###### Machine Learning Methods

After completing this reading, you should be able to: Discuss the philosophical and... **Read More**

**After completing this reading**,** you should be able to**:

- Define risk budgeting.
- Describe the impact of horizon, turnover, and leverage on the risk management process in the investment management industry.
- Describe the investment process of large investors, such as pension funds.
- Describe the risk management challenges associated with investments in hedge funds.
- Distinguish among the following types of risk: absolute risk, relative risk, policy-mix risk, active management risk, funding risk, and sponsor risk.
- Apply VaR to check compliance, monitor risk budgets, and reverse engineer sources of risk.
- Explain how VaR can be used in the investment process and the development of investment guidelines.
- Describe the risk budgeting process and calculate risk budgets across asset classes and active managers.

In this chapter, we explore the benefits of VaR to the investment management fund comprised of mutual funds, pension funds, endowment funds, and hedge funds. VaR is useful in measuring, controlling, and managing underlying risk. It also accounts for leverage, diversification, and volatility.

Risk budgeting is a comprehensive, top-down approach that involves measuring, decomposing, allocating, and managing portfolio risk to maximize the returns of an investment.

It is the investment equivalent of capital budgeting in corporate finance. This is because it involves allocating risk (or risk capital) to investments as opposed to allocating dollar capital to projects.

Risk budgeting provides a top-down, hands-on, and extensive method for managing risk. Further, it makes explicit the current risk profile and acknowledges that it is dynamic.

Banks make up the “sell-side” of the investment industry. They developed VaR and have been using it for many years. On the other side, investors make up the “buy side” of the investment industry. The use of VaR has spread more slowly to the investment management industry. This can be explained by the difference in many core aspects of investment management from the fast-paced trading environment of dealing with banks.

The sell sides, i.e., banks, have a short horizon, rapid turnover, and high leverage. Additionally, the sell side uses VaR and stress tests to measure risk. They also apply position limits, VaR limits, and stop-loss rules to control risk. Banks’ high leverage makes it vital to control their risks.

Conversely, on the buy-side, i.e., investors have a longer horizon, slow turnover, and lower leverage. The buy-side applies asset allocation and tracking errors to measure risk. To control risks, investors utilize diversification, benchmarking, and investment guidelines.

$$ \begin{array}{l|c|c|c|c} \textbf{Feature} & \textbf{Banks (“sell-side”)} & \textbf{Investors (“buy-side”)} \\ \hline \text{Horizon} &\text {Short term (intraday or a day)} & \text{Long-term (months or more)} \\ \hline \text{Turnover} & \text{Quick} & \text{Slow} \\ \hline \text{Leverage} &\text {High} & \text{Low} \\ \hline \text{Risk measures} & \text{VaR} &\text{Asset allocation}\\&\ \text{Stress tests}&\text{Tracking error} \\ \hline \text{Risk controls} & \text{Position limits} & \text{Diversification}\\ & \text{VaR limits} & \text{Benchmarking}\\&\text{Stop-loss rules}&\text{Investment guidelines} \end{array} $$

The investment process involves two steps:

- The first step is for the consultant to provide a strategic and long-term asset allocation study. The first step aims at balancing expected returns and risks using mean-variance portfolio optimization. This step helps in determining the amount to be invested in the available asset classes.
- The second step is where the real management of funds is assigned to a group of active managers, whose performance is measured periodically in terms of tracking error. The active manager’s role is to manage the portfolio to outperform the benchmark portfolio actively. To control risk, there must be conditions that define the entire group of assets to be invested in. These conditions may include restrictions in duration, deviations from the equity sector weights, and amounts of foreign currency to cross hedge. Typically, the risk is measured using historical data.

Investments are becoming more global, complex, and dynamic. VaR models can be utilized to measure the total risk of such investments as it is a straightforward, transparent, and consistent risk measures.

Hedge funds are a heterogeneous group of investment assets that employ different strategies to earn active returns. Some hedge funds are leveraged and have a higher turnover and thus may be more similar to the “sell-side” of the investment industry. The following are some of the risk management challenges with investments in hedge funds:

Hedge funds are not listed on any exchange. Furthermore, investment in hedge funds is restricted to high-net-worth individuals. This creates low investment volumes exposing hedge funds to liquidity risk.

