Lab Test: Portfolio theory and asset pricing · RiskAtoms Academy

How it works

Eight problems from the people who use portfolio theory: a pension scheme's analyst, RiskAtoms Group's risk team, its CRO and its CIO, a wealth manager, an equity analyst, a strategist on the bank's trading desk and a manager selector. Most need several steps and more than one lesson. Answer each part (type numbers only; the unit is shown beside the box), press Check, then step through the worked solution. Numeric answers are marked right within a small tolerance, so round sensibly as you go. Keep a calculator handy.

Illustrative dataRiskAtoms Group's investment portfolio

The group invests £1bn of surplus assets across five asset classes. The expected returns, volatilities and correlations are round, made-up long-run assumptions in pounds, a year, not estimates from market data. The risk-free rate (a one-year Treasury bill) is 3.0%. The same numbers run through the whole module.

Asset class	Expected return	Volatility	Sharpe ratio
Global equities	7.0%	16%	0.250
Emerging-market equities	8.5%	22%	0.250
Government bonds	3.5%	6%	0.083
Corporate bonds	4.5%	8%	0.188
Property	6.0%	18%	0.167

Problems 1, 2, 3, 7 and 8 use these assumptions. The share in Problem 4, the factor portfolios in Problem 5 and the managers in Problem 6 are made up too.

Problem 1Equities, credit and the lowest-risk mix

L3.1.01 · L3.1.03 · L3.1.04 · L3.1.05

Investment analyst at a UK defined-benefit pension scheme

The scheme's growth portfolio holds two funds. On the module's long-run assumptions, global equities are expected to return 7.0% with volatility 16%, sterling corporate bonds 4.5% with volatility 8%, and their correlation is 0.35. The trustees need an expected return of 6.25% a year. One of them also asks how low the risk of the two funds can go.

Lessons: L3.1.01: Portfolio return and risk · L3.1.03: Two correlated assets · L3.1.04: The minimum-variance portfolio · L3.1.05: The efficient frontier

(a)What volatility does the mix that targets 6.25% carry? Answer in %.
%
(b)What weight in global equities gives the lowest possible volatility? Answer in %.
%
(c)What is that lowest volatility? Answer in %.
%
(d)A trustee wants to hold only corporate bonds, “the safest option”. Another warns that in a crisis the correlation could rise to 0.6. Which statement is right?
AAll corporate bonds is the lowest-risk choice, because bonds are the less volatile fundBAll corporate bonds is dominated by the minimum-variance mix; at a correlation of 0.6 the minimum-variance mix would be short equities, because 0.6 is above σ₂/σ₁ = 0.5CAt a correlation of 0.6 diversification stops working and any equities add riskDThe minimum-variance weight depends only on the two volatilities, not on the correlation

Problem 2A proposed mix, the frontier and a crisis matrix

L3.1.05 · L3.1.06 · L3.1.07

Investment risk analyst, RiskAtoms Group

The investment committee proposes a mix of 30% global equities, 50% government bonds and 20% property, on the module's assumptions. Expected returns are 7.0%, 3.5%, 6.0% and the covariance matrix (built as Σ = DRD from volatilities 16%, 6%, 18%) is:

\Sigma = \begin{pmatrix} 0.02560 & -0.00096 & 0.01728 \\ -0.00096 & 0.00360 & 0.00108 \\ 0.01728 & 0.00108 & 0.03240 \end{pmatrix}

For the three assets with short sales allowed, the frontier constants of L3.1.07: Mean-variance optimisation are A = 13.736, B = 0.59013, C = 342.68 and D = 13.536.

Separately, a consultant suggests “crisis correlations” for a stress test: equities and government bonds −0.3, equities and property 0.9, and government bonds and property 0.3.

Lessons: L3.1.05: The efficient frontier · L3.1.06: Many assets and the covariance matrix · L3.1.07: Mean-variance optimisation

(a)What is the volatility of the proposed mix? Answer in %.
%
(b)What is the lowest volatility on the frontier (short sales allowed) for the same expected return? Answer in %.
%
(c)Keeping −0.3 and 0.9, what is the highest government bonds–property correlation for which the crisis matrix is still a valid correlation matrix? Give 3 decimals.
(d)What goes wrong if the consultant's matrix, with 0.3, is used anyway?
ANothing: every correlation is between −1 and 1, so the matrix is validBSome long-short mix of the three assets shows a negative variance, and an optimiser would pile into that seemingly riskless positionCOnly the volatilities change, so portfolio risk is overstated but never negativeDThe matrix can be used if its determinant is negative, because that means the assets hedge each other

