Topics 58-62

Question 1

Three steps in the bond valuation process

Answer 1

There are three steps in the bond valuation process:

Step 1: Estimate the cash flows over the life of the security. For a bond, there are two types of cash flows:

• the coupon payments and
• the return of principal.
• *Step 2:** Determine the appropriate discount rate based on the risk of (uncertainty about) the receipt of the estimated cash flows.
• *Step 3:** Calculate the present value o f the estimated cash flows by multiplying the bond’s expected cash flows by the appropriate discount factors.

Question 2

Bond Price Quotations

Answer 2

Question 3

Identify the components of a U.S. Treasury coupon bond, and compare and contrast the structure to Treasury STRIPS, including the difference between P-STRIPS and C-STRIPS

Answer 3

Zero-coupon bonds issued by the Treasury are called STRIPS (separate trading of registered interest and principal securities). STRIPS are created by request when a coupon bond is presented to the Treasury. The bond is “stripped” into two components: principal and coupon (P-STRIPS and C-STRIPS, respectively).

The Treasury can also retire a STRIP by gathering the parts up to reconstitute, or remake, the coupon bond. C-STRIPS can be put with any bond to reconstitute, but P-STRIPS are identified with specific bonds— the original bond that it was stripped from. What this means is that the value of a P-STRIP comes from the underlying bond. If the underlying was cheap, the P-STRIP will be cheap. If the underlying was rich, the P-STRIP will also be rich.

STRIPS are of interest to investors because:

• Zero-coupon bonds can be easily used to create any type of cash flow stream and thus match asset cash flows with liability cash flows (e.g., to provide for college expenses, house-purchase down payment, or other liability funding). This mitigates reinvestment risk. (The concept of reinvestment risk will be discussed in later topics.)
• Zero-coupon bonds are more sensitive to interest rate changes than are coupon bonds. This could be an issue for asset-liability management or hedging purposes.

STRIPS do have some disadvantages, which include the following:

• They can be illiquid.
• Shorter-term C-STRIPS tend to trade rich.
• Longer-term C-STRIPS tend to trade cheap.
• P-STRIPS typically trade at fair value.
• Large institutions can potentially profit from STRIP mispricings relative to the underlying bonds. They can do this by either buying Treasuries and stripping them or reconstituting STRIPS. Because of the cost involved with stripping/reconstituting, investors generally pay a premium for zero-coupon bonds.

Question 4

Accrued Interest in bond valuation

Answer 4

Question 5

Day-Count Convention in bond valuation

Answer 5

Question 6

Dirty price of a bond

Answer 6

The dirty price is the price that the seller of the bond must be paid to give up ownership. It includes the present value of the bond plus the accrued interest. The clean price is the dirty price less accrued interest:

clean price = dirty price — accrued interest

Note that the dirty price includes the discounted value of the next coupon so that the method of calculating accrued interest does not matter. As long as the clean price is calculated as: dirty price — accrued interest, the sum of the clean price and accrued interest will equal the dirty price.

Question 7

Future value of a bond, holding period return

Answer 7

Question 8

Define par rate and describe the equation for the par rate of a bond.

Answer 8

The par rate at maturity is the rate at which the present value of a bond equals its par value.

Question 9

Assess the impact of maturity on the price of a bond and the returns generated by bonds.

Answer 9

In general, bond prices will tend to increase with maturity when coupon rates are above the relevant forward rates. The opposite holds when coupon rates are below the relevant forward rates (i.e., bond prices will tend to decrease with maturity in this scenario).

Question 10

Yield Curve Shapes

Answer 10

Question 11

Parallel shift of yield curves

Answer 11

Question 12

Yield curve twists

Answer 12

Yield curve twists refer to yield curve changes when the slope becomes either flatter or steeper. With an upward-sloping yield curve, a flattening of the yield curve means that the spread between short- and long-term rates has narrowed. Conversely, a steepening of the yield curve occurs when spreads widen.

Question 13

Yield curve butterfly shifts

Answer 13

Question 14

The net realized return for a bond, reinvestment risk

Answer 14

The net realized return for a bond is its gross realized return minus per period financing costs. Cost of financing would arise from borrowing cash to purchase the bond. Even though borrowing cash to pay for the entire price of the bond would technically reduce the initial cash outlay to zero, convention is to use the initial bond price as the beginning-ofperiod value.

