.

The finance version: William Sharpe's single index model splits a holding's return into a part that moves with the market, scaled by that holding's sensitivity, and a residual part that does not. Add the holdings up. The residual parts, being unrelated, partly cancel. The market parts are the same condition with different multipliers in front of it, so the market parts do not. The market parts add, the residual parts partly cancel, and that is the whole mechanism of diversification.

Unsystematic Risk: what does spreading actually remove?

Unsystematic riskThe part of an investment's variability that comes from things happening to that one holding rather than to everything at once. Also called specific risk, idiosyncratic risk or diversifiable risk. is the variability that belongs to one holding and to nothing else. A contract is lost. A plant stops for three weeks. A chief executive resigns on a Wednesday. None of these is weather; each happened to one name.

Here is the part people carry away wrongly. Spreading reduces this kind of variability for one reason: these events are not synchronised. Some holdings have a bad week while others have a good one, the two partly offset, and the total moves less than the parts did. The entire benefit rests on that lack of synchronisation and nothing else. Diversification is a property of the relationship between holdings, never of the count of them.

The household version lands harder. One salary funds a savings deposit, a monthly equity plan and a home in the town where that employer is the largest, and if the employer contracts all three go wrong in the same month for the same reason. Three product names is not three drivers, and a portfolio of twenty-eight names is not twenty-eight drivers either.

How Diversification Changes Portfolio Risk, and where does the benefit stop?

Take the simplest case. A sleeve holds a number of names, all equally weighted, all with the same volatility, every pair carrying the same average pairwise correlationThe typical correlation between any two holdings in a set, taken as a single summary figure. Real portfolios have a different correlation for every pair; the average stands in for all of them.. None of that is true of a real portfolio, and the point survives the simplification anyway.

The portfolio variance then has exactly two terms. The first is the individual variance divided by the number of holdings. The second is the average pairwise correlation multiplied by the individual variance, multiplied by one less one over the number of holdings. Watch what each term does as the count grows.

The first term shrinks toward zero. At one holding it is the whole individual variance; at four it is a quarter of it; at a hundred it is a hundredth. The second term does the opposite: it starts at zero when there is one holding, and it climbs toward the average correlation times the individual variance and stays there. The first term vanishes and the second does not, so the total does not run down to nothing, it runs down onto the second term and stops.

Six figures carry the whole argument. At an average correlation of 0.30 the variance runs 1.000, 0.650, 0.440, 0.370, 0.325 and 0.307 as the count goes 1, 2, 5, 10, 28 and 100. The first holding added takes 0.350 off it; the ninety after the tenth take 0.063. One over the number falls fastest when the number is small. The first handful of holdings therefore does nearly all the work.

Volatility is the square root of those variances, and volatility is what a committee paper prints. Still at a correlation of 0.30, that multiple reads 1.000 at one holding, 0.608 at ten and 0.554 at a hundred, above a floor of 0.548. The ninety holdings after the tenth bought about a seventh of what the first nine bought, and no arrangement of them could have bought more.

Now put a real number on the benefit. The mandate's own stated assumptions, chosen by the holder: equity at 12.0 per cent expected return and 18.0 per cent volatility, fixed income at 7.5 and 5.0, cash at 6.0 and 0.5, with a correlation of 0.20 between equity and fixed income and cash taken as uncorrelated.

At the policy weights of 60, 30 and 10 the variance is built from four pieces. Equity contributes 0.36 times 324, or 116.6400. The cross term between equity and fixed income contributes twice 0.6 times 0.3 times 0.20 times 18 times 5, or 6.4800. Fixed income contributes 0.09 times 25, or 2.2500. Cash contributes 0.01 times 0.25, or 0.0025. Add them: 125.3725. The square root is 11.196986 per cent, carried in the record as 11.20 per cent.

If the three parts moved perfectly together, the volatility would simply be their weighted average: 0.6 times 18 plus 0.3 times 5 plus 0.1 times 0.5, giving 10.8 plus 1.5 plus 0.05, or 12.35 per cent. The actual figure is 11.196986 per cent. A gap of 1.153 points, carried in the record as 1.15, is the diversification. The gap opens only at a correlation of 0.20 rather than 1.00. Set that correlation to 1.00 and the gap closes to nothing.

Where is the floor, and why is counting holdings the wrong question?

Push the count to infinity in the equally weighted expression and the first term disappears entirely, leaving the average pairwise correlation times the individual variance. Its square root is the diversification floorThe lowest volatility an equally weighted portfolio can reach by adding more of the same kind of holding. It equals one holding's volatility multiplied by the square root of the average pairwise correlation.: one holding's volatility multiplied by the square root of the average pairwise correlation.

That expression is the reason counting holdings answers the wrong question: the count does not appear in it anywhere. Five hundred names at a correlation of 0.60 sit above a floor of 0.775 times one holding's volatility; twenty names at 0.10 sit above 0.316 times it. The second is spread and the first is not, and the count pointed the wrong way both times.