Problem 3One risky portfolio for every client

L3.1.03 · L3.1.07 · L3.1.08

Wealth manager at a private bank

Your firm builds every client's portfolio from two funds and Treasury bills. The global equity fund is expected to return 7.0% with 16% volatility, the corporate bond fund 4.5% with 8%, and their correlation is 0.35. Bills pay 3%, and clients may borrow at the same rate. So:

\boldsymbol\mu - r_f\mathbf 1 = \begin{pmatrix} 0.040 \\ 0.015 \end{pmatrix}, \qquad \Sigma = \begin{pmatrix} 0.02560 & 0.00448 \\ 0.00448 & 0.00640 \end{pmatrix}

Lessons: L3.1.03: Two correlated assets · L3.1.07: Mean-variance optimisation · L3.1.08: The risk-free asset and the capital market line

(a)What is the weight of the equity fund in the tangency portfolio? Answer in %.
%
(b)What is the Sharpe ratio of the tangency portfolio? Give 3 decimals.
(c)A cautious client accepts at most 6% volatility. What is the highest expected return she can get? Answer in %.
%
(d)An adventurous client wants an expected return of 7%. What should you offer?
AEverything in the equity fund: 7% at 16% volatilityBBorrow about 48p for every £1 and hold £1.48 of the tangency portfolio: 7% at about 14.7% volatilityCA risky mix with more equities than the cautious client's, and no borrowing, because adventurous clients need a different risky portfolioDThe minimum-variance mix of the two funds, geared up, because it has the least risk

Problem 4What is the share worth?

L1.5.03 · L3.1.09 · L3.1.10

Equity analyst covering UK industrials

You are valuing a UK-listed engineering company (made-up). Its returns have volatility 28% a year and a correlation of 0.65 with the global equity index, which plays the market: expected return 7%, volatility 16%. The risk-free rate is 3%. Next year's dividend is expected to be £0.48 a share, growing 3% a year for ever. The shares trade at £9.60. Your house adjusts betas with Blume's rule, 0.67 × raw + 0.33.

Lessons: L1.5.03: Valuing equities: the dividend discount model · L3.1.09: CAPM, beta and the security market line · L3.1.10: Using and testing CAPM

(a)What is the share's raw beta? Give 2 decimals.
(b)What is the cost of equity with the Blume-adjusted beta? Answer in %.
%
(c)What is the share worth with the dividend discount model at that cost of equity? Answer in £.
£
(d)Where does the share sit against the security market line at £9.60?
ABelow the SML: its price implies a return under the 7.37% CAPM requires, so it looks overpricedBAbove the SML: at £9.60 it offers about 8.0%, more than CAPM requires, so it looks underpriced (a positive alpha)COn the SML: in CAPM every share is priced correctly, so the gap must be a data errorDAbove the SML, so it should be sold: its large specific risk is not rewarded

Problem 5An arbitrage across factor portfolios

L3.1.11 · L3.1.12

Portfolio strategist on RiskAtoms Bank's trading desk

The desk's factor model, fitted by time-series regressions, explains returns with two factors: surprises in GDP growth (F₁) and in interest rates (F₂). Three well-diversified portfolios have negligible specific risk. The risk-free rate is 3% and the desk can trade £20,000,000.

Portfolio	b₁ (growth)	b₂ (rates)	Expected return
X	1.2	0.4	9.6%
Y	0.6	1.0	7.5%
Z	0.9	0.7	9.3%

Lessons: L3.1.11: Factor models and Fama-French · L3.1.12: Arbitrage pricing theory

(a)Using X and Y, what is the premium λ₁ for growth risk? Answer in %.
%
(b)What expected return should Z offer under APT? Answer in %.
%
(c)What riskless profit does the arbitrage earn in a year on a £20,000,000 position in Z? Answer in £.
£
(d)Which trade locks it in?
ABuy £20m of Z; sell £10m of X and £10m of YBSell £20m of Z; buy £10m of X and £10m of YCBuy £20m of Z and hold it, because its expected return is the highest of the threeDBuy £20m of Z and sell £20m of X

Problem 6Which manager?

