An alternative yield measure – Realised Compound Yield (RCY)
There are several alternative yield measures, however, the alternative measure which is conceptually superior is realised compound yield (RCY). RCY is defined as the compound rate of growth in total during the holding period of the bond is calculated on an annualised rate of interest. Examples of calculation are illustrated below:
Annual Paying Bond
- Assume: F = 1000, C = 80, t =4 years
- Definition of Annual Return
Where
Vt = £ amount at the end,
Vo = £ amount at the beginning,
In this case, Vo is £1000, and t is 4 years. Therefore:
-
To calculate rann we must calculate Vt. To calculate Vt we must account for the reinvestment of the annual 8% coupon (=£80 per annum). Assuming we reinvest these coupons at 8%, we end up with the following cash flows on the bond:
Cash Flows
- In this case,
Thus
There are several reasons why RCY is conceptually superior. These differences are discussed in more depth below.
RCY is superior in comparison to YTM because of its ability to measure the yield for comparison basis for a time period less than maturity. Therefore, RCY takes into account the investors desired holding period. Unlike YTM, which simply assumes bonds are held till maturity. If an investor decides to sell his position at such time earlier than maturity, RCY makes an assumption about the level of bond yields at that point in time. Alternatively, if investors hold their positions till maturity, RCY is still applicable for as a comparative measure of yields.
In addition, RCY also boasts the ability to take into consideration current market interest rates as well as investors investment time plan (providing a more accurate measurement of yield values). It is this point which makes RCY a more superior concept than YTM that has to rely on future predictions as a benchmark for calculations. These forward rates employed are a part of the expectations analyst and can be locked in using financial derivatives such as swaps and Floating Rate Agreement (FRA’s). Swaps are agreements between at least two counter-parties to exchange cash flows in the future according to a pre-specified formula. They can therefore be regarded as portfolios of forward contracts. The most common one is an agreement on the exchange of a fixed rate for a floating rate contract.
FRA’s are interest futures contracts outside the stock market. An interest rate is agreed between the contracting parties, with a predefined currency, nominal sum and reference period. The difference between the agreed interest rate and the interest rate applicable on the date of maturity is settled between the partners.
RCY is extremely useful beneficial as it absorbs all these different factors and calculates the best investment for individual investors. Basically, finds the most suitable investment for each individual investor, depending on their personal expectations. This distinctive and crucial function is not applicable in the YTM measurement scheme.
RCY does not come without faults or limitations. One limitation of RCY is that can be directly affected by changes in key aspects such as changes in forecast of reinvestment rates, holding period, and yield of the bond at the end of the investor’s holding period. Despite its weakness, it is fair to comment that in direct comparison to YTM and its weaknesses; there is great deal of evidence suggesting RCY is a superior form of yield measurement.
Conclusion
YTM whilst having its advantages it is the measurements major disadvantages
that we are more interested in as these are the underlying assumptions which the model relies so heavily on. For example, YTM assumes the bond is held until maturity and that all coupon income can be reinvested at a rate equal to the yield to maturity. In regards to RCY, there are several reasons why this measurement scheme is superior to YTM as it takes into account the ever important forward rates in its calculation process. Not only does this ensure the calculations are more accurate. Whereas, YTM is a projection of future performances. Nevertheless, YTM is still used to date as a measurement tool (a result of ease in use) and am confident it will continue to do so in the near future.
Question (2): The validity of using Modified Duration (MD) to measure interest rate sensitivity in (a) Bonds denominated in a single currency, (b) Bond portfolios consisting of bonds denominated in more than once currency.
The calculation of bond duration takes into account several factors such as present value of bond, the risk of recovering the full reversionary value of the bond and increases with the time to maturity, resulting in one number - a measurement of a bond’s price sensitivity to changes in market interest rates.
A bonds coupon rate (which determines the size of the periodic cash flow), yield (which determines present value of the periodic cash flow), and time to maturity (which weights each cash flow) all contribute to the Duration Factor. Increases in market yield rates causes a decrease in the present value factors of each cash flow. Since duration is a product of the present value of each cash flow and time, higher yield rates also lower duration. Thus, duration varies inversely with yield rates. Increases in coupon rates raise the present value of each periodic cash flow and therefore the market price. This higher market price lowers duration. Thus, duration varies inversely to coupon rate. Increases in maturity increases duration and cause the bond to be more sensitive to changes in market yields. Thus, duration varies directly with time to maturity.
