Risk professionals looking to earn the Financial Risk Manager (FRM®) certification, as well as corporate training programs, professors, and graduate students all rely on the Financial Risk Manager Handbook for the most comprehensive and up-to-date information on financial risk management.

Filled with in-depth insight and practical advice, the Financial Risk Manager Handbook is the core text for risk management training programs worldwide. Presented in a clear and consistent fashion, this completely updated Fifth Edition—which comes with an interactive CD-ROM containing hundreds of multiple—choice questions from previous FRM exams-is one of the best ways to prepare for the Financial Risk Manager (FRM) exam.

Financial Risk Manager Handbook, Fifth Edition supports candidates studying for the Global Association of Risk Professional's (GARP) FRM exam, the global benchmark examination for financial risk management professionals, and prepares you to assess and control risk in today's rapidly changing financial world. Authored by renowned risk management expert Philippe Jorion-with the full support of GARP—this definitive guide summarizes the core body of knowledge for financial risk managers, covering such topics as: * Market, credit, operational, liquidity, and integrated risk management * Quantitative methods * Capital markets * Investment management and hedge fund risk * Relevant regulatory and legal issues essential to risk professionals

The FRM is recognized as the world's most prestigious global certification program—created to measure a financial risk manager's capabilities. Since the FRM exam is an essential requirement for risk managers around the world, the Financial Risk Manager Handbook, Fifth Edition focuses on practical financial risk management techniques and solutions that are emphasized on the test—and are also essential in the real world. Questions from previous exams are explained through tutorials so that you may prepare yourself or your employees for this comprehensive exam and for the risk management challenges you will undoubtedly face at some point in your career.

Financial Risk Manager Handbook

John Wiley & Sons

Chapter One

Risk management starts with the pricing of assets. The simplest assets to study are regular, fixed-coupon bonds. Because their cash flows are predetermined, we can translate their stream of cash flows into a present value by discounting at a fixed interest rate. Thus the valuation of bonds involves understanding compounded interest, discounting, as well as the relationship between present values and interest rates.

Risk management goes one step further than pricing, however. It examines potential changes in the price of assets as the interest rate changes. In this chapter, we assume that there is a single interest rate, or yield, that is used to price the bond. This will be our fundamental risk factor. This chapter describes the relationship between bond prices and yields and presents indispensable tools for the management of fixed-income portfolios.

This chapter starts our coverage of quantitative analysis by discussing bond fundamentals. Section 1.1 reviews the concepts of discounting, present values, and future values. Section 1.2 then plunges into the price-yield relationship. It shows how the Taylor expansion rule can be used to relate movements in bond prices to those in yields. This Taylor expansion rule, however, covers much more than bonds. It is a building block of risk measurement methods based on local valuation, as we shall see later. Section 1.3 then presents an economic interpretation of duration and convexity.

The reader should be forewarned that this chapter, like many others in this handbook, is rather compact. This chapter provides a quick review of bond fundamentals with particular attention to risk measurement applications. By the end of this chapter, however, the reader should be able to answer advanced FRM questions on bond mathematics.

1.1 DISCOUNTING, PRESENT, AND FUTURE VALUE

An investor considers a zero-coupon bond that pays $100 in 10 years. Assume that the investment is guaranteed by the U.S. government, and that there is no credit risk. So, this is a default-free bond, which is exposed to market risk only. Because the payment occurs at a future date, the current value of the investment is surely less than an up-front payment of $100.

To value the payment, we need a discounting factor. This is also the interest rate, or more simply the yield. Define [C.sub.t] as the cash flow at time t and the discounting factor as y. We define T as the number of periods until maturity, e.g., number of years, also known as tenor. The present value (PV) of the bond can be computed as

PV = [C.sub.T]/[(1 + y).sup.T] (1.1)

For instance, a payment of [C.sub.T] = $100 in 10 years discounted at 6 percent is only worth $55.84 now. So, all else fixed, the market value of zero-coupon bonds decreases with longer maturities. Also, keeping T fixed, the value of the bond decreases as the yield increases.

