Portfolio construction

1.Two uncorrelated assets have variances 4 and 1. The long-only minimum-variance portfolio that is fully invested puts what weight on the higher-variance asset?

2.Those same two assets are uncorrelated and the fully invested minimum-variance weight on the low-variance asset is four fifths. What is that portfolio's variance?

3.A book is long 0.5 of a name with beta 1.2 and short 0.5 of a name with beta 0.8. There is no index hedge. What is the net beta?

4.A single risky asset has excess return 0.08 and variance 0.04. Mean-variance utility is excess return minus (lambda/2) times variance, and the weight may exceed 1 by borrowing. The optimal weight is excess return divided by lambda times variance. At lambda = 4, the weight is what?

5.The same asset and the same utility, with lambda doubled from 4 to 8. What happens to the optimal risky weight?

6.All risky assets have the same excess return, and a risk-free asset is available. The maximum-Sharpe mix of the risky assets is which portfolio?

7.One asset has twice the excess return of the other. Volatilities match and correlation is zero. Relative to the minimum-variance mix, the maximum-Sharpe mix does what?

8.Equal weight and minimum variance coincide for two assets in which case?

9.A long-only book never rebalances. In a rally led by one name, that name's weight does what?

10.A fixed-mix rule sells toward the target after that rally. Relative to buy-and-hold, the trade is what?

11.Idiosyncratic variances are uncorrelated across names, and each name also loads on one common factor. As an equal-weight book adds more names, which piece of variance shrinks?

12.A risk model leaves out a factor that the names share. The residuals from the model will show what?

13.The sample covariance of N assets over T days, with T less than N, is what?

14.A factor model or a ridge is applied to that singular covariance before an optimizer runs. What is the optimizer being stopped from doing?

15.Historical means of 100 assets are estimated on 60 months and fed to a mean-variance optimizer with no shrinkage. The weights tend to do what?

16.Implied excess returns in a reverse optimization are proportional to covariance times market weights. If the market is the mean-variance portfolio, those implied returns are what?

17.A view says one asset's excess return is higher than the reverse-optimized value, and the view is blended with the market-implied returns. The new weights do what, relative to the market?

18.Risk contributions are the pieces of portfolio variance attributed to each name, summing to portfolio variance. A risk-parity book sets them how?

19.Correlations are all equal, and volatilities differ. Inverse-volatility weights then do what for risk contributions?

20.Correlations are not equal. Inverse-volatility weights then fail to be risk parity because of what?

21.A name has five times the volatility of the others, equal weight, and similar correlations. Its share of portfolio risk is what, qualitatively?

22.Portfolio volatility is 10 percent. A name's marginal contribution to that volatility, times its weight, summed over names, equals what?

23.Tracking error is the standard deviation of what?

24.Active share is one half the sum of absolute active weights. A portfolio that matches the benchmark exactly has active share what?

25.Two portfolios have the same active share against a benchmark. One's active bets are in names that move almost with the names they replace. The other's bets are in a different sector. Which tends to have the larger tracking error?

26.The information ratio is active return divided by tracking error. A constraint that the optimizer did not want, and that binds, does what to the implemented information ratio relative to the unconstrained paper ratio, before costs?

27.The fundamental law writes the information ratio as skill times the square root of breadth, times a transfer coefficient. Breadth counts what?

28.Doubling the number of names while keeping them perfectly correlated does what to breadth?

29.A turnover cap binds against the unconstrained trade. The cap was not in the objective, and trading costs were ignored in the paper Sharpe. What is a fair reading of the paper Sharpe?

30.Costs are put inside the objective as a penalty on turnover, and the optimizer then chooses smaller trades. Relative to optimizing with no cost and haircutting afterward, this does what?

31.A position cap binds on the optimizer's favorite name. The leftover weight goes where?

32.A sector cap binds, and the best name in that sector is already at the cap. The optimizer still wants sector exposure. It will do what?

33.Gross exposure is the sum of absolute weights. A book that is 100 percent long and 100 percent short has gross and net exposure of what?

34.A 150/50 book is 150 percent long and 50 percent short. Its net exposure is what?

35.Volatility scales linearly with leverage, and variance scales with the square. A book levered from 1x to 2x, with no financing friction, has variance multiplied by what?

36.In a stress window, pairwise correlations move toward 1. A risk model estimated on a calm window, where correlations were 0.2, will have done what to stress portfolio volatility?

37.Variance treats a large gain and a large loss of the same size as the same squared deviation. A conditional-value-at-risk objective, the expected loss in the tail, differs because of what?

38.A drawdown constraint depends on the path, not only on the mean and variance. An optimizer that sees only mean and variance cannot enforce it unless what is added?