Several hedge funds do not disclose information about their positions with the presumption that potential competitors might use it against them. This makes it difficult for clients to measure the risk of their investment.

Some hedge funds invest in illiquid assets. These assets are traded less frequently. Therefore, risk measures do not reflect recent transaction prices, thus giving an inaccurate risk outlook. This creates biases, including:

- Low systematic risk resulting from a lowered correlation with other assets.
- Low overall risk as a result of lowered volatility.

Risk is defined as the chance of occurrence of loss relative to the expected return. It is measured in base currency. There are various types of risks, as discussed below.

The absolute risk, also referred to as asset risk, is the possibility of a dollar loss over the horizon, and its rate of return is:

$${ \text{R} }_{ \text{asset}}=\sum{{w}_{i}{R}_{i}}$$

Where:

\(w_i\) is the weight of asset \(i\).

\(R_i\) is the asset return.

Relative risk is defined as the dollar difference between the portfolio return and benchmark portfolio return. It is measured by the tracking error.

Tracking error is the excess return of the asset over the benchmark. It is given by the formula:

$$ \text{TE}={\text{R} }_{\text{asset}}-{\text R}^{\text b} $$

The tracking error is generally referred to as the extra return of the asset over the benchmark.

Policy mix risk is the possibility of the base currency loss in relation to the benchmark selected by the fund. This type of risk reflects a passive strategy since it can be implemented by investing in passive funds.

Policy mix risk is the largest source of risk relative to active management VaR and asset VaR as the most risk in portfolio performance can be attributed to the choice of a mix of assets (stocks and bonds).

The active management risk is the sum of profits or losses from all active managers in relation to the benchmark. It results from individual managers deviating from the set weights for different assets.

The absolute risk can be expressed in terms of the portfolio mix risk and asset management risk as per the following equation:

$$ { \text{R} }_{ \text{asset} }={ \text{R} }_{ \text{policy mix} }+{ \text{R} }_{ \text{active mgt} }=\sum _{ \text{i} }^{ }{ { \text{w} }_{ \text{i} }^{ \text{b} }{ \text{R} }_{ \text{i} }^{ \text{b} } } +\sum _{ \text{i} }^{ }{ \left( { { \text{w} }_{ \text{i} }{ \text{R} }_{ \text{i} }-\text{w} }_{ \text{i} }^{ \text{b} }{ \text{R} }_{ \text{i} }^{ \text{b} } \right) } $$

Where \({ \text{R} }_{ \text{i} }^{ \text{b} }\) is the return on the benchmark for fund \(i\) and \({ \text{w} }_{ \text{i} }^{ \text{b} }\) is the portfolio weight.

We can find the total VaR of the fund from the policy mix VaR, active management VaR and the cross product.

Active management risk is minimal for well-managed funds, as explained by diversification through a prudent choice of various styles or many fund managers and investing in indexed or closely indexed funds.

Funding risk is the possibility of loss arising when the value of assets is insufficient to cover the liabilities of the fund. In this scenario, risks are looked at in terms of assets and liabilities.

Surplus is defined as the difference between the value of assets and liabilities. Change in the surplus is given by:

$$ \Delta { \text{S} }= \Delta { \text{A} }- \Delta { \text{L} } $$

When we normalize by the initial value of assets we get:

$$ { \text{R} }_{ \text{s} }=\cfrac { \Delta \text{S} }{ \text{A} } =\cfrac { \Delta \text{A} }{ \text{A} } -\left( \cfrac { \Delta \text{L} }{ \text{L} } \times \cfrac { \text{L} }{ \text{A} } \right) ={ \text{R} }_{ asset }-{ \text{R} }_{ liabilities }\left( \cfrac { \text{L} }{ \text{A} } \right) $$

Where \({ \text{R} }_{ \text{liabilities} }\) is the rate of return on the liabilities.

Surplus at risk (SaR) occurs when the surplus is **negative**. The fund sponsor is obliged to provide additional contributions if the surplus is negative.

ABC Public Service Fund has assets worth $500 million and liabilities worth $450 million. Suppose that the expected return on the surplus, scaled by assets is 5%. The volatility of the surplus is 12%.