In order to compute the realized return for a bond over multiple periods, we must keep track of the rates at which coupons received are reinvested. When a bondholder receives coupon payments, the investor runs the risk that these cash flows will be reinvested at a rate that is lower than the expected rate. For example, if interest rates go down across the board, the reinvestment rate will also be lower. This is known as reinvestment risk.

Question 15

Bond equivalent yield (BEY)

Answer 15

The yield to maturity calculated above (2 x the semiannual discount rate) is referred to as a bond equivalent yield (BEY), and we will also refer to it as a semiannual YTM or semiannual-pay YTM.

Question 16

Answer 16

Question 17

The Limitations of Traditional Yield Measures

Answer 17

• Reinvestment risk is a major threat to the bond’s computed YTM, as it is assumed in such calculations that the coupon cash flows can be reinvested at a rate of return that’s equal to the computed yield (i.e., if the computed yield is 8%, it is assumed the investor will be able to reinvest all coupons at 8%).
• Reinvestment risk becomes more of a problem with longer term bonds and with bonds that carry larger coupons. Reinvestment risk, therefore, is high for long-maturity, high-coupon bonds and is low for short-maturity, low-coupon bonds.
• The realized yield on a bond is the actual compound return that was earned on the initial investment. It is usually computed at the end of the investment horizon. For a bond to have a realized yield equal to its YTM, all cash flows prior to maturity must be reinvested at the YTM, and the bond must be held until maturity. If the “average” reinvestment rate is below the YTM, the realized yield will be below the YTM. For this reason, it is often stated that: The yield to maturity assumes cash flows will be reinvested at the YTM and assumes that the bond will be held until maturity.

Question 18

Coupon Effect

Answer 18

If two bonds are identical in all respects except their coupon, the bond with the smaller coupon will be more sensitive to interest rate changes. That is, for any given change in yield, the smaller-coupon bond will experience a bigger percentage change in price than the larger-coupon bond. All else being equal:

• The lower the coupon rate, the greater the interest-rate risk.
• The higher the coupon rate, the lower the interest-rate risk.

Question 19

Total price appreciation for a bond

Answer 19

The total price appreciation for a bond is equal to its price at t minus its price at t— 1. Given the passage of time, total price appreciation results from moving along the original term structure, R , from t— 1 to rand accounts for changes in bond spread, s, from t— 1 to t.

Mathematically, it can be represented as follows:

total price appreciation = BV_t(R_t, s_t) - BV_t-1 (R_t-1,s_t-1)

Question 20

The carry-roll-down, rate changes, spread change components for a bond

Answer 20

Question 21

Identify the most common assumptions in carry roll-down scenarios, including realized forwards, unchanged term structure, and unchanged yields

Answer 21

Traders make investment return calculations based on their expectations, and many traders will consider scenarios where rates do not change. Given this expectation, term structure choices for no change scenarios include: realized forwards, unchanged term structure, and unchanged yields.

• The realized forward scenario assumes that forward rates are equal to expected future spot rates, and over the investment horizon, these forward rates will be realized. This implies, for example, that an investor could earn the same return by investing in a 3-year bond or by investing in a 3-year bond and then a 2-year bond after the 3-year bond expires. This means that the one-period gross realized return will equal the prevailing one-period rate.
• The unchanged term structure scenario simply assumes that the term structure will remain unchanged over the investment horizon. This means that the gross realized return will depend greatly on the relationship between the bond’s coupon rate and the last forward rate before the bond matures. This scenario implies that there is a risk premium built into forward rates.
• As the name suggests, the unchanged yields scenario assumes that bond yields remain unchanged over the investment horizon. This means that the one-period gross realized return will equal a bond’s yield (i.e., its yield to maturity). Thus, this scenario assumes that bond coupon payments are reinvested at the YTM. As stated earlier, there are limitations to this reinvestment assumption since the term structure is unlikely to be flat and remain unchanged.