Conversely, we can compute the future value (FV) of the bond as

FV = PV x [(1 + y).sup.T] (1.2)

For instance, an investment now worth PV = $100 growing at 6 percent will have a future value of FV = $179.08 in 10 years.

Here, the yield has a useful interpretation, which is that of an internal rate of return on the bond, or annual growth rate. It is easier to deal with rates of returns than with dollar values. Rates of return, when expressed in percentage terms and on an annual basis, are directly comparable across assets. An annualized yield is sometimes defined as the effective annual rate (EAR).

It is important to note that the interest rate should be stated along with the method used for compounding. Annual compounding is very common. Other conventions exist, however. For instance, the U.S. Treasury market uses semi-annual compounding. Define in this case [y.sup.S] as the rate based on semiannual compounding. To maintain comparability, it is expressed in annualized form, i.e., after multiplication by 2. The number of periods, or semesters, is now 2T. The formula for finding [y.sup.S] is

PV = [C.sub.T]/[(1 + [y.sup.S]/2).sup.2T] (1.3)

For instance, a Treasury zero-coupon bond with a maturity of T = 10 years would have 2T = 20 semiannual compounding periods. Comparing with (1.1), we see that

(1 + y) = [(1 + [y.sup.S] /2).sup.2] (1.4) Continuous compounding is often used when modeling derivatives. It is the limit of the case where the number of compounding periods per year increases to infinity. The continuously compounded interest rate [y.sup.C] is derived from

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

where [e.sup.(·)], sometimes noted as exp(·), represents the exponential function.

Note that in all of these Equations (1.1), (1.3), and (1.5), the present value and future cash flows are identical. Because of different compounding periods, however, the yields will differ. Hence, the compounding period should always be stated.

Example: Using Different Discounting Methods

Consider a bond that pays $100 in 10 years and has a present value of $55.8395. This corresponds to an annually compounded rate of 6.00% using PV = [C.sub.T]/[(1 + y).sup.10], or (1 + y) = [([C.sub.T]/PV).sup.1/10].

This rate can be transformed into a semiannual compounded rate, using [(1 + [y.sup.S] /2).sup.2] = (1 + y), or [y.sup.S] /2 = [(1 + y).sup.1/2] - 1, or [y.sup.S] = ([(1 + 0.06).sup.(1/2)] - 1) x 2 = 0.0591 = 5.91%. It can be also transformed into a continuously compounded rate, using exp([y.sup.C]) = (1 + y), or [y.sup.C] = ln(1 + 0.06) = 0.0583 = 5.83%.

Note that as we increase the frequency of the compounding, the resulting rate decreases. Intuitively, because our money works harder with more frequent compounding, a lower investment rate will achieve the same payoff at the end.

1.2 PRICE-YIELD RELATIONSHIP

1.2.1 Valuation

The fundamental discounting relationship from Equation (1.1) can be extended to any bond with a fixed cash-flow pattern. We can write the present value of a bond P as the discounted value of future cash flows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

where: [C.sub.t] = the cash flow (coupon or principal) in period t t = the number of periods (e.g., half-years) to each payment T = the number of periods to final maturity y = the discounting factor per period (e.g., [y.sup.S]/2)

A typical cash-flow pattern consists of a fixed coupon payment plus the repayment of the principal, or face value at expiration. Define c as the coupon rate and F as the face value. We have [C.sub.t] = cF prior to expiration, and at expiration, we have [C.sub.T] = cF + F. The appendix reviews useful formulas that provide closed-form solutions for such bonds.

When the coupon rate c precisely matches the yield y, using the same compounding frequency, the present value of the bond must be equal to the face value. The bond is said to be a par bond. If the coupon is greater than the yield, the price must be greater than the face value, which means that this is a premium bond. Conversely, if the coupon is lower, or even zero for a zero-coupon bond, the price must be less than the face value, which means that this is a discount bond.

