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risk-free rate of return can be defined as

the rate at which money is borrowed or lent when there is no credit risk, so that the money is certain to be repaid.
Assets issued by national governments in their own currency were traditionally assumed to be “risk free”, and so the risk-free rate was often referred to as the “Treasury rate”.


In practice, and depending on the particular context, the risk-free return might therefore be interpreted as the rate of return offered by:

  Treasury bills 

  short-term conventional government bonds 

  long-term conventional government bonds 

  index-linked government bonds. 
This assessment was based on the thinking that national governments can always print money to ensure that their debts will be paid. Thus the concept of national debt being risk-free was only valid in a closed economy, where all financial transactions are in the currency of that economy. This feature was seen in much of the early part of the twentieth century, where large economies of China and the Soviet Union were effectively closed to outside influence. 
However, as recent events have demonstrated, the ability of governments to print money is a route not available to governments that use a shared currency (such as the euro), or those who wish to tie their exchange rate closely to something else (like the dollar). 
However in practice at the current time, all economies interact with each other, and therefore currency risk needs to be taken into account. Printing money deflates its value against other currencies, and thus the holder of overseas government debt may not receive a return that is free of currency risk, or is risk free in real terms if the inflation rates in the overseas territory and in the domestic economy of the holder differ. 
For this reason the main rating agencies assign a credit rating to the government debt of most countries, generally assessed against the US Dollar. One key measure to be considered is the ratio of outstanding debt to GDP, as well as the maturity of the existing debt, since this will determine the cost of servicing the debt and the need to raise refinancing capital as existing debt matures. 


Asset liability modelling

Perhaps the most important practical actuarial technique for use in determining an investment strategy nowadays is asset liability modelling.
Asset models and the fundamentals of Asset Liability Modelling were discussed in the Core Reading for Subject A301.


investor’s objectives will normally be stated with reference to both

assets and liabilities. In setting an investment strategy to control the risk of failing to meet the objectives a method of taking into account the variation in the assets simultaneously with the variation in the liabilities is required. This can be done by constructing a model to project the asset proceeds and liability outgo into the future.
In other words, by constructing and using an asset liability model.
The outcome of a particular investment strategy is examined with the model and compared with the investment objectives. The investment strategy is adjusted in the light of the results obtained and the process repeated until the optimal strategy is reached.
The optimal strategy being that which best meets the objectives of the investor.
Models can either be deterministic or stochastic. In both deterministic and stochastic models it is important that the assumptions made in constructing the model are consistent with each other.
It is essential to ensure that the assumptions and parameter values underlying both the assets and the liabilities are consistent if sensible results are to be obtained. In practice, this may be achieved by modelling of the appropriate dynamic links between assets and liabilities. For example, in many models inflation is assumed to influence both the asset returns and the liabilities.


Advantages of asset liability modelling

encourages investors to formulate explicit objectives. The objectives should include a quantifiable and measurable performance target, defined performance horizons and quantified confidence levels for achieving the target.


a typical objective might be to:

  maximise the expected solvency level 

  at the end of a 3-year period 

  subject to the probability of insolvency at any time over that period being no more than 0.1% 
For an institution such as a pension fund or insurance company the objectives might be specified in terms of the results of a valuation carried out at a specified time in the future. In practice there is likely to be feedback between the model output and the setting of the objectives. This may be because the additional insight provided by the model projections may help the investor to refine its investment objectives. 
The success of the strategy is monitored by means of regular valuations. The valuation results will be compared with the projections from the modelling process and adjustments made to the strategy to control the level of risk if necessary. The model itself may also be refined in the light of how the actual experience compares to that which it predicted.


Deterministic asset liability modelling

A deterministic model is based on a single set of assumptions about future experience. Using the assumptions we project the asset proceeds and liability outgo for each future time period. We can then compare the incidence of income and outgo to check that the outgo can always be met. We will also want to know the value of assets and the value of liabilities at each future time period according to one or more valuation bases.


Stochastic asset liability modelling

In contrast to a deterministic model, a stochastic model treats the key parameters as random variables with a given mean and a defined probability distribution. For example, future inflation might be defined by a normal distribution with mean of 5% and standard deviation of 2%, and real interest rates (ie the excess of nominal short- term interest rates over inflation) might be distributed between –2% and +8%, say.


What assumptions would be required in an asset model used to estimate the future income stream and asset values of an investment portfolio?