39.Gaussian value-at-risk uses a multiple of portfolio volatility. If the true loss distribution has a heavier tail than a Gaussian, that value-at-risk does what?

40.Lending is at 2 percent and borrowing is at 5 percent. The usual single tangency portfolio, which assumes one risk-free rate, does what?

41.Short sales are banned. The minimum-variance portfolio relative to the unconstrained one does what?

42.Cash is riskless in the model. In a plot of risk and expected return, cash sits where?

43.The part of the risky frontier below the minimum-variance portfolio is called inefficient because of what?

44.A benchmark-relative book uses active weights. Those active weights sum to what, if the portfolio and the benchmark are both fully invested?

45.The beta of a portfolio against a factor is the weighted sum of the name betas, with portfolio weights. A hedge that sells factor exposure equal to that beta does what?

46.Residual risks are uncorrelated across names. Their contribution to portfolio variance is the sum of weight squared times residual variance. Spreading a fixed gross over more names does what to that sum, if no name's residual variance blows up?

47.A covariance estimated on a very short window treats a chance difference in sample volatility as a precise fact. The optimizer then does what?

48.Shrinking the sample covariance toward a structured target, such as a constant-correlation or a factor model, is done to do what?

49.The diversification ratio is the weighted average of standalone volatilities divided by portfolio volatility. It equals 1 when what is true?

50.A name with a negative correlation to the current book is added in a small long weight. Near zero weight, that addition does what to portfolio risk?

51.The same name is highly correlated with the book. A small long addition does what?

52.Integer share lots are imposed on a continuous optimizer solution. The rounded book does what relative to the continuous one?

53.A rebalance band does nothing until a weight drifts past a threshold, then it trades back. Relative to trading every day back to the exact target, the band does what?

54.An impact cost that rises with the square root of dollars traded is put in the objective. As the book's assets grow, optimal participation in a thin name does what?

55.The objective ignores impact, and the same thin-name weight is applied at a much larger asset size. The implemented book then faces what the optimizer did not see?

56.Kelly sizing for a repeated edge maximizes long-run growth and leads to large drawdowns. A mean-variance desk with a high risk aversion, relative to full Kelly, does what?

57.The average of many optimized portfolios, each fit on a resampled mean and covariance, is not the same as optimizing once on the average mean and covariance. Why?

58.A long-only constraint and a sector constraint both bind. The shadow price of a constraint is what?

59.Two constraints bind, and one has a shadow price near zero. Relaxing the near-zero one does what?

60.A portfolio is split into a benchmark holding plus an active long-short overlay. The overlay's dollar net, when both pieces are fully invested, is what?

61.The benchmark is the market portfolio in a model where the market is efficient. An active overlay with a positive expected return in that model implies what?

62.Expected returns are estimated with so much noise that the desk sets them all equal and optimizes only the covariance. The resulting risky portfolio is what?

63.The desk instead trusts the means and sets the risk aversion near zero. The fully invested solution does what?

64.A factor-neutrality constraint forces the book to have zero exposure to a factor the signal was accidentally loading on. The expected return of the book after the constraint, if part of the signal was that factor, does what?

65.Specific risk is diversified, and factor risk is not, as N grows in an equal-weight book. A risk limit written only on specific risk will do what as N grows?

66.The covariance between the portfolio and a new name, divided by portfolio variance, is the name's beta to the portfolio. A name with that beta equal to 0.5, added in small size, contributes marginal risk in proportion to what?

67.A mean-variance problem is stated in dollars of active weight with a quadratic penalty. Doubling every active weight doubles expected active return and multiplies active variance by 4. The information ratio of the scaled book does what?

68.Costs are proportional to turnover, and turnover scales with the active-weight scaling. Once costs are subtracted, the net information ratio does what as the book is scaled up?

69.A covariance is forecast one day ahead with a shrinkage intensity chosen to minimize the previous day's error, and that same intensity is then frozen for a later month. Choosing the intensity on the previous day and freezing it is what kind of use?

70.The optimizer's objective uses a covariance forecast that itself was fit on the same window as the means, and the reported in-sample Sharpe uses those same days. That Sharpe is what?

71.A robust version maximizes the worst expected return over a set of means inside a box around the point estimate. Relative to optimizing the point estimate, the robust solution does what?

72.A constraint says the portfolio variance is at most a budget. The solution lies where, when the budget binds?

73.The variance budget is loosened, and the shadow price of the cap was positive. Expected return of the optimal book does what?

74.Names are grouped, and a constraint forces each group's weight to match the benchmark group weight. The active book can still differ how?