One warning about the control's default. The record locks a beta of 1.08, a benchmark volatility of 10.4 per cent and a portfolio volatility of 11.8 per cent for the stated year. The three together give a portfolio against benchmark correlation of 1.08 times 10.4 divided by 11.8, or 0.9519. The figure of 0.9519 is not an average pairwise correlation between the twenty-eight equity names. No such correlation appears in the record, and none is supplied in its place.

Work that default in words before touching it. At a correlation of 0.9519 the curve reads 1.000 at one holding, 0.978 at ten and 0.976 at a hundred, against a floor of 0.976: a hundred holdings still leave 97.6 per cent of one holding's volatility. Drag the control to 0.30 and the three readings become 1.000, 0.608 and 0.554 against a floor of 0.548.

How much of this portfolio was market exposure in the stated year?

One number belongs first in any conversation about spreading and is almost always computed last. Three inputs settle it, all invented and all for the one stated year: the beta of 1.08, the benchmark volatility of 10.4 per cent and the portfolio volatility of 11.8 per cent.

The systematic part of the portfolio's volatility is the beta times the benchmark's volatility: 1.08 times 10.4, or 11.232 per cent. Squared, that is the systematic variance, 126.157824. The portfolio's own variance is 11.8 squared, or 139.24. Divided out, 126.157824 over 139.24 is 0.906046, so 90.6 per cent of the portfolio's variance in that year was market exposure and 9.4 per cent was everything else.

The beta had already decided nine tenths of that year's variability before a single holding was chosen, leaving about a tenth for any diversification decision to touch. None of that is an argument against spreading. The argument is for knowing the size of the market block before a committee meeting is spent on it.

Read those inputs a second way, and a misunderstanding stops before it starts. Ninety point six per cent of the variance is not 90.6 per cent of the volatility: the systematic volatility of 11.232 per cent is 95.2 per cent of the portfolio's 11.8, and the residual volatility is the square root of 13.082176, about 3.62 per cent. Squares add; the volatilities do not, and 11.232 plus 3.62 is nowhere near 11.8.

One more thing, or a reader will make a wrong connection later. The residual volatility of 3.62 per cent is not the tracking error, even though the record's 3.7 per cent looks like the same number rounded. The residual is the part of the portfolio's movement not explained by the beta-scaled benchmark; tracking error is the spread of the portfolio's return minus the benchmark's, one for one.

Tracking error is fixed by the other three figures rather than chosen alongside them, so the identity is worth showing. Tracking error squared is the portfolio variance plus the benchmark variance less twice the beta times the benchmark variance: 139.24 plus 108.16 less 233.6256, or 13.7744. The square root is 3.7114 per cent, printed in the record as 3.7. Change any one of the volatility figures or the beta and the tracking error must move with it, so a record showing four independently chosen numbers here is a record with an error in it.

How Concentration Risk Builds in a Portfolio, and what does a holdings list miss?

Concentration riskThe risk that comes from a portfolio depending heavily on a small number of positions or on one shared condition, so that a single event moves an outsized share of the whole. builds in three ways: a single holding grows large, or a group turns out to share an exposure so that ten lines behave as one, or the tail is too small to matter and the portfolio has the administration of many names and the behaviour of few.

Every line can sit comfortably inside its cap and the group behind them can still move together. The second way is the one a holding level report cannot see. A list of twenty-eight positions shows twenty-eight compliant weights. The shared exposure is a fact about the set rather than any one name, and a report organised one line at a time has no row to put it in.

The street version makes that middle case obvious. A landlord's rent roll shows ten tenants in one mall, none above a tenth of the rent. The anchor store closes, footfall halves, and all ten struggle in the same quarter. Nothing on the roll was wrong; it simply had no column for the mall.

What does a cap on a single holding actually prevent?

The Anantara mandate carries a single holding capA stated limit on how much of a portfolio may sit in any one holding. It is written against a named base, usually the whole portfolio, and it binds one line at a time. of 5 per cent of the portfolio, an equity band of 50 to 70 per cent, a ban on unlisted holdings and a minimum credit standing on the fixed income sleeve. All four are that mandate's own limits, invented for teaching.

Run the arithmetic rather than admiring the limit. Equity is 60 per cent of the portfolio and the cap is 5 per cent of it, so 60 divided by 5 is twelve. The cap alone permits the entire equity sleeve to sit in twelve names. A cap bounds the worst single case and describes no spread at all. A portfolio can satisfy every cap every day of the year and still be concentrated.

Set that permitted shape beside the shape actually held. The equity sleeveThe part of a portfolio held in equities, treated as a group. A sleeve has its own base, so a weight measured against the sleeve is a different number from the same weight measured against the whole portfolio. of Rs 300 crore sits across 28 names, so equal weighting would put 2.14 per cent of the portfolio in each, against the largest holding's 4.6 per cent and a cap of 5.

