To price a swap, we need to determine the present value of cash flows for each leg of the transaction. In an interest rate swap, the fixed leg is fairly straightforward because the cash flows are specified by the coupon rate set at the time of the agreement. Pricing the floating leg is more complex because, by definition, the cash flows change with future changes in interest rates. The forward rate for each floating payment date is calculated by using the forward curves.

### Yield to Maturity

Yield to maturity (YTM) is the expected rate of return for a bond that is held until its maturity, assuming that all coupon and principal payments are made in full when due and that coupons are reinvested at the original YTM.

YTM is perhaps the most familiar pricing concept in bond markets. To understand this concept, it is important to clarify how YTM is related to spot rates and a bond’s expected and realized returns.

**How is the yield to maturity related to spot rates?**

In bond markets, most bonds outstanding have coupon payments and many have various options, such as a call provision. The YTM of these bonds with maturity *T* would not be the same as the spot rate at *T*. But, the YTM should be mathematically related to the spot curve. Because the principle of no arbitrage shows that a bond’s value is the sum of the present values of payments discounted by their corresponding spot rates, the YTM of the bond should be some weighted average of spot rates used in the valuation of the bond.

The following example addresses the relationship between spot rates and yield to maturity.

** Is the yield to maturity the expected return on a bond? **

In general, it is not, except under extremely restrictive assumptions. The expected rate of return is the return one anticipates earning on an investment. The YTM is the expected rate of return for a bond that is held until its maturity, assuming that all coupon and principal payments are made in full when due and that coupons are reinvested at the original YTM. However, the assumption regarding reinvestment of coupons at the original yield to maturity typically does not hold. The YTM can provide a poor estimate of expected return if (1) interest rates are volatile; (2) the yield curve is steeply sloped, either upward or downward; (3) there is significant risk of default; or (4) the bond has one or more embedded options (e.g., put, call, or conversion).

If either (1) or (2) is the case, reinvestment of coupons would not be expected to be at the assumed rate (YTM). Case (3) implies that actual cash flows may differ from those assumed in the YTM calculation, and in case (4), the exercise of an embedded option would, in general, result in a holding period that is shorter than the bond’s original maturity.

The realized return is the actual return on the bond during the time an investor holds the bond. It is based on actual reinvestment rates and the yield curve at the end of the holding period. With perfect foresight, the expected bond return would equal the realized bond return.

### Yield Curve

The yield curve is a graphical representation of the different interest rates at various maturities.

There are different types of yield curves: spot, forward, par etc. No matter what type of yield curve, it represents different interest rates (at different maturities) at *a single point of time*.

## Slope of the Yield Curves

In developed markets, yield curves are most commonly upward sloping with diminishing marginal increases in yield for identical changes in maturity; that is, the yield curve “flattens” at longer maturities. Because nominal yields incorporate a premium for expected inflation, an upward-sloping yield curve is generally interpreted as reflecting a market expectation of increasing or at least level future inflation (associated with relatively strong economic growth). The existence of risk premiums (e.g., for the greater interest rate risk of longer-maturity bonds) also contributes to a positive slope.

An inverted yield curve is somewhat uncommon. Such a term structure may reflect a market expectation of declining future inflation rates (because a nominal yield incorporates a premium for expected inflation) from a relatively high current level. Expectations of declining economic activity may be one reason that inflation might be anticipated to decline, and a downward-sloping yield curve has frequently been observed before recessions.

A flat yield curve typically occurs briefly in the transition from an upward-sloping to a downward-sloping yield curve, or vice versa.

A humped yield curve, which is relatively rare, occurs when intermediate-term interest rates are higher than short- and long-term rates.

## Links

### Par Curve

The **par curve** is a hypotenetical yield curve for coupon-paying Treasury securities that assumes all securities are priced at par. It represents the yields to maturity on coupon-paying government bonds, priced at par, over a range of maturities. In practice, recently issued (“on the run”) bonds are typically used to create the par curve because new issues are typically priced at or close to par.

The par curve is important for valuation in that it can be used to construct a zero-coupon yield curve. The process makes use of the fact that a coupon-paying bond can be viewed as a portfolio of zero-coupon bonds. The zero-coupon rates are determined by using the par yields and solving for the zero-coupon rates one by one, in order from earliest to latest maturities, via a process of forward substitution known as bootstrapping (a statistical method for estimating a sample distribution based on the properties of an approximating distribution).

### No-arbitrage Principle

The no-arbitrage principle says that tradable securities with identical cash flow payments must have the same price.

### Forward Rate

A forward rate is an interest rate that is determined today for a loan that will be initiated in a future time period. It can be think of as the spot rate in the future.

The term structure of forward rates for a loan made on a specific initiation date is called the forward curve.