The magnitude of duration is an index to the sensitivity of the bond to changes in market interest rates. Bonds with high duration factors experience greater increases in values when rates decline, and greater losses when rates increase, compared to bonds with lower duration. To calculate an approximate percent change in market value as the result of a percent change in yield, Macaulay derived Modified duration, which is simply duration times the factor we removed from the formula below.
Where
DMac = Macaulay Duration in years
CFt = Cash flow at time t (coupons and redemption)
t = Time period when cash flow is due
T = Final time period
y = Yield to Maturity per cash flow period where k is the number of coupons per year
P0 = Present value of the bond (dirty price)
Modified duration (DM) is calculated as:
Remember that where k is the number of coupons per year.
(Source: Terry Watsham lecture handout, pg. 11)
Modified duration is a measure of the sensitivity of a bond’s price to interest-rate changes, assuming that the expected cash flow does not change with interest rates. Modified duration assumes a flat term structure of interest rates, because we use the yield-to-maturity as the discount rate.
Modified duration has several weaknesses, and it is exactly these weaknesses which make up the validity arguments of using modified duration to measure interest rate changes in bond portfolios. At the end of the day, the modified duration of the bond is only an approximation of the resultant change in the bond’s price. The approximation error occurs because the price: yield function is not linear but convex as seen in appendix 1. This is a severe problem because bonds are not linear, as a lot of them have embedded options in them. Despite modified duration has the ability to calculate bonds with bonds with embedded options. The underlying assumptions behind the linear relationship does not hold true.
As the price yield curve of a bond is convex, the Modified Duration will understate the rise in price of a bond for a given fall in yield, and overstate the fall in price for a given rise in yield. The impact of convexity increases in the event of large changes in interest rate levels. Despite the issue of convexity, modified duration grants adequate results for smaller interest rate level changes (+/- 100 basis points). Issue of convexity can be largely removed by the role of the Price Value of a Basis Point (PV01). Recalculation of PV01 diminishes the need to make adjustments for convexity, as long as these recalculations are continuous. Viewing of PV01’s formulae can be viewed in appendix 2.
To sum up, there are several validity issues to modified duration as a measure of interest rate sensitivity. Firstly, its assumptions of flat term structure and that yield curve only shifts in parallel. In reality, the term structure of interest rates is not flat at all. As shifts are not in parallel, it does not give an accurate valuation of the change in overall portfolio value. The model is based primarily on this assumption, and as the assumption does not hold true, signalling important weaknesses in the practical application. Secondly, it uses yield to maturity as a discount rate, which as mentioned in the previous question has several limitations. Critically, the discount rates are only an approximation/ future project and thus could be misleading. Thirdly, modified duration is only a reasonable approximation of the interest rate sensitivity for small changes in yield, as large movements in yield are inaccurately reflected in price change, unless PV01 is employed to reduce the convexity issue.
In relation to a bond portfolio consisting of bonds portfolio denominated in more than one currency, it holds an assumption of perfect correlation between exchange rates, as we all know is completely untrue. To price a bond portfolio, one must employ the following formulae:
Modified Duration of a Bond Portfolio:
Where
= Duration of a bond portfolio
PVi = Present value of bond i
= Modified Duration of bond i
PVP = Present value of the bond portfolio
However, the problem arises when the portfolio consists of different currency bonds. Reasons being currencies and interest rates on different bonds fluctuate constantly, therefore, making it unbelievably difficult to calculate the changing interest rates effect on the bond portfolio.
In conclusion, for bonds denominated in a single currency, modified duration would be reasonable as a measurement tool providing the changes were small, cause if the changes were drastic, modified duration would give a distort interpretation. For bonds denominated in more than one currency, the daily fluctuations in currencies and the different interest rates for individual bonds makes it extremely difficult to calculate modified duration.
The sterling component of my bond portfolio consists of three components:
- 25% holding in Gilt Strips with a modified duration of 3;
- 50% holdings of Gilt Strips with a modified duration of 6; and
- 25% holding of Gilt Strips with a modified duration of 10.
Calculate the modified duration of the whole portfolio.
Calculate the change in value of the portfolio if the term structure of interest rates changes with the spot rates.
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.
Explain how to measure the exposure to foreign exchange risk
Exposure is the risk which an investor accepts when buying and selling in foreign currency-hedged financial instruments. This exposure can be measured in percentage terms of a foreign currency asset by using the following equation:
Where
Price relative of the foreign currency asset in domestic currency terms
Price relative of the exchange rate
The above formulae is a linear regression equation where the coefficient, B, represents the sensitivity of the domestic currency returns, Pt+1/Pt of the foreign asset to changes in the exchange rate, St+1/St.
