Further assumptions involve the level of bond yields at the time of sale and reinvestment of redemption proceeds, depending on the life of the bond and how it lies in accordance with the desired holding period.
To calculate the Realised Compound yield we derive the total coupon payments and interest upon reinvestment. The method uses the forward yield curve to reflect expectations on future interest rates. The price of the bond and whether it is equal to the redemption price lies upon the holding period and whether that is shorter or longer than the life of the bond.
The final step to the realized compound yield calculation is the derivation of the terminal value of the investment.
The formula used for this calculation is:
With TV being the terminal value of the investment, P the original price of the bond at which it was purchased and k the number of coupon paying periods. The following example shows the importance of forward rates. It considers two 4-year bonds. One pays 8% pa, is valued at 65.32 with a yield to maturity of 24.5%, and the other pays 4% pa, and is valued at 56.19 with a yield to maturity of 22.7%.
Assume that the respective cash flows from each bond are reinvested at the following forward rates, 10.08%, and 12 .08% and 14.65%. The first coupons are reinvested for 3 years, the second coupon for 2 years and the third for 1 year. The terminal values will be as follows:
8% Coupon Bond:
8% * (1+0.1008) * (1+0.1208) * (1+0.1465) = 11.316
8% * (1+0.1208) * (1+0.1465) = 10.279
8% * (1+0.1465) = 9.172
Redemption and Final Coupon = 108.0
Terminal Value = 138.77
4% Coupon Bond:
4% * (1+0.1008) * (1+0.1208) * (1+0.1465) = 5.658
4% * (1+0.1208) * (1+0.1465) = 5.140
4% * (1+0.1465) = 4.586
Redemption and Final Coupon = 104.0
Terminal Value = 119.39
The Realised Compound Yield of each of the two bonds is
In order to identify the best bond we must focus on the investors’ expectations.
Specifically, it depends on the interest rate at which the coupon interest payments can be reinvested until the end of the investors’ planned investment horizon. In addition, for bonds with a maturity longer than the investment horizon, it depends on the investors’ expectations about required yields in the market at the end of the planned investment horizon. Consequently, any of these bonds can be the best alternative, depending on some reinvestment rate and some future required yield at the end of the planned investment horizon.
The realized compound yield measure considers these expectations and determines the best investment for the investor, depending on personal expectations. It uses implicit forward rates to reflect expectations about the future spot rates using current data. If these expectations are not fulfilled the term structure of interest rates changes future predictions.
To sum up, Realised Compound Yield allows the portfolio manager to project the performance of a bond based on the planned investment horizon and expectations concerning reinvestment rates and future market yields. This permits the portfolio manager to evaluate which of several potential bonds considered for acquisition will perform best over the planned investment horizon. This cannot be done using the yield to maturity as a measure of relative value.
QUESTION 2
Discuss the validity of using Modified Duration to measure interest rate sensitivity in:
a) Bonds denominated in a single currency
b) Bond portfolios consisting of bonds denominated in more than one currency.
Modified Duration
Modified duration is a measure of bonds’ volatility. It is related to the approximate percentage change in price for a given change in the yield or one percent parallel shift in the yield curve or the term structure of interest rates, assuming the term structure is flat due to the use of the yield to maturity as the discount rate. The fundamental principle that bond prices move in the opposite direction of the change in interest rates implies that there is an inverse relationship between modified duration and the approximate percentage in price for a given yield change.
The diagram shows the convex relationship between bond prices and interest rates. A tangent line has been drawn against the price function. The tangent’s gradient reflects the bond’s price change due to interest rate changes.
Diagram: Duration as approximation for bond price changes
Suppose that yield rises by dy, the price of the bond P0 falls to P1 along the function P(YtM), describing the inverse relationship between bond price and interest rates. Modified Duration is the slope of the tangent divided by the dirty price of the bond. The restriction of this illustration is that it is only an approximation of the resultant change in the bond’s price (dP). The price:yield function is not linear but convex as is shown above. The impact of the Convexity increases in the event of large changes in the interest rate level. Nonetheless, the Modified Duration provides sufficiently adequate results for smaller interest rate level changes (± 100 basis points).
The ways in which modified duration behaves depend on the bond, the coupon payments and its maturity. “The higher the coupon on a bond the lower the modified duration. The reason is that the higher the coupon the larger the proportion of total cash flows received as coupons prior to redemption and thus the larger the proportion of cash flows received early compared to an equivalent lower coupon bond. Duration shortens as time to maturity shortens. Between coupon dates the duration shortens with time. However, immediately after coupon payment, duration increases slightly as the effect of an imminent cashflow is removed from total weighted average time to receipt of the cash flows”.