L3.1.08 · L3.1.11 · L3.1.13

Manager-selection analyst at a pension scheme

Three active equity managers pitch for a mandate. Over the same five years the index returned 9% a year with 15% volatility, and the risk-free rate was 3%. Their figures, a year (made-up):

Manager	Style	Average return	Volatility	Beta	Tracking error
North	Low-beta, broadly spread	8.1%	9.5%	0.60	6.7%
South	About 30 high-conviction picks	13.1%	20.4%	1.10	12.1%
East	Index exposure geared with futures	11.6%	21.1%	1.40	6.2%

Lessons: L3.1.08: The risk-free asset and the capital market line · L3.1.11: Factor models and Fama-French · L3.1.13: Measuring performance

(a)What is North's M² (its Sharpe ratio expressed at the index's volatility)? Answer in %.
%
(b)What is South's Jensen's alpha? Answer in %.
%
(c)What is East's information ratio against the index? Give 2 decimals.
(d)South's Fama-French three-factor regression gives loadings of 1.05 on the market, 0.7 on size (SMB) and 0.4 on value (HML). Over the five years the three factors averaged 6%, 2% and 2.5% a year. What is South's three-factor alpha? Answer in %.
%
(e)The scheme will hold the winner as one of twelve managers in a broad equity portfolio. Which reading is best?
ANorth, because it has the best Sharpe ratio and M²BEast, because it has the best information ratioCJudge on Treynor or alpha, where South leads; but only 1.4% of its 3.5% alpha survives the three-factor model, and five years cannot tell even that from luckDSouth, because it had the highest raw return

Problem 7Where the risk sits

L2.5.06 · L3.1.06 · L3.1.14

Chief risk officer, RiskAtoms Group

RiskAtoms Insurance holds £600m of investments: 50% government bonds, 30% corporate bonds and 20% property. On the module's assumptions (volatilities 6%, 8%, 18%; correlations 0.7, 0.1 and 0.4) the covariance matrix is:

\Sigma = \begin{pmatrix} 0.00360 & 0.00336 & 0.00108 \\ 0.00336 & 0.00640 & 0.00576 \\ 0.00108 & 0.00576 & 0.03240 \end{pmatrix}

The board asks where the risk comes from. VaR is parametric: one year, 99% (z = 2.3263), mean zero.

Lessons: L2.5.06: Value at risk: the idea · L3.1.06: Many assets and the covariance matrix · L3.1.14: Risk contributions and risk budgeting

(a)What is the portfolio's volatility? Answer in %.
%
(b)What share of the portfolio's volatility does property contribute? Answer in %.
%
(c)What is property's component VaR? Answer in £ millions.
£m
(d)The investment team proposes moving £30m (5% of the portfolio) from government bonds into property. Using the marginal contributions, by how many percentage points would volatility change?
points

Problem 8How much to trust the inputs

L3.1.01 · L3.1.07 · L3.1.15

Chief investment officer, RiskAtoms Group

You are setting a strategic mix of global equities and government bonds. A consultant offers expected returns estimated from 20 years of annual data; equities' volatility is 16%. You prefer Black-Litterman. The market weights are 60%/40%, the market's expected excess return is 3%, and the covariance matrix is the module's:

\Sigma = \begin{pmatrix} 0.02560 & -0.00096 \\ -0.00096 & 0.00360 \end{pmatrix}

You hold one view: government bonds will beat cash by 1.5% a year, with 50% confidence. Use τ = 0.05 and set Ω so that a lone view gets weight equal to its confidence, as in L3.1.15: Estimation error, shrinkage and Black-Litterman.

Lessons: L3.1.01: Portfolio return and risk · L3.1.07: Mean-variance optimisation · L3.1.15: Estimation error, shrinkage and Black-Litterman

(a)How wide is a 95% interval for equities' mean return estimated from 20 years? Give the half-width (the ± part) in %.
%
(b)What is the implied (equilibrium) excess return of global equities, Π? Answer in %.
%
(c)What is government bonds' posterior expected excess return? Answer in %.
%
(d)What happens to global equities' expected excess return in the posterior?
ANothing: the view is only about bondsBIt rises: better bonds lift the whole marketCIt falls slightly, to about 4.65%, because equities and bonds are negatively correlated, so good news for bonds reads as slightly bad news for equitiesDIt falls to zero, because the view replaces the equilibrium

Score 0 / 330 of 8 problems checked

Back · L3.1.15

Estimation error, shrinkage and Black-Litterman

Next module · L3.2.01 · coming soon

What a trading desk does