Size the cap against something a committee can feel. Losing the largest holding entirely costs 4.6 per cent of the portfolio, about 47 per cent of the 9.7 per cent worst drawdown it took in the stated year, peak to trough inside that window. A different window gives a different figure, and the window is quoted every time for that reason. The cap is doing real work, and it is not the same work as diversification.

Why does one concentration fact produce two numbers that sound nothing alike?

Because a weight is a fraction, and a fraction has a denominator that somebody chose. The largest holding is 4.6 per cent of the Rs 500 crore portfolio, or Rs 23,00,00,000/-. The same Rs 23,00,00,000/- is 7.7 per cent of the Rs 300 crore equity sleeve. Neither figure is wrong; they answer different questions.

The group behaves the same way. The top ten holdings are Rs 155 crore: 31.0 per cent against the portfolio, 51.7 per cent against the sleeve, a little over half the sleeve in ten names out of twenty-eight. Naming the base beside every weight is the cheapest defect prevention in this subject, and an account that moves between the two without saying which has misled its reader about concentration.

Here is a check that catches a wrong concentration figure with no extra data at all. Suppose the ten largest holdings out of twenty-eight came to less than ten twenty-eighths of the sleeve. The average of the ten largest would then sit below the average of all twenty-eight, and holding eleven would have to be larger than holding ten. So the ten largest can never fall below ten twenty-eighths of the sleeve. Ten twenty-eighths is 35.71 per cent. Any top ten figure below 35.71 per cent of the sleeve is arithmetically impossible whatever else the report says, and that test can be run in the head before anybody opens a position file.

The recorded 51.7 per cent clears that floor comfortably: ten names are 35.7 per cent of the count and 51.7 per cent of the money, a ratio of 1.45 that no choice of base can move.

The top ten average Rs 15.50 crore each, 3.10 per cent of the portfolio apiece, against Rs 8.06 crore and 1.61 per cent for each of the other eighteen. The sleeve is really two groups, the leading ten carrying about twice the weight per name of the eighteen behind them.

What does the record support here, and where does it stop?

The record runs out at a specific place. The record carries enough for the floor check, the group comparison and every weight against a named base, but not enough for an effective number of holdingsA single figure that says how many equally sized holdings a portfolio behaves like. The figure is computed from every individual weight, so group totals alone cannot produce it.. The individual weights inside each group are not recorded.

The effective number needs the sum of the squares of all twenty-eight weights, and no group total produces it: a group of ten can be ten identical holdings or one large one and nine small ones, and those give very different answers. All the record supports is that the effective count sits below twenty-eight, and producing a substitute would be inventing the very figure that ought to be demanded.

The curve above is constructed for the same reason. The record locks no correlation matrix across the twenty-eight names and no per-name returns, so using the 0.9519 where an average pairwise correlation belongs would be making up the most important input in the subject.

What does diversification never protect against?

Three things. Diversification does not remove market exposure, put by the arithmetic above at 90.6 per cent of the stated year's variance. Spreading does not help when correlations rise together. And spreading is not protection against loss in any period.

The second deserves a moment. The benefit comes from holdings not being synchronised, and conditions that frighten everybody at once synchronise them. The benefit therefore shrinks in exactly the weeks it was counted on, and a portfolio designed on a comfortable correlation is run on a worse one when it matters. The arithmetic said so all along, in the term that carries the correlation.

The third, stated once and left alone. The deviations are squared before they are averaged, so volatility counts a rise and an equal fall alike. A narrower spread means outcomes bunched more tightly around the middle. A statement about the chance or the size of a fall is a different statement altogether, and no diversification arithmetic ever produced protection against losing money.

How does a committee actually use this on a Tuesday?

Three numbers and one question, read in that order before the holdings file is opened. The share of variance that was market exposure comes first, and it tells the room how large the subject under discussion really is. Every weight then carries its base, so nobody silently compares 31.0 against 51.7. The floor check takes five seconds and has already caught one error here. The question comes last because the numbers cannot answer it.

A lender does the same thing when it looks at a borrower with several revenue lines and asks how many customers those lines depend on. An analyst does it when a company's segments all sell to the same five buyers. A household does it with a pen: write down what each savings decision depends on, and if the same word appears three times, the count of products was never the count of drivers. Stop counting the lines and start counting what the lines rest on.

Whose ideas are these, and where do they sit?

Two attributions belong here. Harry Markowitz's insight is that a combination's variance rests on how its parts move together rather than on the parts one at a time. The cross term above is the only part of the arithmetic that responds to spreading, for exactly that reason. The split of a return into a market part and a residual is William Sharpe's single index model.

A third is a pointer only. The relationship giving an expected return at a stated beta was developed independently by William Sharpe, John Lintner, Jan Mossin and Jack Treynor in the middle 1960s. Here the beta is used to size an exposure and no further; splitting the year's gross excess of 1.6 points into an exposure part and a residual belongs to the risk-adjusted measures.