In addition, duration changes as bond price changes and is positively related to the price of a bond and negatively related to the yield. If interest rates (yields) fall, a parallel shift in the yield curve (Δytm) changes the price of the bond (ΔP/P0).
ΔP = change of value (price)
P0 = current value (dirty price)
Δytm = change of interest rate (yield)
There is a consistency between the properties of bond price volatility and the properties of modified duration. With all other factors constant, the longer the maturity the greater the price volatility. Thus, modified duration can be interpreted as the approximate percentage change in price for a 100-basis-point change in yield.
As illustrated and explained above using the diagram, price yield curve of a bond with no embedded options is non-linear (convex), yet the first derivative of price with respect to yield is the tangent of the price yield curve. Thus, modified duration is only a reasonable approximation of the interest rate sensitivity for small changes in yield and it can only be applicable to measuring price risk over very small, and instantaneous changes in yield.
Furthermore, we use yield to maturity as discount rate. Consequently, all future cashflows are discounted at the same rate.
This assumes that the term structure of interest rates is flat and only shifts in parallel, which is contrary to empirical evidence provided by researchers who have developed models that use the observed spot rates as the discount rates and allow for non-parallel shifts in the term structure.
When there are large movements in the required yield, modified duration is not adequate to approximate the price reaction. Duration will overestimate the price change when the required yield rises, thereby underestimating the new price. When the required yield falls, duration will underestimate the price change and thereby underestimate the new price.
Relying on duration as the sole measure of the price volatility of bonds denominated in a single currency may mislead investors.
First, deriving price volatility using modified duration assumes that all cash flows for the bonds are discounted at the same discount rate. The assumption of flat term structure and parallel shifts imposes a limitation of applying duration when the assumption does not hold, and the yield curve does not shift in a parallel fashion. This is extremely important when we use a portfolio’s duration to quantify the responsiveness of a portfolio’s value to a change in interest rates.
Duration of a portfolio is determined by a linear combination of the present value weighted durations of the individual bonds in the portfolio. Interest rate risk is included in the portfolio as the unexpected fluctuations in the bond price that affects all bonds in the portfolio.
The duration of each bond in the portfolio is weighted by its percentage within the portfolio. If a portfolio has bonds, denominated in more than one currency, with different maturities, the duration measure may not provide a good estimate for unequal changes in interest rates of different maturities. The justification to that lies in the assumption of perfect correlation between exchange rates around the world being far from rational.
This limitation can be dealt with using rate duration. Rate duration is the approximate change in the value of a portfolio or bond to a change in the interest rate of a particular maturity assuming that the interest rate for all other maturities is held constant. Therefore, interest rates are allowed to change individually and disproportionately causing maturities to vary by different numbers of basis points.
QUESTION 2 Part b
- The modified duration of the portfolio would be:
Modified duration = (25% holding in Gilt Strips * modified duration of 3) + (50% holding in Gilt Strips * modified duration of 6) + (25% holding in Gilt Strips * modified duration of 10) =
= (25% * 3) + (50% * 6) + (25% * 10) = 6.25
- The change in the value of the portfolio would be:
Change in the value of the portfolio = (25% holding in Gilt Strips * modified duration of 3 * 1% fall in spot rates) + (50% holding in Gilt Strips * modified duration of 6 * no change in spot rates) + (25% holding in Gilt Strips * modified duration of 10 * 1% increase in spot rates) =
= (25% * 3 * (-1%)) + (50% * 6 * 0%) + (25% * 10 * 1%) = 1.75%
QUESTION 3
Explain how to measure the exposure to foreign exchange risk in the above bond portfolio.
Discuss various ways in which that currency exposure can be managed through hedging or through insurance.
Foreign exchange risk lies in diversified international bond portfolios because of the existence of bonds paying cashflows in foreign currencies. The cashflows are dependent on the exchange rate at the time the payments are received.
The return to investors from investments in bonds that are denominated in a foreign currency consists of two components: the return on the security measured in the currency in which the bond is denominated (local currency return), which results from coupon payments, reinvestment income and capital gains or losses; and changes in the foreign exchange rate.
From the perspective of an investor, the cashflows of assets denominated in a foreign currency expose the investor to uncertainty as to the cash flow in home currency. This uncertainty is in a sense raised by foreign exchange risk.