In practice, we might develop a stochastic asset model in which each of the key variables is modelled as a random variable, typically depending upon

  past values of itself 

  present and past values of other economic and/or investment variables. 
Having run the model a large number of times, we would scan the results from each simulation to gauge how likely it is for an unsatisfactory result to emerge. For example, we might define an unsatisfactory result as the company becoming insolvent. 
If say seven out of 1,000 simulations resulted in insolvency at some point over the projection period, then we might conclude that the probability of insolvency or ruin over that period is about 0.7%. A further set of simulations could then be performed based on a different investment strategy to see how the probability of ruin compared. By repeating the projections for a suitable number of different investment strategies, we can ultimately determine which is optimal in terms of best meeting our investment objectives. 
It is important that the particular structure of the model and value of the parameters depend on the purpose of the model. Although there are powerful statistical techniques to aid in the construction of the model, it will be necessary to apply judgement in establishing the most appropriate form and appropriate value of the parameters. If the results are to be used in an intelligent and critical manner, the actuary should be aware of the strengths and weaknesses of a particular model before interpreting the output. 
Using a stochastic model is, arguably, the most appropriate way of allowing for the volatility and uncertainty underlying the assets and liabilities. However, this is highly dependent on whether we can specify probability distribution functions that reflect the true level of uncertainty. 
Asset liability modelling is discussed further in Chapter 18.


What is the main problem with relying solely on past data to develop a stochastic asset model?



What are the two main risks involved in the development of an asset liability model?



List five “actuarial” techniques that may be used to develop an appropriate investment strategy and that take into account the liabilities.



Financial risks

The financial risks faced by an institutional investor can be considered under five headings, according to the source of the risk:
1. Market risk is the risk relating to changes in the value of the portfolio due to movements in the market value of the assets held. 

2. Credit risk is the risk that a counterparty to an agreement will be unable or unwilling to fulfil their obligations. 

3. Operational risk is the risk of loss due to fraud or mismanagement within the fund management organisation itself. 

4. Liquidity risk is the risk of not having sufficient cash to meet operational needs at all times. It is related to market risk in as much as the liquidity of the overall portfolio is need to be taken into account in portfolio selection. 

5. Relative performance risk is the risk of under-performing comparable institutional investors. 


Monitoring and controlling market risk Introduction

For many institutional investors, variations in the market values of its assets represent the main risk of failing to meet its investment objectives. For example, a fall in asset values to levels far below those anticipated may threaten the financial integrity of the institution. It is therefore crucial to try and control market risk. The first steps in the control process will be to:
  identify and define the risks 

  develop techniques and/or models by which to quantify such risks.


12 sources of operational risk for a small domestic company.



Defining market risk

The first stage in monitoring and controlling market risk is to define what is meant by risk. A suitable measure might be the variance of the return on the portfolio over a specified period of time or the maximum loss that could be suffered with say 95 or 99% probability within the timescale. The latter measure is commonly known as Value at Risk (VaR).
We discuss the use of Value at Risk further in Chapter 21, when we consider measures of downside risk.
The returns and losses may be measured in absolute terms or relative to some suitable performance benchmark such as an index, an industry median fund or the value of the liabilities.
The timescale chosen will depend on the institution. For a long-term investor such as a pension fund the period may be measured in months or years, whereas for a bank it is likely to be measured in hours.
For example, the estimation of market risk for the purposes of the Basel regulations that are intended to ensure the solvency of banks is based on a 2-week timescale and a 99% probability level. However, banks are likely to use different time periods and probability levels for their own internal risk assessment calculations. In contrast, insurance companies and pension funds typically use time periods of several years.


Modelling risk

Having defined risk it will be necessary to establish a mathematical model which will allow the risk at any point to be calculated. For example, in the mean variance framework, given the expected returns on all the assets in a portfolio and the variance/covariance matrix, the mean and variance of the return on the portfolio can be calculated. These can be used to calculate the desired risk measure.
In practice, banks often perform market risk calculations based on the assumption that the individual investment returns are normally distributed. Using estimates of means, variances and covariances of the returns for individual securities, it is then a simple matter to calculate the distribution of portfolio returns, which will likewise be normally distributed, and hence calculate the Value at Risk.


normality of returns may not be an appropriate assumption in practice, as

investment returns may demonstrate fat tails or negative skew.
An alternative approach is to carry out use simulations based on time series modelling. Thus an insurance company or a pension fund is more likely to use a suitable asset liability model to estimate the variance of expected surplus or the Value at Risk based on the surplus of assets over liabilities.
One desirable feature of any model used is that the factors that determine risk level are understandable.


