CHAPTER 4

Background on Traded Instruments

Introduction

• Market risk: – the possibility of losses resulting from unfavorable

market movements. – It is the risk of losing money because the perceived

value of an instrument has changed.

• The difference between credit risk and market risk – In credit risk there needs to be a default or failure by a

counterparty to fulfill an obligation– In market risk, we deal simply with changes in the

prices that investors are prepared to pay

Introduction

• Before launching into the mathematics of risk measurement, it is necessary to have a reasonable understanding of the types of trades and instruments that give rise to the risk

• This chapter will discuss the main instruments that banks trade—along with an explanation of how to value each of the instruments

• Valuation is very important in risk measurement because risk is all about potential changes in value

Introduction

• Typically, a bank trades the following instruments:– Debt instruments, also known as fixed income or bon

ds– Forward rate agreements– Equities, also known as stocks– Foreign exchange, also known as FX or currency– Forwards and futures– Swaps– Options

DEBT INSTRUMENTS

• Debt instruments are securities that provide interest payments but no ownership claim on the issuer

• Four important features for debt instruments:– Maturity– Issuer credit rating– Payment structure– Currency

DEBT INSTRUMENTS

• Maturity – the time left until final payment

• Issuer Credit Ratings– the credit risk of most bonds is rated by a thir

d-party company called a rating agency, such as Standard & Poor's

– The ratings generally correspond to probabilities of default

– Bonds with high probabilities of default trade at lower prices than risk-free bonds

DEBT INSTRUMENTS

– As we discus later in the chapter, a lower price implies a higher interest rate

– The difference between the interest rate for a risky bond and the interest rate for a risk-free bond of the same maturity is called the credit spread

DEBT INSTRUMENTS

• Payment Structure– The payments of interest on a bond are called

coupons and are either fixed or floating– Fixed-rate bonds pay the same percentage every

time, and the rate is fixed when the bond is first issued

– Floating-rate bonds pay a variable percentage and the interest rate is reset periodically to a prevailing market rate

• Currency– Debt instruments can be denominated in any currency

The Valuation of Bonds

• A Single Payment– Any bond promising to pay a certain amount

at a future point in time has a value– The value of a bond is the price that investors

are prepared to pay today to own that bond– If we consider a bond that promises to make a

single payment, at time t, then the current value of the bond and the future payment amount can be used to define a discount factor (DF):

The Valuation of Bonds

• The discount factor will be less than one, as it is almost always better to have cash now than the same amount promised in the future

• The discount factor includes all the effects that cause the value to be less than the promised amount, including the effects of inflation and the possibility of default.

• If the cash flow is certain (e.g., a fixed-rate government bond), then the discount is called the risk-free discount

The Valuation of Bonds

• If the cash flow is risky, e.g., if there is a possibility of default, the value will be less, and therefore, the discount factor will be smaller

• The discount factor for a given maturity, t, can be used to define a discount rate, rt, which is more familiarly known as an interest rate.

• The usual expression for the discount rate is as follows:

Zero-coupon bonds

• Zero-coupon bonds are bonds that do not pay interest explicitly.

• The bond is sold at a discount to face value, with the difference between the face value and the sale price implicitly being the interest payment.

• The value of a risk-free zero-coupon bond is simply the amount of the payment multiplied by the risk-free discount factor:

Coupon-paying bonds

• Coupon-paying bonds have a series of fixed-intermediate-interest payments.

• The value of a coupon-paying bond is the sum of the value of the individual payments:

Measuring Interest Rate Sensitivity Using Duration

• Duration is a measure of the interest rate sensitivity of the value of a bond or loan.

• This is useful because once we know the duration and the possible movement of interest rates, we have a measure of how much value the bank could lose in its bond portfolio

• If a bond had a duration of seven, it would mean that that value of the instrument would decline by seven percent if interest rates rose one percent

Duration

Macauley duration

• There are two types of duration: Macauley duration and modified duration

• Macauley duration is easier to calculate, but modified duration is more accurate

• Macauley duration is simply the average time for cash flows, weighted by their present values:

Modified duration

Modified duration

• If we had not divided by V we would have had the absolute price sensitivity, which is called duration dollars:

Modified duration

Durations

• The unit of duration is time

• This may seem strange, but it is because duration is the sensitivity with respect to interest rates, and interest rates are in units of increase per time period

• Therefore, since duration is change in percentage value per change in interest rate, the unit of duration is time

Durations

• Why we bother with duration rather than just using the derivative directly?

• The simple answer is that it is market custom.

• Many bank financial reports will give "duration," but few will give "the derivative of value with respect to interest rates.“

• Therefore, it is necessary to understand the meaning of duration

Limitations of Durations

• Duration has significant limitations as a measure of risk.

• It is the first derivative of value with respect to rates.

• As such, it is a linear measure and only describes changes in the value of a bond based on small parallel shifts in the yield curve.

• It does not describe the value changes that could result from the convexity of bond prices or complex shifts in the yield curve

Limitations of Durations

• Convexity is the nonlinear relationship between the price of a bond and its yield, as illustrated in Figure 4-2.

• As duration is a linear description of value changes, it becomes increasingly inaccurate as the rate change becomes larger

• To account for convexity, we need to give up the linear approximation of duration and use full bond valuation.

Limitations of Durations

OPTIONS

• An option is a type of derivative contract• An option gives the holder the right but not the

obligation to buy or sell an underlying asset at a future time, at a predetermined price

• The predetermined price is called the strike price.

• The party holding the right to choose is said to be long the option, and the counterparty who sold or wrote the option is said to be short the option

OPTIONS

• Options can be categorized according to the rights given to the holder:– Puts: A put option gives the holder the right t

o sell the underlying security and receive a predetermined strike price.

– Calls: A call option gives the holder the right to buy the underlying security by paying a predetermined strike price.

OPTIONS

• Options are similar to futures, but with one important difference: the holder can choose whether to exercise the option at the time of contract expiration

• Options carry an "aura of mystery" because they can be difficult to value

• There are many option classifications

The Black-Scholes Equation

• The Black-Scholes equation was developed by two professors who were later awarded a Nobel Prize for their accomplishment.

• The Black-Scholes equation calculates the current value of holdingan option on a stock

• The equation assumes that the option can only be exercised at maturity, and that there are no dividends or transaction costs

• It also assumes that the percentage change in the stock price has Normal distribution

The Black-Scholes Equation

• With these restrictions, the value of a call option on a stock is a function of the following five variables– S = Spot price of the underlying security– T = Time left to maturity of the option– X = Strike price of option– r = Risk-free interest rate to the time of option

expiration– σ= Annual volatility for the underlying security.

The Black-Scholes Equation