In other words, if the foreign currency depreciates relative to the home currency, the value of the cashflows will be proportionately less.
At this point, it should be noted that it is very crucial to distinguish between risk and exposure to risk. In a situation where there are two Sterling-based investors in US dollar assets, whereas they are both subject to the same degree of fluctuations in the dollar/sterling exchange rate i.e. the risk, the exposure to that risk varies depending on the size of their investment.
Another difference between risk and exposure lies on whether the asset or liability that exhibits currency exposure is traded or not.
With such differences in exposure, it is important to be clear about the objectives of any currency risk management transactions. For example, if we wish to remove the uncertainty of currency rates when foreign cash flows are to be transformed into domestic currency, then the foreign currency exposure is measured as the number of currency units that are expected to be transformed.
However, if we wish to trim down the variability of the translated value of a tradable foreign currency asset during the period of investment, a more complex approach to measuring currency exposure is required.
Adler and Dumas (1984) and Adler and Simon (1986) have derived the following equation for currency exposure in percentage terms embodied in a foreign currency asset.
where: Pt+1/Pt is the price relative of the foreign currency asset in terms of the home currency. St+1/St is the price relative of the exchange rate.
The equation represents a linear regression where b coefficient represents the sensitivity of the home currency returns, Pt+1/Pt of the foreign asset to changes in the exchange rate, St+1/St. It is that part of the random returns in the home currency that is linearly related to the exchange rate returns. The intercept a, and the error term e, represent the non-currency determinants of the home returns.
The results given by changing currency exposure highlight problems of establishing and managing mean- variance hedges.
This concept of exposure can be applied to our portfolio consisting of both foreign and domestic currency assets. Multiple regression is used to regress upon the currency returns for each of the currencies embodied in the foreign assets:
where: rfx1 represents the returns to the ith currency included in the portfolio. The betas represent the exposure of the whole portfolio, including domestic assets, to the respective currencies.
Using multiple regression comes with problems of autocorrelation in the foreign exchange returns data as well as multicollinearity in some of the variables as they are similarly influenced by the strength or weakness of the basis currency.
This currency exposure can be hedged using linear derivatives such as foreign exchange spot, forwards, futures, or insured using options instruments.
HEDGING
Hedging is the taking up of a position, either financial or operational, that allows offsetting the effects of exchange rate changes. It is defined as offsetting a particular currency exposure by establishing an opposite currency position. In other words, investors match a foreign currency asset (liability) with a foreign currency liability (asset). Hedging incurs costs, and may incur losses.
There are different types of currency exposure hedges: Risk Shifting, Forward market hedging, Money market hedging, Swaps, Futures, Options.
Risk Shifting
This method involves an invoice in the home currency – shifting the risk to the counterparty. This may make it difficult to get contracts, and a rational counterparty will demand a better price in return for taking on the currency risk.
Forward Market
Foreign exchange is, of course, the exchange of one currency for another. Trading or "dealing" in each pair of currencies consists of two parts, the spot market, where payment (delivery) is made right away, and the forward market. The rate in the forward market is a price for foreign currency set at the time the transaction is agreed to but with the actual exchange, or delivery, taking place at a specified time in the future. While the amount of the transaction, the value date, the payments procedure, and the exchange rate are all determined in advance, no exchange of money takes place until the actual settlement date. This commitment to exchange currencies at a previously agreed exchange rate is usually referred to as a forward contract.
Forward contracts are the most common means of hedging transactions in foreign currencies. Investors may use the forward market to tie down the home currency value of the foreign currency payable or receivable. Contract is entered into immediately, but not fulfilled until time has elapsed. In forward markets 100% hedge is possible.
The trouble with forward contracts, however, is that they require future performance, and sometimes one party is unable to perform on the contract. When that happens, the hedge disappears, sometimes at great cost to the hedger. This default risk also means that many companies do not have access to the forward market in sufficient quantity to fully hedge their exchange exposure. For such situations, futures may be more suitable.
Furthermore, forward market contracts may not exist for all countries, or for all time periods (usual limit is one year) and may even be illegal for some currencies.
Money Market
In the money market, investors can match their foreign currency liability (asset) with a foreign currency asset (liability), by depositing (borrowing) in foreign currency today, borrowing (depositing) in their home currency and doing the currency transaction in the spot market.
There are different procedures depending on whether they have a payable or receivable. Another thing to be taken into account is that interest rates are expressed on a per annum basis. Where different borrowing and deposit rates are given, they borrow at the higher rate and deposit at the lower rate – the difference is the bank’s reward for financial intermediation.