Forward rates and forward curves can be mathematically derived from the current spot curve.

### Spot Rate

A **spot rate**, or a **spot interest rate**, is a rate of interest on a security that makes a single payment at a future point in time.

### Term Structure of Interest Rates

The **term structure of interest rates** is a mathematical description of the relationship among market interest rates at various maturities. The graphical representation of the term structure is the yield curve.

### Value at Risk

Value-at-risk (VaR) is the minimum loss that would be expected a certain percentage of the time over a certain period of time given the assumed market conditions. It can be expressed in either currency units or as a percentage of portfolio value. VaR can be viewed as a probabilistic measure of the range of values a firm’s portfolio could lose due to market volatility. This volatility includes effects from changes in interest rates, exchange rates, commodities prices, and other general market risks.

A typical reporting of VAR would be the following statement:

The 5% VaR of a portfolio is $2 million over a one-day period.

The following three points are important in understanding the concept of VaR:

– VaR can be measured in either currency units (in this example, the USD) or in percentage terms. In this example, if the portfolio value is $100 million, the VaR expressed in percentage terms would be 2% ($2 million/$100 million = 0.02).

– VaR is a minimum loss. This point cannot be emphasized enough. VaR is often mistakenly assumed to represent how much one can lose. If the question is, “how much can one lose?” there is only one answer: the entire portfolio. In a $100 million portfolio, assuming no leverage, the most one can lose is $100 million.

– A VaR statement references a time horizon: losses that would be expected to occur over a given period of time. In this example, that period of time is one day. (If VaR is measured on a daily basis, and a typical month has 20–22 business days, then 5% of the days equates to about one day per month.)

These are the three explicit elements of a VaR statement—the frequency of losses of a given minimum magnitude expressed either in currency or percentage terms. Thus, the VaR statement can be rephrased as follows: A loss of at least $2 million would be expected to occur about once every month.

A 5% VaR is often expressed as its complement—a 95% level of confidence. Commonly, the notion, the 5% VaR, is used, but we should be mindful that it does imply a 95% level of confidence.

## History

J.P. Morgan Chairman Dennis Weatherstone demanded a simple report at the end of each day on how the firm’s position could change due to market risk. Morgan analysts came up with VAR as this measure and incorporated it into what became known as the 4:15 report, the report was given to Weatherstone daily at that time to summarize the day’s market events.

The Washington-based Group of Thirty, in a 1993 study headed by Weatherstone, *Derivatives: Practices and Principles*, recommended using VAR as a means of identifying a firm’s overall market risk. Since then, VAR has skyrocketed in popularity and there are few financial institutions which do not envisage making it part of their day-to-day reporting. In fact, many financial regulators, such as the European Economic and Monetary Union, require banks to report their VARs on a regular basis.

Because there are a variety of ways to calculate VAR, J.P. Morgan sought to make its method the industry standard. In 1994 it began giving away copies of RiskMetrics, a program it developed to calculate VAR. Morgan’s Internet web site includes a RiskMetrics section, where individuals or firms can download parts of the program. This initial move into the VAR market did not stop other investment banks from creating their own VAR calculation programs and peddling them to other financial organizations. Many of these programs differ from RiskMetrics, and while RiskMetrics is widely recognized in VAR measurement, the industry has yet to settle on a single calculation method.

### Investment Strategy

An **investment strategy** is an approach to investment analysis and security selection. In the broadest sense, investment strategies are **passive**, **active**, or **semiactive**.

In a passive investment approach, portfolio composition does not react to changes in capital market expectations (passive means not reacting). For example, a portfolio indexed to the MSCI-Europe Index, an index representing European equity markets, might add or drop a holding in response to a change in the index composition but not in response to changes in capital market expectations concerning the security’s investment value. Indexing, a common passive approach to investing, refers to holding a portfolio of securities designed to replicate the returns on a specified index of securities. A second type of passive investing is a strict buy-and-hold strategy, such as a fixed, but non-indexed, portfolio of bonds to be held to maturity.

In contrast, with an active investment approach, a portfolio manager will respond to changing capital market expectations. Active management of a portfolio means that its holdings differ from the portfolio’s benchmark or comparison portfolio in an attempt to produce positive excess risk-adjusted returns, also known as positive alpha. Securities held in different-from-benchmark weights reflect expectations of the portfolio manager that differ from consensus expectations. If the portfolio manager’s differential expectations are also on average correct, active portfolio management may add value.

A third category, the semiactive, risk-controlled active, or enhanced index approach, seeks positive alpha while keeping tight control over risk relative to the portfolio’s benchmark. As an example, an index-tilt strategy seeks to track closely the risk of a securities index while adding a targeted amount of incremental value by tilting portfolio weightings in some direction that the manager expects to be profitable.