Calculate the VaR and the deficit that would occur in case there is a loss associated with the VaR at the 99% confidence level after one year.

The surplus is expected to grow by:

Expected surplus growth = \(5\%×500=$25 \text{ million}\)

Expected surplus in a years’ time = \($500m-$450m+$25m=$75m\)

$$VaR=Z_c×σ×P=2.33×0.12×$500=$139.8m$$

Deficit= Expected Surplus – VaR = \($75m -139.8m = -$64.8m\)

This means that there is a 1% chance that the surplus will turn into a deficit of $64.8 million or more over the next year.

The risk profile of a firm is defined as the tradeoff between the deficit and the expected surplus growth.

The following information relates to Grimond Employees Retirement Fund (in $ million):

$$ \begin{array}{l|c|c} \textbf{Item} & \textbf{Value} \\ \hline \text{Assets} & {$600} \\ \hline \text{Liabilities} & {$480} \\ \hline \text{Return on assets} & {6.0\%} \\ \hline \text{Volatility of assets} & {8\%} \\ \hline \text{Return on liabilities} & {3\%}\\ \hline \text{Volatility of liabilities} & {2\%}\\\hline \text{Correlation between assets and liabilities} & {0.50} \\ \hline \end{array} $$

Calculate the 99% surplus at risk of the fund.

Expected surplus growth =\(($600×6\%)-($480×3.0\%)=$21.6\)

Expected surplus=\(($600-$480)+$21.6=$141.6\)

To calculate the variance of the surplus, recall:

\(Variance (V)=X_A^2 V_A+X_B^2 V_B+2X_A X_B ρ_{AB} σ_A σ_B\)

In this case:

$$V(A-L)=A^2 V_A+L^2 V_L-2ALρ_{AL} σ_A σ_L$$

$$V=600^2×0.08^2+480^2×0.02^2-2×600×480×0.50×0.08×0.02= 1935.36$$

The volatility of surplus growth = \(\sqrt{1935.36}=43.99\)

Recall:

$$VaR =-μδt+σZ_α \sqrt{σt}$$

Similarly,

$$SaR_{99\%}= -21.6+43.99×2.33=$80.90$$

The concept of surplus can be applied to the risk to the bearer of the pension fund, that is, the plan sponsor. The plan sponsor is the one who ultimately bears responsibility for the pension fund. Therefore, we need to know the difference between the following risks:

**The cash flow risk**: This is the risk arising from yearly changes in the contribution to the pension fund. Owners of a pension fund who can absorb substantial variations in funding costs can adopt a more volatile risk profile.**The economic risk**: This is the risk that arises when the financial earnings of the plan sponsor fluctuate.

From the fund’s sponsor’s perspective, the risk is measured by movements in assets, surplus, and the effect on the firm’s economic value.

VaR system is vital in enabling the investors to check that their managers follow the guidelines and manage the market risks.

VaR can be used to catch rogue traders in large firms. These are managers who delegate investment decisions to other active managers. Additionally, it can be utilized to identify unauthorized trading in and out of positions. This is because huge deviations from stated policies can easily be caught using VaR systems due to a centralized repository for all investments

Investors can easily manage market risk by use of the VaR system. Active portfolio management changes the risk appearance of the fund. For example, if the investor realizes an untimely increase in the fund’s VaR, the possible cause of such a change needs to be identified.

The following questions must be considered:

**Is the manager taking more risk**? Managers can monitor dynamic risk with a risk budget using VaR. In case of an exceedance of the budget line, the VaR limit will be flagged and examined. Suppose it is an unauthorized trade, the fault can be corrected, or a discussion with the manager might be necessary.**Are different managers taking similar bets**? This arises from cases where the managers increase allocations in more attractive sectors. In this case, to minimize portfolio risk, managers should be given appropriate instructions.**Are the markets more volatile**? VaR increases with high volatility. It is the work of the plan sponsor to decide whether or not to accept the higher volatility. An increase in volatility is a result of a decrease in the assets causing higher expected returns.

We can **reverse engineer VaR** to understand where the risk emanates. For example, component VaR can be used to identify individual positions with the largest effect on the overall portfolio risk given that the risk management system captures all the risks.