Currency futures
Outside the interbank forward market, the best-developed market for hedging exchange rate risk is the currency futures market. In principle, currency futures are similar to foreign exchange forwards in that they are contracts for delivery of a certain amount of a foreign currency at some future date and at a known price. In practice, they differ from forward contracts in important ways.
One difference between forwards and futures is standardization. Forwards are for any amount, as long as it is big enough to be worth the dealer's time, while futures are for standard amounts, each contract being far smaller that the average forward transaction. Futures are also standardised in terms of delivery date. The normal currency futures delivery dates are March, June, September and December, while forwards are private agreements that can specify any delivery date that the parties choose. Both of these features allow the futures contract to be tradable.
Another difference is that forwards are traded by phone and telex and are completely independent of location or time. Futures, on the other hand, are traded in organized exchanges such the LIFFE in London, SIMEX in Singapore and the IMM in Chicago.
However, the most important feature of the futures contract is not its standardization or trading organization but in the time pattern of the cash flows between parties to the transaction. In a forward contract, whether it involves full delivery of the two currencies or just compensation of the net value, the transfer of funds takes place once: at maturity. With futures, cash changes hands every day during the life of the contract, or at least every day that has seen a change in the price of the contract. This daily cash compensation feature largely eliminates default risk.
Thus, forwards and futures serve similar purposes, and tend to have identical rates, but differ in their applicability. Most big companies use forwards; futures tend to be used whenever credit risk may be a problem.
Debt instead of forwards or futures
Debt -- borrowing in the currency to which the firm is exposed or investing in interest-bearing assets to offset a foreign currency payment -- is a widely used hedging tool that serves much the same purpose as forward contracts. Such a tool is termed a money market hedge. According to the interest rate parity theorem, the interest differential equals the forward exchange premium, the percentage by which the forward rate differs from the spot exchange rate. Therefore, the cost of the money market hedge should be the same as the forward or futures market hedge, unless the firm has some advantage in one market or the other. The money market hedge suits many companies because they have to borrow anyway, so it simply is a matter of denominating the company's debt in the currency to which it is exposed. However, if a money market hedge is to be done for its own sake, the firm ends up borrowing from one bank and lending to another, thus losing on the spread. This is costly, so the forward hedge would probably be more advantageous except where the firm had to borrow for ongoing purposes anyway.
INSURANCE
Currency options
Many companies, banks and governments have extensive experience in the use of forward exchange contracts. With a forward contract one can lock in an exchange rate for the future. There are a number of circumstances, however, where it may be desirable to have more flexibility than that which a forward provides. In such a situation, the use of forward or futures would be inappropriate. What is called for is a foreign exchange option: the right, but not the obligation, to exchange currency at a predetermined rate.
A foreign exchange option is a contract for future delivery of a currency in exchange for another, where the holder of the option has the right to buy (or sell) the currency at an agreed price, the strike or exercise price, but is not required to do so. The right to buy is a call; the right to sell, a put. For such a right he pays a price called the option premium.
The option seller receives the premium and is obliged to make (or take) delivery at the agreed-upon price if the buyer exercises his option. In some options, the instrument being delivered is the currency itself; in others, a futures contract on the currency.
American options permit the holder to exercise at any time before the expiration date; European options, only on the expiration date.
An option, contrary to future and forward agreements, gives one party the right but not the obligation to buy or sell an asset under specified conditions while the other party assumes an obligation to sell or buy that asset if that option is exercised.
Options provide the most convenient means of insurance or positioning "volatility risk." Indeed the price of an option is directly influenced by the outlook for a currency's volatility: the more volatile, the higher the price.
A currency call or put option's value is affected by both direction and volatility changes, and the price of such an option will be higher, the more the market's expectations favor exercise and the greater the anticipated volatility.
Finally, one can justify the limited use of options by reference to the deleterious effect of financial distress. Unmanaged exchange rate risk can cause significant fluctuations in the earnings and the market value of an international firm. A very large exchange rate movement may cause special problems for a particular company, perhaps because it brings a competitive threat from a different country. At some level, the currency change may threaten the firm's viability, bringing the costs of bankruptcy to bear.
To prevent this, it may be worth buying some low-cost options that would pay off only under unusual circumstances, ones that would particularly hurt the firm. Out-of-the-money options may be a useful and cost-effective way to insure against currency risks that have very low probabilities but which, if they occur, have disproportionately high costs to the company.