Systems, reporting and benchmarks

The next stage is to ensure that the computer systems and data inputs are in place to calculate the risk exposure as often as is required.
The model can be used to assess the market risk associated with different asset allocation strategies and thereby determine which strategy is optimal given the objectives of the investor. The levels of market risk should be monitored at regular intervals, and the results of the analysis disseminated to all parties involved in the investment decision process, so as to enable them to make properly informed decisions.
Individual fund managers will need to be continuously aware of their risk exposures and the reports should also be made regularly to senior management.
In practice, fund managers will not only need to know their risk levels but will also need the tools to understand the effect of their actions on the risk of the portfolio. Guidelines will need to be established which can be translated into practical benchmarks and limits for departure from the benchmarks. A typical risk control system might therefore give the fund manager a benchmark asset distribution, expressed as a percentage of the portfolio in specified asset categories.
For a risk control system to be effective it is also desirable that it is clearly documented and that the personnel responsible for monitoring risk are independent of the fund managers.


State two ways in which allowance might be made for non-normal investment returns.



Load differences and load ratios

Limits for departure from the benchmark would also be set. Limits can be expressed in two ways.
A load difference specifies the range over which the percentage allocation to a specific class can vary, for example limiting overseas equities to between 5 and 15% of the total portfolio.
A load ratio specifies the maximum variation of the allocation to a specific asset class expressed as a percentage of the benchmark allocation to that class. This has the advantage that a constant load ratio permits a smaller absolute variation in the lower weighted asset classes.


A fund manager is set a benchmark for overseas investment of 20%, with a permissible range of 15% to 25%. Express this benchmark as a load ratio.



What type of base output would you expect from a simple risk control system for an institutional investor?



You have been asked to develop a risk monitoring system. List the desirable features and requirements of such a system.



How might risk be defined or measured for a manager responsible for a unitised fund, eg the pension managed fund run by a life insurance company?



Monitoring and controlling credit risk

key factors in managing credit risk are:
  the creditworthiness of the counterparties with which an institution deals; 

  the total exposure to each counterparty. 



Creditworthiness can be controlled by placing limits on the credit ratings (as published by the major rating agencies) of counterparties with which an institution may deal. Different limits might be imposed depending of the potential length of exposure. For example investing in a company’s corporate debt is a different risk to a settlement period risk with a broker or market maker. 
Credit risk can be also controlled in derivatives transactions by dealing on a recognised exchange with a central clearing house that stands as counterparty to all deals, rather than over-the-counter. The clearing house will seek to protect itself by requiring the counterparties to deposit “margin” with it. These margin payments are a particular example of the use of collateral provided by a counterparty as a tool against credit risk. 
More generally, any institution may demand collateral or margin payments as a way of protecting itself against the credit risk it faces under a particular transaction. The operation of a clearing house was described in more detail in Chapter 1. 


Credit exposure

It is important to monitor and place limits on the credit exposure to any single counterparty. 
An example of such a limit might be that the investor is limited to a total investment in a bond with a credit rating of BBB of no more than the maximum of 5% of the total fund size or $250m. Of course the upper limit could be zero for bonds with a low credit rating. 
This will involve aggregating exposures in different areas. For example a pension fund may hold both equity and debt issued by a bank as well as having cash on deposit with the same bank and having them as a counterparty to a derivatives deal. It will also be necessary to be aware of the particular relationships between different companies within the same group. 
In addition, credit risk can be hedged using the credit derivatives discussed in Chapter 3. 

Allowance for credit risk is often made in the model used to analyse Value at Risk. In practice, many quantitative investment models allow for both elements of risk simultaneously.


Monitoring and controlling operational risk

Operational risk may not be as easy to quantify and measure as credit and market risk but it has arguably been responsible for more spectacular corporate losses. For example, the collapse of Barings Bank in the 1990s. 
Given that operational risk refers to losses due to internal fraud or mismanagement, 
control of operational risk essentially depends on good management practices including having established and documented chains of reporting and responsibility. 
In addition, those with responsibility should have suitable qualifications and experience. 
Two areas that have been highlighted in recent years are:
1. the need for management to understand the nature of the complex deals undertaken by traders 

2. the need for the separation of “front office” and “back office” functions. 

. Front office functions are making and recording deals, back office functions are settlement and accounting.


Monitoring and controlling liquidity risk

For all companies liquidity risk is the risk that cashflows from assets are insufficient to meet liabilities in all future periods. The focus is on cash, which is a different test from the company having assets in excess of its liabilities. 
For financial services institutions liquidity risk is the risk of not being able to raise funds (by borrowing or sale of assets) at a reasonable cost at all times and, therefore, the risk that a market does not have the capacity to handle the volume of desired transactions when needed. 
Techniques that can be used for identifying and measuring liquidity risk include the cash budgeting / short-term financial planning techniques considered in Chapter 6 of this course.


Which types of security give rise to potentially large credit risk as well as increased probability of operational risk?