Centralized risk management is the principle behind VaR. The easiest way of centralization is by using one global custodian. Numerous investors accumulate their portfolios under a single custodian. This leads to a clearer consolidated picture of the overall fund exposure.

However, not all investors might be for the idea of aggregating portfolio holdings under one custodian. Therefore, some plans have decided to develop an internal risk management system. In this case, the claim is that control over risk measures is strict.

When it comes to managing money, the manager is compelled by the clients to assure them of a sound risk management system. The clients will ask for risk analysis, supposing that there is dissatisfaction with the quarterly performance report. To avoid a competitive disadvantage, managers must have a comprehensive risk management system.

VaR systems can be used to enhance investment guidelines for active managers, manage risks, and help in the investment process.

Managers’ guidelines that limit notionals and sensitivities are insufficient in the presence of leverage and new instruments. This is because they fail to account for risk variations and correlations. Additionally, these guidelines focus on individual positions.

Well-designed VaR systems can replace such ad hoc guidelines because VaR limits account for risk, leverage, diversification, and derivatives. Further, VaR limits are comparable across assets.

The most crucial step in the investment process is the strategic asset allocation decision. This decision is based on the mean-variance framework to identify the portfolio with the best risk-return trade-off.

VaR can also be used to allocate funds across assets since it is consistent with the mean-variance framework. Marginal VaR can be used to select new assets to add to a portfolio. If we have two assets with the same expected return, the one with the lowest marginal VaR should be selected provided that the goal is to achieve the lowest portfolio risk.

Moreover, the excess return to the marginal VaR ratio can be used to decide whether to increase the allocation of an existing asset over another. In this case, the asset with the higher ratio is increased.

Risk budgeting is a **top-down process **that involves the following steps:

- Setting risk budgets on the quantity of risk associated with each portfolio component. (asset class, portfolio manager, and security)
- Determining asset allocations based on the risk budgets
- Continually comparing the risk budgets to the risk measures due to each factor.
- Adjusting the asset allocations to maintain the risks within the risk budget

Risk budgeting across asset classes should incorporate **diversification effects**.

A pension fund wants to allocate $1500 million to only two asset classes. The fund’s board of trustees has settled on a total volatility profile for the fund of 12%. The following table shows the estimated volatilities of the three asset classes from which the fund manager can choose.

$$ \begin{array}{l|c|c} \textbf{Asset} & \textbf{Volatility} \\ \hline \text{A} & {16\%} \\ \hline \text{B} & {18\%} \\ \hline \text{C} & {10\%} \\ \hline \end{array} $$

The fund manager decides to invest $750 million in asset class A and the rest in either B or C, whichever maintains the fund VaR below $300 million at the 95% confidence level. Assume that the correlation between A and B is 0.6 while A and C are uncorrelated.

Which of the two asset classes, B and C, should the fund manager invest the rest of $750 million in, to keep the fund within its risk budget?

**Solution**

The portfolio VaR composed of A only is determined as:

$$VaR_P=Z_c σP$$

$$VaR_A=1.65×16\%×750=$198 \text{million}$$

The volatility of the portfolio after adding asset B is:

$$Volatility =\sqrt{X_A^2 V_A+X_B^2 V_B+2X_A X_B\rho_{AB} σ_A σ_B }$$

$$σ_{A+B}=\sqrt{(0.5)^2 (0.16^2)+(0.5^2)(0.18^2 )+(2)(0.5)(0.5)(0.6)(0.16)(0.18)}=15.21\%$$

$$VaR_{A+B} =1.65×15.21\%×1500\approx$376\text{million}$$

The volatility of the portfolio after adding asset C is:

$$σ_{A+C}=\sqrt{(0.5)^2 (0.16^2)+(0.5^2)(0.1^2 )+(2)(0.5)(0.5)(0)(0.16)(0.10)}=9.43\%$$

$$VaR_{A+C}=1.65×9.43\%×1500\approx$233\text{million}$$

We can see that asset class C keeps the total portfolio risk within the $300 million risk budget.

Active managers are evaluated using their **tracking error** (TE), which is given by the difference between the **active return** and the benchmark return. The tracking error is then used to determine the **information ratio, **used to assess managers’ performance.