Swaps
FX Swaps: This is one deal combining two currency transactions, one purchase, one sale, at two different dates. FX swaps have two basic uses: to switch a deal from one currency to another on a hedged basis and to move a deal in a given currency backwards or forward in time.
Currency Swaps: These involve exchanging interest payments in one currency with interest payments in another, which is like a package of (spot and) forward contracts.
They are used where forward markets do not exist, and it is hard to get access (at competitive rates) to foreign money markets. Currency swaps are often used for large transactions, where the counterparty is a central bank, but banks may act as financial intermediaries, matching swap trades.
The currency swap involves three stages:
- Initial exchange of principal (either a notional or physical exchange)
- Exchange of interest, either fixed or floating (LIBOR + so many basis points), with rates and terms agreed, based on the principal.
- Re-exchange of principal, at the maturity date of the swap, usually at the original spot exchange rate.
Interest payments, if they are to be made in foreign currency may have to be hedged.
This enables us to borrow in a currency in which we have a good credit record, and transform it into the currency you need.
If we hold the swap until maturity, as all exchange rates are pre-agreed, there is no currency risk.
Foreign exchange swaps have overtaken spot market transactions as the most important type of contract in the last ten years.
QUESTION 4
Discuss the strengths and weaknesses of using VaR as an alternative method of identifying interest rate risk in bond portfolios.
Value-at-risk (VaR) is a category of risk measures that describe probabilistically the market risk of a trading portfolio.
VaR is a powerful tool for assessing market risk, but it also poses a challenge. Its power is its generality. It has become very popular because traditional risk measures had several weaknesses. They could not be aggregated over different types of risk factors/securities. They did not measure capital at risk neither did they facilitate top-down control of risk exposures.
Unlike market risk metrics such as the Greeks, duration and convexity, or beta, which are applicable to only certain asset categories or certain sources of market risk, VaR is general. It is based on the probability distribution for a portfolio's market value. All liquid assets have uncertain market values, which can be characterized with probability distributions. All sources of market risk contribute to those probability distributions. Being applicable to all liquid assets and encompassing, at least in theory, all sources of market risk, VaR is an all-encompassing measure of market risk.
VAR provides a succinct, dollar-based summary measure of risk which allows management to aggregate risks.
It is worth distinguishing between three concepts:
-
A VaR measure is an algorithm with which we calculate a portfolio's VaR.
-
A VaR model is the financial theory, mathematics, and logic that motivate a VaR measure. It is the intellectual justification for the computations that are the VaR measure.
- A VaR metric is our interpretation for the output of the VaR measure.
VaR measures have traditionally been categorized according to the transformations procedures they employ. There are four basic forms of transformation in widespread use:
- linear transformations,
- quadratic transformations,
- Monte Carlo transformations, and
- Historical transformations.
Linear VaR measures employ linear transformations. Many names have been used to describe linear VaR measures, such as parametric, variance-covariance, closed-form, or delta-normal VaR measures. There are shortcomings with most of these names. While linear VaR measures are parametric, so are most VaR measures. While linear VaR measures use variances and covariances, so do all VaR measures, with the exception of historical VaR measures. While some linear VaR measures employ a delta remapping, most do not. Also, while a normal assumption is common with linear VaR measures, it is by no means universal. Linearity describes the one characteristic that is common to all linear transformations: they are applicable to portfolios whose portfolio mapping function is a linear polynomial. Such portfolios include portfolios of equities, physical commodities, or futures. The market value of such portfolios depends linearly upon applicable key factors. Other portfolios are so nearly linear that they can reasonably be approximated with a linear polynomial. These include portfolios of forwards (including foreign exchange forwards) and most non-callable debt. Linear VaR measures are generally not applicable to portfolios that hold options or instruments with embedded options. These include callable bonds, mortgage-backed securities and many structured notes.
The Analytic Variance-Covariance Method can be simpler to estimate since we do not need the entire distribution of factor values. It specifies distributions and payoff profiles (e.g., normal and linear) and decomposes securities into simpler transactions/buckets. It then derives the Variances/Covariances of standard transactions. VaR is then calculated based on the standard definition of variance. Analytic Method is intuitively simpler and does not require any pricing models but it is not conducive to sensitivity analysis and cannot handle non-linear payoff profiles such as options.
Historical Simulation involves identifying factors that affect market values of securities in the portfolio. It simulates future values of the factors using historical data. Then the simulated factors are used to estimate the value of the portfolio several times. Finally, a histogram is created of the portfolio’s expected change in value and the relevant probability levels for the VaR calculation are identified (e.g., the change in portfolio that occurs at the lowest 1% of the distribution). Historical Simulation does not assume specific distributions for the securities and uses real-world data but it requires pricing models for all instruments and allows limited sensitivity analysis.