This regression coefficient B represents that part of the random returns in the domestic 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 domestic returns. It is assumed when using OLS regression that these other elements are independent of the currency element.
Changing currency exposure highlights problems that arise in establishing and managing mean-variance hedges; that is the problem of the stability of the covariance matrix. If the covariances are not stable, the regression coefficient will not provide a reliable hedge ratio. This problem can be easily overcome through the use of a rolling one or three month hedge that is rebalanced to recalculate the levels of exposure will captor the major changes in exposure.
Explain the difference between hedging and insurance.
The methodology for determining the place of foreign assets in portfolios is similar to that for determining the place for domestic assets. The decision is based upon expected returns, expected variances and expected co-variances. The return to a foreign currency asset comprises of the returns that the local investor would enjoy plus the returns for bearing currency risk. The extent of these foreign exchange returns will depend upon how the exposure to currency risk is managed. The more the currency exposure is hedged, the less will be the influence of the foreign exchange returns and the more the returns will look like the returns that the local currency investor would enjoy. Without hedging, the greater will be the influence of foreign exchange returns.
Fluctuations in foreign exchange affect total returns accordingly. Depreciation in the domestic currency against a foreign currency over the holding period will show a positive foreign exchange return. An appreciation in domestic currency against a foreign currency over the holding period will have the opposite effect and result in a negative foreign exchange return. The effect on foreign currency movement varies with the financial instrument, with stocks being enormously affected. However, the greatest effect can be noticed on bonds.
Modern derivative instruments such as forwards, futures, swaps, and options facilitate the management of exchange and interest rate volatility. They are employed to offset the risks in portfolios because their cash flows and values change with changes in interest rates and foreign currency prices. Among other things, they can be used to make offsetting bets to "cancel out" the risks in a portfolio. Derivative instruments are ideal for this purpose, because many of them can be traded quickly, easily, and with low transactions costs, while others can be tailored to customers' needs.
Hedging and insurance are both strategies to avoid an exposure to adverse movements in the price of an asset. The fundamental difference between the two is that hedging positions and insurance. A key aspect of hedging is the cost of, or price received for, the underlying asset is ensured. However, there is no assurance that the outcome with hedging will be better than the outcome without hedging. Whereas with insurance using an option, the purchaser decides what the outcome will be: either to exercise the option or to pay premium (cost of the option). An option is the right but not the obligation to do something. A future is the obligation to do something once a contract has been constructed.
Forward contracts are designed to neutralise the risk by fixing the price that the hedger will pay or receive for the underlying asset. Options contracts by contrast, provide insurance. They offer a way for investors to protect themselves against adverse price movements in the future while still allowing them to benefit from favourable price movements. Unlike forwards, options involve the payment of an up-front fee.
An example of hedging foreign exchange exposure:
Import Co, an US company knows it has to pay £10 million in three months time. Import Co could hedge its foreign exchange exposure by entering into a three month forward contract. A forward contract is a cash market transaction in which a seller agrees to deliver a specific to a buyer at some point in the future. Forward contracts are privately negotiated and are not standardized. Further, the two parties must bear each other's , as the contracts are not exchange traded, there is no marking to market requirement, which allows a buyer to avoid almost all capital outflow initially (though some counterparties might set collateral requirements). The forward price makes the forward contract have no value when the contract is written. However, if the value of the underlying commodity changes, the value of the forward contract becomes positive or negative, depending on the position held.
Forward contracts are the most common means of hedging transactions in foreign currencies. 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.
Outside of 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's 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 standardized 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.
But 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.
Another mean of hedging takes the form of a swap. In practical terms this means that on purchasing a foreign bond the fund manager should enter a swap contract. This means that on the near leg of the swap they can buy the foreign currency (to pay for the bond) and the second leg is for the sale of the foreign exchange currency to neutralise the FX risk. Once the hedges are due for settlement the fund simply executes another swap. The near leg is equal and opposite to the maturing deals (except the rate is now the spot rate) and the second leg reinstates the forward hedge.
The portfolio manager also needs to decide which currency to keep constant, domestic or the foreign currency. In general it is the foreign currency which should be kept constant in line with the value of the bond holdings. If the foreign hedge amount is kept constant be aware that each roll date will give rise to a profit or loss on the hedge. Also at each roll date the value of the foreign bond needs to be taken into account and the hedge adjusted accordingly to make sure the portfolio is not over or under hedged.
