FI-2 Lecture 2

Yield to Maturity

Definition

YTM is defined to be the single interest rate used to discount all of the bond’s future payments. The YTM satisfy

Remark:

YTM does not account for what an investor does with the coupon payments.

If two different bonds with the same maturity (but different coupon rates) have different YTM, it’s not necessarily the case that the security with the higher YTM represents a “better investment”.

Even if a bond is held until maturity, the YTM does not necessarily represent “the return” on the investment. When cash flows arrive at multiple dates, it is not clear how to define a return on the investment.

If the YTM remains unchanged between two successive coupon payments, then this common value represents the yield associated with holding the bond over that period of time.

It is theoretically possible to have a negative YTM, but hardly ever occurs in practice in the US.

If the spot-rate curve is not flat, then the YTM cannot be used to discount individual cash flows of a bond (expcet in the case of ZCB where there is only one payment)

The YTM can always be used to discount all of the bond’s cash flows, add the results together, and obtain a price for the bond. That is, YTM applies to the bond as a whole rather than to individual payments.

Computation

Coupon Bonds

Let , then we have

solve the equation and recover through .

For Zeros with maturity , the YTM is simply the spot rate

Annuities

If the current price of an annuity is , then the YTM satisfies

where . It can be observed that

Annuities’ YTM is independent of the payment size

Annuities’ YTM is depend on the maturity

Approximation:

Perpetuities

Derive from the price of Annuities,

therefore

which has straightforward intuition. For perpetuities making payments times per year and having YTM (expressed according to compounding times per year), the same logic gives

Effective Yield to Maturity

If there is a security that will make payments at each of the time then the effective yield to maturity is the unique number satisfying

where is the current price of security.

Continuously Compounded YTM

If we have a security that will make payments at each of the times , then the continuously compounded yield to maturity is the unique number satisfying

where is the current price of the security.

Remark:

if the payments can be both positive and negative, then it can happen that there is no YTM or there can be more than one or there is exactly one.

Proposition

Price / FV and Coupon Rate / YTM

For coupon bonds, we have

Proof:

In practice, prices of coupon bonds are often quoted by giving the YTM.

YTM and Spot Rates

The YTM is always between the lowest spot rate and the highest spot rate for times up to maturity of the bond

If the term structure is flat (i.e. the spot rates are constant), then the common spot rate is equal to the YTM

Pull to Par: as the time to maturity shortens, the prices of premium bonds fall to par and the prices of discount bonds rise to par.

YTM and Coupon Rate

Assuming that the bond has just been issued or that a coupon payment has just been made, the YTM of a coupon bond is the unique number satisfying

where . Using the formula for the price in terms of the discount factors, we have

Cancelling out in the equation, the YTM of a coupon bond generally depends on the coupon rate and on the maturity .

In general, there is no simple relationship between and YTM. However, if the spot rates are monotonic (i.e. always increasing or always decreasing) we can order the YTM of coupon bonds having the same maturity fi we know the ordering of the coupon rates.

Consider two coupon bonds having the same maturity , but different coupon rates (both are assumed to pay coupons twice per year, with the next coupon payment at time ). Let and denote the YTM of these bonds.

If for , then

If for , then

Yield Curves

For US Treasury securities, there are three yield curves we will look at:

Zero-Coupon Yield Curve: Same as the spot-rate curve.

Annuity Yield Curve: Plot of the yield to maturity for annuities making payments every 6 months, versus maturity of the annuity. (also called the self-amortizing yield curve).

Par-Coupon Yield Curve: Plot of the yield to maturity for par coupon bonds versus maturity of the bond.

Remarks:

Each one of these yield curves completely determines the other two.

The par-coupon yield is always between the zero-coupon yield and the annuity yield, but whether the zero-coupon yield is above or below the annuity yield depends on the shape of the zero-coupon yield curve.

The Fed publish par-coupon yield curve.

Rather than interpolating to fit the yield curve (or discount function) exactly at all observed points, it is sometimes better to use a “least-squares” type of approximation based on a family of curves whose shapes are adapted to the term structures produced by certain types of models. A very important example of such a family of curves is the Nelson-Siegel family with four parameters

Accrued Interest

We can divide the bond’s market price (full price or invoice price) into two parts: flat price and accrued interest.

flat price = “clean price”, full price = “dirty price”

It is the flat price that is quoted

For a coupon bond the accrued interest is computed using a simple interest convention by the formula

where is the fraction of a coupon period that elapses between the last coupon payment and settlement. For US treasury notes and bonds, the actual/actual convention is used, thus

where is the actual number of days between the last coupon payment and the next coupon payment and is the number of days between the last coupon payment and settlement.

Remarks:

Current Yield Between Coupon Payments: Defined to be for bonds in between coupon payments.

Coventions for T-Bills

Prices of T-Bills are often quoted in terms of the so-called discount yield . The formula is

where is the price per 100 face and is the number of days between settlement and maturity.

Remark for the definition of :

it divides the dollar gain by 100 rather than (eaiser to compute)

it assumes a 360-day year

The bond-equivalent yield (also called the coupon-equivalent yield) corrects these two defects. For , we have

Sensitivity Analysis

Even those fixed income securities having no risk of default and whose payment dates and amounts are known with certainty at the time of issue can face interest rate risk. Roughly speaking, as interest rates risk, the market prices of previously issued bonds will drop.

Basic Idea

Suppose the price of a security is given by

where is some interest rate factor (or variable).

The first-order approximation

The second-order approximation

DV01

The abbreviation DV01 the change in dollar value in price resulting from a change in the interest rate factor of one basis point (so the denominator is ).

PVBP stands for price value of a basis point, meaning the same thing as DV01.

Remark:

DV01 if often quoted for some specified amount of face. And sometimes the opposite sign convention is used.