Gap analysis

A simple “balance sheet” model of liquidity is useful for establishing liquidity policies and operating limits. All assets are allocated to one of two categories - liquid or illiquid. All liabilities are classified as either stable or volatile. The focal point of the analysis is the concept of “net liquid assets” or the “liquidity gap”.
A six-month remaining maturity criterion is usually adopted in classifying assets and liabilities. So, assets maturing in six months or less would be classified as liquid, and liabilities that are due in six months or less would be classified as volatile.
Liquidity gap or net liquid assets are defined as:
the difference between the level of liquid assets and volatile liabilities. In analysing the net liquid assets position, allowance should be made for the liquidation costs associated with converting items to cash. These costs will be a function of brokerage and investment banking fees and the basic bid-offer spread in the market for the assets involved, as well as the time available for conversion.
So, the "liquidity gap" (or "net liquid assets") method considers the difference between liquid assets and volatile liabilities. This gap represents a liquidity risk.


Duration analysis

A more rigorous approach involves the concept of liquidity duration or liquidity risk elasticity (LRE), where the impact of changes in market conditions is considered.
Extending the liquidity gap approach, it looks to quantify the potential cost of such a gap by considering the impact of a change in interest rates (specifically, an increase in the cost of raising funds).
The process consists of two steps:
1. Calculate the present value of assets and liabilities using the “cost of funds” rate as the discount rate 
The basic gap (or institution's equity) is calculated as the present value of assets minus the present value of liabilities. The effect of an increase in interest rates at all durations is then measured. This rate of change is called the LRE. 

2. Measure the change in the market value of the institution’s equity (LRE) from a change in the cost of funds (due to an increase in the risk premium paid to raise money). 

If the LRE is zero, the institution has zero liquidity risk (by this measure).
If the duration of the assets is longer than that of the liabilities, the LRE will be negative. This is because the value of assets will decrease more than the value of the liabilities following such a increase in interest rate.
So, if the LRE is negative then increases in interest rates will pose liquidity problems under stress situations where the cost of finance is high.
If the LRE is sharply negative, it will pay the institution to shorten the maturity of its assets and lengthen the maturity of its liabilities, thereby increasing liquidity.


Monitoring and controlling relative performance risk

The techniques for monitoring and controlling relative performance risk are essentially the same as those for controlling market risk except that performance is measured relative to the performance of the institution’s competitors rather than in absolute terms or relative to the whole market.

The techniques for monitoring and controlling relative performance risk are essentially the same as those for controlling market risk except that performance is measured relative to the performance of the institution’s competitors rather than in absolute terms or relative to the whole market.
Typical objectives here might be to aim to minimise the risks of achieving:
  below median investment returns over one or more specified terms, eg 1 year, 3 year, 5 year and/or 10 year terms 

  median or above investment returns less than 90% of the time (for any particular term) 

  returns below those yielded by a particular investment index. Ways to minimise relative performance risk might therefore include:
  commercial matching ̧ ie holding similar investment portfolios to your competitors 

  index tracking.



is the investment of the assets in such a way that the present value of the assets minus the present value of the liabilities is immune to a general change in the rate of interest. Immunisation requires that:
1. The present values of the liability outgo and asset proceeds are equal. 

2. The discounted mean term of the value of the asset proceeds must equal the mean term of the value of the liability outgo. 

3. The spread about the mean term of the value of the asset proceeds should be greater than the spread of the value of the liability outgo. 


Problems with immunisation:

1. the amounts of the liabilities may not be known with certainty 

2. it removes the possibility of mismatching profits, apart from a very small, second-order effect 

3. it works only for small changes in interest rates 

4. it assumes a flat yield curve at all times 

5. it requires constant rebalancing 

6. assets of a suitably long discounted mean term may not exist 

7. the timing of asset proceeds and liability outgo may not be known. 


Portfolio theory

Portfolio theory enables the investor to identify its optimal portfolio – the one that maximises expected utility as a function of the mean and variance of investment returns.
The assumptions underlying portfolio theory are as follows:
●  All expected returns, variances and covariances of pairs of assets are known. 

●  Investors make their decisions purely on the basis of expected return and variance. 

●  Investors are non-satiated. 

●  Investors are risk-averse. 

●  There is a fixed single-step time period. 

●  There are no taxes or transaction costs. 

●  Assets may be held in any amounts, ie short-selling, infinitely divisible holdings, no maximum investment limits. 
The opportunity set is the set of combinations of means and variances that the investor is able to obtain by constructing portfolios containing the available securities. 
A portfolio is efficient if there is no other portfolio with either a higher mean and the same or lower variance, or a lower variance and the same or higher mean. The efficient frontier is the set of efficient portfolios in E -V space. 
Indifference curves join points of equal expected utility in E - V space.
The optimal portfolio is the portfolio that maximises the investor’s expected utility as a 
function of the mean and variance of investment returns.
Portfolio theory can be used to explain how risk can be diversified away.