An alternative hedging method is to borrowing in the currency to which the firm is exposed or investing in interest-bearing assets to offset a foreign currency payment. This is a widely used hedging tool that serves much the same purpose as forward contracts. 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. This is costly, so the forward hedge would probably be more advantageous except where the firm had to borrow for ongoing purposes anyway.
An example of insuring foreign exchange exposure:
An US company due to receive UK (£) at a known time in the future can insure its risk by buying put options on sterling that matures at that time. It is called a foreign exchange option: the right, but not the obligation, to exchange currency at a predetermined rate. This strategy insures that the value of the sterling will not be less than the exercise price while allowing the company to benefit from any favourable exchange rate movements. Unlike a forward contract locks in the exchange rate in the future transaction.
Finally, one can see that there are several different means of hedging. Individually having own advantages and disadvantages. Insurance is a seen as a more flexible type of hedging foreign exchange exposure as the option holder has the right but not the obligation to exercise the option. This feature does however, come at a cost.
Question (4): Discuss the strengths and weaknesses of using VaR as an alternative method of identifying interest rate risk in bond portfolios.
The sheer numbers and complexity of some of derivative instruments, the magnitudes of the risks in companies' portfolios often are not obvious. Resulted in a demand for portfolio level quantitative measures of market risk such as "value at risk." The flexibility of derivative instruments and the ease with which both cash and derivative instruments can be traded and retraded to alter companies' risks also has created a demand for a portfolio level summary risk measure that can be reported to the senior managers charged with the oversight of risk management and trading operations.
Value at risk is a single, summary, statistical measure of possible portfolio losses. Specifically, value at risk is a measure of losses due to "normal" market movements. Losses greater than the value at risk are suffered only with a specified small probability. Subject to the simplifying assumptions used in its calculation, value at risk aggregates all of the risks in a portfolio into a single number suitable for use in the boardroom, reporting to regulators, or disclosure in an annual report. Once one crosses the hurdle of using a statistical measure, the concept of value at risk is straightforward to understand. It is simply a way to describe the magnitude of the likely losses on the portfolio.
VaR on a portfolio is the max loss one might expect over a given holding or horizon period, at a given level of confidence. Measurements depend on two arbitrarily chosen parameters. First is a holding period, the period of time over which one measures the portfolio profit or loss. Second is the confidence level, indicating the likelihood that one will get no outcome worse than the VaR. The VaR is dependant on the choice of confidence level and holding period, and will generally change when confidence levels and/or the holding period change.
In order to calculate VaR, one must choose a confidence level (CL). For example, a CL at 95% means the VaR is given by the negative (-’ve) of the point on the x-axis that cuts off the top 95% of the Profit and Loss (P/L) observations from the bottom 5%of tail observations. The negative P/L value corresponds to a positive VaR, illustrating the worst outcome at this level of confidence is a loss of whatever amount.
In practice, the point on the x-axis corresponding to the VaR will normally be negative, and will correspond to a loss and a positive VaR. This point can also be positive, in which case, it illustrates a profit instead of a loss, meaning VaR will be negative.
The higher the CL, the smaller the tail, a cut off point further to the left and thus, a higher VaR. Providing other things remaining the same, VaR tends to increase as confidence levels increase.
The theory provides little guidance about the choice of CL. It is determined primarily by how the designer and/or user of the risk management system want to interpret the value at risk number: is an "abnormal" loss one that occurs with a probability of 1 percent, or 5 percent?
In order to provide a more accurate measure, need to combine the two parameters together to form a VaR surface, enabling the user to read off the value of VaR for any two given parameters (CL & holding period). The shape of the surface reveals how the VaR varies was the underlying parameters vary, and expresses a lot of information. This shape increases as both variables, revealing where the portfolio is most vulnerable, as both parameters approach their maximum values.
VaR is a generally acceptable means of calculating financial risk as it encompasses the use of standard deviation, deviation, price value of a basis point, delta, gamma, Vega and rho. In addition to incorporating the probability of a loss, it has the advantage that it is applicable across all products and encompasses the probability of loss based upon the empirical distributions of the value of the portfolio. The other valuation models mentioned above, however, do not encompass the probability of a loss.
There are several approaches to calculating VaR, one is historical simulation. Involves creating a database consisting of the daily movements in all market variables over a period of time. Second method is the building-approach. Is relatively straightforward if two assumptions can be made. First, the change in the value of the portfolio is linearly dependent on a number of variables. Second, the variables are multivariate and normally distributed. In this case, the probability of the portfolio is normal, and there are analytic formulas for relating the standard deviation of the portfolio to the volatilities and correlations of the underlying market variables.