Question 22

Define and compute the DV01 of a fixed income security given a change in yield and the resulting change in price

Answer 22

Question 23

Define, compute, and interpret the effective duration of a fixed income security given a change in yield and the resulting change in price

Answer 23

Macaulay duration is an estimate of a bond’s interest rate sensitivity based on the time, in years, until promised cash flows will arrive. Since a 5-year zero-coupon bond has only one cash flow five years from today, its Macaulay duration is five.

Periodic market yield = YTM / number of coupon periods per year​

Question 24

Compare and contrast DV01 and effective duration as measures of price sensitivity.

Answer 24

Question 25

Proxy for convexity of a bond

Answer 25

Question 26

Price Change Using Both Duration and Convexity

Answer 26

Question 27

Explain the impact of negative convexity on the hedging of fixed income securities

Answer 27

With callable debt, the upside price appreciation in response to decreasing yields is limited (sometimes called price compression).

When the price begins to rise at a decreasing rate in response to further decreases in yield, the price-yield curve “bends over” to the left and exhibits negative convexity.

Thus, in Figure 2, so long as yields remain below level y*, callable bonds will exhibit *negative convexity*, however, at yields *above level y, those same callable bonds will exhibit positive convexity. In other words, at higher yields the value of the call options becomes very small, so that a callable bond will act very much like a noncallable bond. It is only at lower yields that the callable bond will exhibit negative convexity.

Question 28

Construct a barbell portfolio to match the cost and duration of a given bullet investment, and explain the advantages and disadvantages of bullet versus barbell portfolios

Answer 28

A barbell strategy is typically used when an investment manager uses bonds with short and long maturities, thus forgoing any intermediate-term bonds. A bullet strategy is used when an investment manager buy bonds concentrated in the intermediate maturity range.

The advantages and disadvantages of a barbell versus a bullet portfolio are dependent on the investment manager’s view on interest rates. If the manager believes that rates will be especially volatile, the barbell portfolio would be preferred over the bullet portfolio.

Question 29

Yield curve risk

Answer 29

It is more realistic to recognize that rates in different regions of the term structure
are not always correlated. The risk that rates along the term structure move differently (i.e., nonparallel shifts) is called yield curve risk.

Question 30

Describe key-rate shift analysis

Answer 30

The most common key rates used for the U.S. Treasury and related markets are par yield bonds — 2-, 5-, 10- and 30-year par yields. If one of these key rates shifts by one basis point, it is called a key rate shift.

A key rate ‘01 is the effect of a dollar change of a one basis point shift around each key rate on the value of the security.

Question 31

Describe the key rate exposure technique in multi-factor hedging applications

Answer 31

For every basis point shift in a key rate, the corresponding key rate ‘01 provides the dollar change in the value of the bond. Similarly, key rate duration provides the approximate percentage change in the value of the bond. Key rate duration works off 100 basis point changes, so it is the percentage of price movement for every 100 basis point change in rates.

Question 32

Calculate the key rate exposures for a given security, and compute the appropriate hedging positions given a specific key rate exposure profile

Answer 32

Question 33

Relate key rates, partial ‘01s and forward-bucket ‘01s, and calculate the forward bucket ‘01 for a shift in rates in one or more buckets

Answer 33

With more complex portfolios that contain swaps, partial ‘01s and forward-bucket ‘01s are often used instead of key rates. These approaches are similar to the key rate approach, but instead divide the term structure into more parts. Risk along the yield curve is thus measured more frequently, in fact daily.

A partial ‘01 is the change in the value of the portfolio from a one basis point decrease in the fitted rate and subsequent refitting of the curve. Forward-bucket ‘01s are computed by shifting the forward rate over several regions of the term structure, one region at a time, after the term structure is divided into various buckets.

Question 34

Construct an appropriate hedge for a position across its entire range of forward bucket exposures

Answer 34

In order to set up a proper hedge for a swap position across an entire range of forward bucket exposures, the hedger will determine the various forward-bucket exposures for several different swaps and select the hedge that contains the lowest forward-bucket exposures in net position.

Question 35

Apply key rate and multi-factor analysis to estimating portfolio volatility

Answer 35

Multifactor approaches to hedging, such as key rate and bucket shift approaches, can be used to estimate portfolio volatility effects because they incorporate correlations across a variety of interest rate effects.