The information ratio is given by the formula:

$$IR=\frac{µ}{w}$$

Where:

\(μ\) is the expected excess returns.

\(w\) is the tracking error volatility (TEV).

An IR of 0.50 means good performance. Managers with excellent performance are allocated a higher risk budget. The goal of the active manager is to maximize total portfolio IR with respect to the tracking error volatility (TEV) constraint.

Therefore, the added value for the portfolio P is given by:

$$μ_p=∑_{i}x_{i}μ_{i}=∑_{i}x_{i} (IR_{i} \times w_{i})$$

Where:

\(x_i \)= Proportion allocated to manager \(i\).

\(w_i\) = Tracking error of manager \(i\).

\(μ_i\) = Excess return.

\(μ_p\) = Value added for the portfolio \(p\).

If the excess returns for the managers are independent, the portfolio TEV is fixed at:

$$w_p=\sqrt{∑_{i}x_i^2 w_i^2}$$

After maximizing *IR* in relation to the constraint (TEV), we get:

$$x_i w_i=IR_i (\frac{1}{IR_p}w_p)$$

Therefore, to achieve optimal allocation to managers, weights can be allocated to managers using the formula:

$$x_i=\frac{IR_i}{IR_p} \frac{w_p}{w_i}$$

Where:<br>

\(x_i\) = Proportion of portfolio managed by manager \(i\).

\(IR_i\) = Information ratio for manager \(i\).

\(IR_P\) = Portfolio information ratio.

\(w_p\) = TEV of the portfolio.

\(w_i\) = TEV of the manager.

The relative risk budgets are proportional to the IR.

Assume that a pension fund wants to allocate $600 million to two active fund managers with the aim of maximizing the IR of the fund with a TEV of 3% as the constraint.

$$ \begin{array}{l|c|c|c|c}\textbf{}&\textbf{TEV\( w_i\)}&\textbf{Information Ratio \(IR_i\)} & \textbf{Weight \(x_i\)} & \textbf{Excess Return \( x_iμ_i\)} \\ \hline \text{Manager A} & {5\%} & {0.56} & {42\%} & {1.7\%} \\ \hline \text{Manager B} & {5\%} & {0.44} & {33\%} & {0.7\%} \\ \hline \text{Benchmark} & {0\%} & {0.00} & {25\%} & {0.0\%} \\ \hline \text{Portfolio} & {3\%} & {0.80} & {100\%} & {2.4\%} \end{array} $$

At the 95% confidence level:

Risk budget \(=Z_c σP=1.65×3\%×$600m=29.7m\)

From the table, notice that the managers have a TEV of 5% each. To achieve a TEV of 3%, some investment should be made in the benchmark. The benchmark has a TEV of 0.

The active managers have different information ratios. This implies that they have different capabilities. Given the expected excess return on the portfolio as 2.4%, the information ratio is determined as follows:

$$IR_P=\frac{\text{Expected return on the portfolio}}{\text{Portfolio TEV}}$$

$$IR_P=\frac{2.4\%}{3\%}=0.80$$

The information ratio of the portfolio is higher than that of each of the individual managers. This can be explained by the substantial diversification effects from the independence assumption of active returns.

The optimal weight allocated to each manager is calculated as:

$$x_i=\frac{IR_i}{IR_p}\frac{w_p}{w_i}$$

$$x_A=\frac{0.56}{0.80}×\frac{3\%}{5\%}=42\%$$

$$x_B=\frac{0.44}{0.80}×\frac{3\%}{5\%}=33\%$$

Difference =100%-(42%+33%)=25%

The 25% difference is invested in the benchmark.

## Practice Question

Don Parker is evaluating the employees’ pension fund which reports total assets at $23.4 billion and total liabilities as measured by an independent actuary at $16.1 billion.

If the surplus has a normal distribution with volatility of 15% per annum, what will the 95% surplus at risk (SaR) be over the next year?

A. $2.16 billion.

B. $1.81 billion.

C. $5.79 billion.

D. $3.98 billion.

The correct answer is

B.The fund’s surplus is the excess of assets over liabilities, which is:

$$ $23.4-$16.1=$\text{7.3 billion}$$

Surplus at risk at the 95% level over one year is:

$$ 1.65\times $7.3=$\text{1.81 billion}$$