As the bonds sometimes have embedded options in them, this is necessary to be taken into account. When a portfolio includes options, the portfolio is not linearly related to the variables. Nevertheless, can derive at an approximate quadratic relationship between portfolio and variables. After this relationship has been derived, one can utilise the Monte Carlo simulation to estimate VaR.
Value-at-Risk is scientifically rigorous in that it utilizes statistical techniques that have evolved in physics and engineering. VaR like all risk measurement tools has some limitations too. It is questionable in that it makes assumptions in order to use these statistical techniques. Chief assumption is the return of financial prices is normally distributed with a mean of zero. Empirical evidence would suggest otherwise. The return of a financial price may be thought of as the capital gain/loss that one might expect to accrue from holding the financial asset for one day. VaR estimates can be subject to error, which VaR systems can be subject to model risk or implementation risk of errors.
One weakness of using VaR as a measure of interest rate risk in bond portfolio is the issue of parameter t, determined by the entity's holding period. Those which actively trade their portfolios, such as financial firms, typically use 1 day, while institutional investors and non-financial corporations may use longer holding periods. A value at risk number applies to the current portfolio, so a (sometimes implicit) assumption underlying the computation is that the current portfolio will remain unchanged throughout the holding period. This may not be reasonable, particularly for long holding periods.
In interpreting value at risk numbers, it is crucial to keep in mind the probability x and holding period t. Without them, value at risk numbers are meaningless. The choice of holding period can have an even larger impact, for the value at risk computed using a t-day holding period is approximately times as large as the value at risk using a one day holding period. Absent appropriate adjustments for these factors, value at risk numbers are not comparable across entities. In addition, many researchers also point out that VaR’s single number to describe a risk in a portfolio may tempt traders to choose a portfolio with a return distribution that does not accurately reflect all information.
The only environment in which value at risk numbers will be used alone is at the level of oversight by senior management. Even at this level, the value at risks numbers often will be supplemented by the results of scenario analyses, stress tests, and other information about the positions. In addition to calculating VaR, many companies carry out what is known as a stress test of their portfolio (see appendix 3).
To conclude, despite its apparent advantages and disadvantages, value at risk is not a panacea. It is a single, summary, statistical measure of normal market risk and is only suitable for measuring interest rate risk providing the correct calculation model is utilised. At the level of the trading desk, it is just one more item in the risk manager's or trader's toolkit. The traders and front-line risk managers will look at the whole panoply of Greek letter risks, i.e. the delta's, gamma's, Vega’s, et cetera, and may look at the portfolio's exposures to other factors such as changes in correlations. In many cases they will go beyond value at risk and use simulation techniques to generate the entire distribution of possible outcomes, and will supplement this with detailed analyses of specific scenarios and "stress tests." Even if the correct models were used, in my opinion, I think VaR is a model more suited for calculating other things beside interest rate risk. However, at the same time, VaR can be useful in some scenarios.
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Hull J C., (2002), Fundamentals of Futures and Options, Prentice Hall, 4th Ed.
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Fabozzi, F J., (2000), Bond Markets, analysis and strategies, Prentice Hall, 4th Ed.
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Fabozzi, F J., (2002), Interest rate, term structure and valuation modelling, John Wiley
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Galitz L, (1994), Financial engineering : tools and techniques to manage financial risk, Pitman
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Watsham T J., (1998), Futures and Options in Risk Management, International Thompson Business Press, 2nd Ed.
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List of Appendices
Appendix 1: Modified Duration
Source: Terry Watsham Lecture Handout, pg. 9
Appendix 2: Price Value of a Basis Point
An alternative way of measuring the interest rate sensitivity of a bond, largely employed by practitioners, is known as the price value of a basis point (PVO1). This method gives the monetary amount by which the bond price will change for a one basis point change in yield. The formula for calculating the PV01 is
Source: Terry Watsham Lecture Handout, pg. 16
Appendix 3: Stress Testing
“Stress testing involves estimating how the portfolio would have performed under some of the most extreme market moves in the last 10 to 20 years. To test the effect of extreme movements in UK interest rates, the company might set the percentage changes in all market variables to a previous date. Stress testing can be considered as a way of taking into account extreme events that do occur from time to time but are virtually impossible according to probability distributions assumed for market variables. Whatever, the method used for calculating VaR; an important reality check is back testing.” (Fundamentals of Futures and Options, pg. 348)
The End
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