75.That within-group active book has zero group weight and a positive expected return from a name signal. Its contribution to the group factor, if the group factor is the group itself, is what?

76.A portfolio's Herfindahl, the sum of squared weights, is 0.2. An equal-weight book with that Herfindahl has how many names?

77.A second equal-weight book has a Herfindahl of 0.01. How many names is that?

78.The first of those two books, Herfindahl 0.2, and the second, Herfindahl 0.01, are compared on concentration. Which is more concentrated?

79.A risk model uses a single index. The residual of the portfolio is the return minus beta times the index. Hedging with a futures position of size equal to beta times portfolio value does what to the index piece?

80.The futures hedge is sized to yesterday's beta, and today's beta has jumped because the book traded. Using yesterday's hedge today leaves what?

81.A long-only portfolio and a long-short portfolio are built from the same name scores. The long-only book cannot underweight a name it does not like except by holding zero. Relative to the long-short book, its transfer of the score into weights is what?

82.The benchmark already holds a name the signal dislikes. Long-only can underweight it down to zero, which is an active weight of minus the benchmark weight. A long-short mandate with no short ban can underweight it further. The extra room matters when?

83.Expected active return is 2 percent and tracking error is 4 percent. The information ratio is what?

84.That information ratio of 0.5 and a breadth of 16 independent bets, with a transfer coefficient of 1, are plugged into the fundamental law in the form IR = IC × square root of breadth. The implied skill, IC, is what?

85.The same numbers, but the transfer coefficient is 0.5 because of constraints. If the formula is IR = IC × sqrt(breadth) × TC, and the measured IR is still 0.5, the IC consistent with that formula is what?

86.A covariance forecast is multiplied by 2 by mistake, and means are unchanged. Minimum-variance weights, which do not use means, do what?

87.The same doubled covariance is used in a mean-variance problem that also has a linear mean term and a fixed risk aversion, with weights free to borrow. The risky weights do what?

88.A portfolio is rebalanced to target weights at the close, using closing prices to compute the trades, and the trades are assumed to fill at those same closes. The assumption fails when?

89.Target weights are computed from yesterday's close, and the rebalance trades at today's open. That timing is what?

90.A sector-neutral constraint and a beta-neutral constraint are imposed together. A signal that is mostly a sector bet and a beta bet will produce a book that is what?

91.The signal is rewritten as a residual to sector and beta before the optimizer runs, and the optimizer has no neutrality constraints. Relative to constraining the optimizer, this pre-residualizing does what when the optimizer would also have imposed those constraints?

92.A name's weight is positive, its expected return is revised down, and no constraint binds on it. In a mean-variance book the optimal weight does what?

93.A name's variance forecast is revised up, and its expected return is unchanged. No constraint binds. The optimal weight does what?

94.Two names are substitutes, with correlation near 1, and the first has a slightly higher expected return. The unconstrained solution does what?

95.A position limit of a few percent is added to that pair trade. The limit does what?

96.Portfolio variance is w' Sigma w. If every weight is multiplied by k, variance is multiplied by k squared. If the desk quotes volatility and then doubles weights, the quoted volatility should be multiplied by what?

97.A benchmark return is subtracted inside the objective, and the covariance used is the covariance of active returns rather than total returns. This setup is aimed at what?

98.The same objective is run with total-return covariance by mistake, while the mean is still an active mean. The risk term then charges what it should not?

99.A name is added that is already in the benchmark at weight 3 percent. The portfolio holds it at 3 percent. Its active weight is what?

100.The portfolio holds that name at 5 percent against a 3 percent benchmark weight. The active weight is what?

101.Summing absolute active weights and dividing by 2 gives active share. For a single 2 percent overweight funded by a 2 percent underweight elsewhere, active share is what?

102.A risk budget says no name may contribute more than 5 percent of portfolio variance. A name that wants 20 percent of variance under the unconstrained solution is cut back. The freed variance budget is then what?

103.All name-level variance caps bind, and the sum of the capped contributions is below the risk the objective wants. The book then does what?

104.A covariance matrix has a negative eigenvalue in a computed file. Using it as a risk model allows what?

105.The fix for that negative eigenvalue, before optimizing, is what?

106.A book is optimized in dollars, then scaled to a volatility target by leveraging up or down. The volatility target is applied after the relative weights are chosen. This separates what?

107.That final scale assumes the covariance forecast is right. If the forecast volatility is half the realized volatility, the scaled book comes out how, relative to the target?

108.A desk runs the optimizer once a month and holds the weights. Within the month a name's price doubles and its weight drifts. The risk model used at the start of the month now does what?