Monte Carlo Simulation involves the same steps as in the Historical Simulation method except that we use Monte Carlo techniques to obtain the simulated Factor values. It can value complex derivatives properly by generating a large number of trials to determine the distribution of possible outcomes. It is often used to deal with extreme cases and predict future results. These characteristics make it the most appropriate method of dealing with non-linear returns and measuring risk in bond portfolios that use embedded options. Monte Carlo Simulation makes it easier to do sensitivity analysis but requires the analyst to specify asset distributions as well as pricing models.
Empirical Tests have proved that to date, tests of the three methods suggest that the approaches can yield similar results when portfolio payoffs are linear, 95% confidence level is used and there are not many large outliers in the historical data set. The biggest differences can occur between the 2 simulation approaches and the analytic method when non-linear payoffs are a significant share of the portfolio and they do not cancel out (e.g., long a large number of put options), large number of outliers in the historical data set, 99% or higher confidence level is used.
If the portfolio has linear (or weakly non-linear) payoffs, then the Analytic method might be best. If the portfolio has non-linear payoffs, then the two Simulation methods are better. If stress-testing and sensitivity analysis are needed, then Monte Carlo Simulation is the preferred method. However, it can be very complex to remove all possible arbitrage opportunities from the simulation.
In some respects, VaR is a natural progression from earlier portfolio theory including, however, many different aspects and improvements. Although the variance-covariance analysis has the same theoretical basis as Portfolio Theory, the other two approaches of VaR, historical simulation and Monte Carlo simulation do not. One of its advantages is that it interprets risk in bond portfolios in terms of the maximum likely loss. Another advantage of the VaR approach is that it can be applied to a much broader range of risk problems by taking into account the correlation between them and present greater flexibility in terms of the approaches applied to different circumstances. Perhaps the greatest advantage is that VaR approaches are better at accommodating statistical problems such as non-normal returns. By doing so, it provides management of firms with more accurate details on risks leading to more informed and better risk management. It also provides a consistent, integrated treatment of risks across the institution, leading to greater risk transparency and a more consistent treatment of risks across the firm. Its approaches enhance decision-making by supplementing new operational decision rules to guide investment, hedging and trading decisions. Moreover, systems based on VaR methodologies can be used to measure other risks such as credit, liquidity and cashflow risks, as well as the market risks measured by VaR systems proper. This leads to a more integrated approach o the management of different kinds of risks, and to improved budget planning and better strategic management. It gives insight to firms on how to comply with capital adequacy regulations whilst retaining their portfolios to minimise the burden that such regulations impose on them. As with most statistical approaches for dealing with risk in bond portfolios, VaR methods also have weaknesses and limitations.
One of them is that all VaR systems are backward-looking. As they are based on, the assumption that what has taken place in the past will occur in the future, they attempt to forecast likely future losses using past data.
Although its assumptions provide it with flexibility, on the other hand, they inherit the risk that these assumptions might not stand in any given circumstances, hence give irrelevant results. If the VaR approach is used to evaluate risk in bond portfolios, including embedded options, it would not be wise to use a model that assumes normal returns and should be replaced by one that allows for non-normality.
Finally, another limitation is that the method is not foolproof. It is only to be used by experts who know how to use the tools that go with it.
Dr Terry Watsham, The Role and Structure of Interest Rates in the Economy, Lecture Notes on Bond Pricing, p41, University of Brighton,2004.
Fabozzi, Bond Markets: Analysis and Strategies, 4th ed, Prentice Hall, Chapter 3,p46.
Dr Terry Watsham, Risk in Bonds, p9, University of Brighton,2004.
Dr Terry Watsham, Risk in Bonds, Properties of Duration, p29, University of Brighton,2004.
Fabozzi, Bond Markets: Analysis and Strategies, 4th ed, Prentice Hall, Chapters 4,14,17.
Watsham, T.J, Options and Futures in International Portfolio Management, p290, London : Chapman & Hall, 1992.
Watsham, T.J, International Portfolio Management, Chapter 6, pp 126-130, London : Longman, c1993 1993.
Beyond Value at Risk, Kevin Dowd, Ch. 1,3,4,5 John Willey & Sons,1998.
Value at Risk, Philippe Jorion, Ch. 7,9, McGraw Hill Int, 2nd edition,2001.