Equation (1.6) describes the relationship between the yield y and the value of the bond P, given its cash-flow characteristics. In other words, the value P can also be written as a nonlinear function of the yield y:

P = f (y) (1.7)

Conversely, we can set P to the current market price of the bond, including any accrued interest. From this, we can compute the "implied" yield that will solve this equation.

Figure 1.1 describes the price-yield function for a 10-year bond with a 6% annual coupon. In risk management terms, this is also the relationship between the payoff on the asset and the risk factor. At a yield of 6%, the price is at par, P = $100. Higher yields imply lower prices. This is an example of a payoff function, which links the price to the underlying risk factor.

Over a wide range of yield values, this is a highly nonlinear relationship. For instance, when the yield is zero, the value of the bond is simply the sum of cash flows, or $160 in this case. When the yield tends to very large values, the bond price tends to zero. For small movements around the initial yield of 6%, however, the relationship is quasilinear.

There is a particularly simple relationship for consols, or perpetual bonds, which are bonds making regular coupon payments but with no redemption date. For a consol, the maturity is infinite and the cash flows are all equal to a fixed percentage of the face value, [C.sub.t] = C = cF. As a result, the price can be simplified from Equation (1.6) to

P = cF [1/(1 + y) + 1/[(1 + y).sup.2] + 1/[(1 + y).sup.3] + ···] = c/y F (1.8)

as shown in the appendix. In this case, the price is simply proportional to the inverse of the yield. Higher yields lead to lower bond prices, and vice versa.

Example: Valuing a Bond

Consider a bond that pays $100 in 10 years and a 6% annual coupon. Assume that the next coupon payment is in exactly one year. What is the market value if the yield is 6%? If it falls to 5%?

The bond cash flows are [ITLITL.sub.1] = $6, [ITLITL.sub.2] = $6, ..., [ITLITL.sub.10] = $106. Using Equation (1.6) and discounting at 6%, this gives the present value of cash flows of $5.66, $5.34, ..., $59.19, for a total of $100.00. The bond is selling at par. This is logical because the coupon is equal to the yield, which is also annually compounded. Alternatively, discounting at 5% leads to a price of $107.72.

1.2.2 Taylor Expansion

Let us say that we want to see what happens to the price if the yield changes from its initial value, called [y.sub.0], to a new value, [y.sub.1] = [y.sub.0] + [DELTA]y. Risk management is all about assessing the effect of changes in risk factors such as yields on asset values. Are there shortcuts to help us with this?

We could recompute the new value of the bond as [P.sub.1] = f ([y.sub.1]). If the change is not too large, however, we can apply a very useful shortcut. The nonlinear relationship can be approximated by a Taylor expansion around its initial value

[P.sub.1] = [P.sub.0] + f' ([y.sub.0]) [DELTA]y + 1/2 f" ([y.sub.0])[([DELTA]y).sup.2] + ··· (1.9)

where f'(·) = dP/dy is the first derivative and f"(·) = [d.sup.2]P/d[y.sup.2] is the second derivative of the function f(·) valued at the starting point. This expansion can be generalized to situations where the function depends on two or more variables. For bonds, the first derivative is related to the duration measure, and the second to convexity.

Equation (1.9) represents an infinite expansion with increasing powers of [DELTA]y. Only the first two terms (linear and quadratic) are ever used by finance practitioners. They provide a good approximation to changes in prices relative to other assumptions we have to make about pricing assets. If the increment is very small, even the quadratic term will be negligible.

Equation (1.9) is fundamental for risk management. It is used, sometimes in different guises, across a variety of financial markets. We will see later that this Taylor expansion is also used to approximate the movement in the value of a derivatives contract, such as an option on a stock. In this case, Equation (1.9) is

[DELTA] P = f'(S) [DELTA] S + 1/2 f"(S)[([DELTA]S).sup.2] + ··· (1.10)

where S is now the price of the underlying asset, such as the stock. Here, the first derivative f'(S) is called delta, and the second f"(S), gamma.