The “change” in price is approximated, based on that is small.

The quantity represents the slope of a secant line to get the graph of the price function . If we have an analytical formula for we can calculate the derivative , and thus

Duration

Another measure of interest-rate risk, defined as

Duration measures the relative change in price resulting from a change in (not basis point).

Remark:

A change in of one basis point should be entered as , and a change in of one percent should be entered as .

If , then the duration equals the DV01.

Convexity

Convexity measures how the interest rate sensitivity changes when the interest rate factor changes. Definition

This assumes that we have an explicit expression for the price as a function of . Without such a formula, convexity must be approximated numerically.

Remark:

Some authors define convexity as

Some terminology used in relation to convexity:

Positive Convexity: Long on Volatility
Negative Convexity: Short on Volatility

Portfolios’ Sensitivity

Suppose that a portfolio is being built using fixed-income securities , and each of the securities has dollar price . An investor builds a portfolio with total initial capital , buying shares of , thus

(some of the can be negative here).

For the entire portfolio we have

where are the DV01, duration, and convexity of the i-th security.

Coupon Bonds’ Sensitivity

DV01 for Coupon Bonds

Since

thus

For par coupon bonds ()

Duration for Coupon Bonds

Macaulay Duration

For securities with deterministic cash flows, the price is

thus

The Macaulay duration is defined as

The numerical value of the is fairly close to the numerical value of the duration. But the Macaulay duration has the very nice propery that it is excatly equal to the maturity for a zero coupon bond.

For coupon bonds, the Macaulay duration is

For fixed , we have

Duration for special Coupon Bonds

For par coupon bonds ()

For Annuities

For Perpetuities

thus

Convexity for Coupon Bonds

Leveraging equation to calculate and Convexity , we have

Special Coupon Bonds

Zero Coupon Bonds

Remarks:

Using Calculus, we can show that (when the yield is held fixed at some value ), the DV01 of a ZCB increases with increasing maturity until reaches

and decreases for larger than this value, tending to be zero as . The phenomenon of DV01 decreasing as maturity increases for a ZCB can certainly occur in the actual markets.

The Macaulay duration of a ZCB always equals the maturity.

For fixed , the convexity of a ZCB increases (quadratically) with increasing .

For fixed , the convexity of a ZCB decreases with increasing .

Par Coupon Bonds ()

Perpetuities

For a perpetuity that pays twice per year

Rules of Thumb for Coupon Bonds

Note: Unless the yield curve is flat, changing the coupon rate or maturity of a bond will generally change the yield, so it may not be realistic to think of changing bond parameters “one at a time”. Besides, some of these rules (e.g. duration increases with maturity) are valid under typical market conditions, but could be violated under extreme conditions.

DV01: Rules of Thumb for Coupon Bonds

Roughly speaking:

DV01 increases with increasing maturity (could be violated for discount bonds having large maturities)

DV01 increases with increasing coupon

DV01 decreases with increasing yield

To state more precisely, for , let denote the DV01, then:

For fixed ,

For fixed with , increases with increasing

For fixed with , there is a critical maturity such that increases with until reaches and then decreases for larger than

For fixed , increases with increasing

For fixed , decreases with increasing

Duration: Rules of Thumb for Coupon Bonds

Roughly speaking:

Duration increases with increasing maturity (could be violated under extreme conditions)

Duration decreases with increasing coupon rate

Duration decreases with increasing yield to maturity

To state more precisely, for , let denote the Macaulay duration of a bond with coupon rate , yield to maturity , and maturity . Then we have the following results:

For fixed ,

For fixed , and , increases with increasing

For fixed , and , there is a critical maturity (depending on ) such that increases with when and then decrease as . (Typically, the is much larger than the maturities of traded bonds)

For fixed , decreases with increasing

For fixed , decreases with increasing

Convexity: Rules of Thumb for Coupon Bonds

Roughly speaking:

Convexity increases with increasing maturity (this can be violated for deeply discounted bonds)

Convexity decreases with increasing coupon

Convexity decreases with increasing yield to maturity

Barbells Versus Bullets

In the asset-liability context, barbelling refers to the use of a portfolio of short-term and long-term bonds rather than intermediate term bonds.

For fixed , define the function by

and assume that the term structure is flat at .

Notice that is a convex function because . Let be given and . Let be given. Suppose that we build a barbell portfolio by investing in a zero with maturity and in a zero with maturity .

The duration and convexity of the barbell are given by

The bullet portfolio having the same price (namely 1) and the same duration is obtained by investing $1 in a zero with maturity

The convexity of the bullet portfolio is

Since is a convex function, we have

that is

In fact, the inequality if strict because is strictly convex.

Remark:

Trades in which an intermediate-maturity security is purchased (or sold) and two securities whose maturities straddle the intermediate maturity are sold (or purchased) are referred to as butterfly trades.

Some Other Issues

Callable Bond

A callable bond is a bond that the issuer has the right to buy back at some fixed set of prices on some fixed dates before maturity.

Suppose that there is a bond with and maturity callable in one year at par. (For simplicity, we assume that there is only one possible call date.) The value of the callable bond should be the difference in value between the underlying bond and a European call option on the underlying bond struck at par with exercise data year. (Here the underlying is a non-callable coupon bond with and )

In general, callable bonds exhibit positive convexity when market rates are high (relative to the coupon rate) becuase there is little likelihood of the bond being called. However, when the current market rates are low, the bond is subject to “price compression” and the price-yield curve lies below its tangent lines, having negative convexity but positive duration.

For callable bonds, a yield to call is frequently quoted, especially when current conditions suggest that the bond may be called. It is simply the yield to maturity computed under the assumption that the bond will be called at the earliest future call date.