The Taylor expansion allows easy aggregation across financial instruments. If we have [x.sub.i] units (numbers) of bond i and a total of N different bonds in the portfolio, the portfolio derivatives are given by

f'(y) = [N.summation over (i=1)] [x.sub.i] [f'.sub.i](y) (1.11)

1.3 BOND PRICE DERIVATIVES

For fixed-income instruments, the derivatives are so important that they have been given a special name. The negative of the first derivative is the dollar duration (DD):

f'([y.sub.0]) = dP/dy = D* x [P.sub.0] (1.12)

where D* is called the modified duration. Thus, dollar duration is

DD = D* x [P.sub.0] (1.13)

where the price [P.sub.0] represent the market price, including any accrued interest. Sometimes, risk is measured as the dollar value of a basis point (DVBP),

DVBP = DD x [DELTA]y = [D* x [P.sub.0]] x 0.0001 (1.14)

with 0.0001 representing an interest rate change of one basis point (bp) or one hundredth of a percent. The DVBP, sometimes called the DV01, measures can be easily added up across the portfolio.

The second derivative is the dollar convexity (DC):

f"([y.sub.0]) = [d.sup.2]P/d[y.sup.2] = C x [P.sub.0] (1.15)

where ITLITL is called the convexity.

For fixed-income instruments with known cash flows, the price-yield function is known, and we can compute analytical first and second derivatives. Consider, for example, our simple zero-coupon bond in Equation (1.1) where the only payment is the face value, [C.sub.T] = F. We take the first derivative, which is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.16)

Comparing with Equation (1.12), we see that the modified duration must be given by D* = T/(1 + y). The conventional measure of duration is D = T, which does not include division by (1 + y) in the denominator. This is also called Macaulay duration. Note that duration is expressed in periods, like T. With annual compounding, duration is in years. With semiannual compounding, duration is in semesters. It then has to be divided by two for conversion to years. Modified duration D* is related to Macaulay duration D

[D* = D/(1 + y) (1.17)

Modified duration is the appropriate measure of interest rate exposure. The quantity (1 + y) appears in the denominator because we took the derivative of the present value term with discrete compounding. If we use continuous compounding, modified duration is identical to the conventional duration measure. In practice, the difference between Macaulay and modified duration is usually small.

Let us now go back to Equation (1.16) and consider the second derivative, which is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.18)

Comparing with Equation (1.15), we see that the convexity is C = (T + 1)T/ [(1 + y).sup.2]. Note that its dimension is expressed in period squared. With semiannual compounding, convexity is measured in semesters squared. It then has to be divided by 4 for conversion to years squared. So, convexity must be positive for bonds with fixed coupons.

Putting together all these equations, we get the Taylor expansion for the change in the price of a bond, which is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.19)

Therefore duration measures the first-order (linear) effect of changes in yield and convexity the second-order (quadratic) term.

(Continues...)

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Excerpted from Financial Risk Manager Handbook by Philippe Jorion Copyright © 2009 by John Wiley & Sons, Ltd. Excerpted by permission.
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Philippe Jorion is Professor of Finance at the Paul Merage School of Business at the University of California at Irvine. He has also taught at Columbia, Northwestern, the University of Chicago, and the University of British Columbia. He holds an MBA and a PhD from the University of Chicago and a degree in engineering from the University of Brussels. Dr. Jorion has authored more than ninety publications-directed towards academics and practitioners-on the topic of risk management and international finance. He is on the editorial board of a number of financial journals and was editor of the Journal of Risk. His work has received several prizes for research. Dr. Jorion has written the first four editions of Financial Risk Manager Handbook (Wiley), as well as Financial Risk Management: Domestic and International Dimensions; Big Bets Gone Bad: Derivatives and Bankruptcy in Orange County; and Value at Risk: The New Benchmark for Managing Financial Risk. He is also a managing director at Pacific Alternative Asset Management Company (PAAMCO), a global fund of hedge funds.