### Applying Duration, Convexity, and DV01

After completing this reading you should be able to:

• Describe a one-factor interest rate model and identify common examples of interest rate factors.
• Define and compute the DV01 of a fixed income security given a change in yield and the resulting change in price.
• Calculate the face amount of bonds required to hedge an option position given the DV01 of each.
• Define, compute, and interpret the effective duration of a fixed income security given a change in yield and the resulting change in price.
• Compare and contrast DV01 and effective duration as measures of price sensitivity.
• Define, compute, and interpret the convexity of a fixed income security given a change in yield and the resulting change in price.
• Explain the process of calculating the effective duration and convexity of a portfolio of fixed income securities.
• Describe an example of hedging based on effective duration and convexity.
• 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.

## Interest Rate Factors

An interest rate factor is a random variable that influences individual interest rates along the yield curve. These factors determine price changes for bonds. They include:

### The Fed

The Federal Reserve influences rates through its policy statements and overnight lending to banks at what’s called the Federal Fund’s Rate. The Fed Fund’s rate is a short-term rate that reflects the long-term one, although this need not be the case as we’ve seen in recent years.

### Inflation

As inflation increases, so do interest rates to maintain the purchasing power of money

### Flight to Quality

This is what investors do in large numbers when they sell higher risk securities like stocks and buy lower risk securities like U.S. T-bonds. Higher demand for U.S. Treasuries pushes up their price, thus reducing interest rates.

Measures of interest rate sensitivity allow investors to evaluate bond price changes as a result of interest rate changes. The ability to measure price sensitivity can be useful, for example, when hedging a position. The hedger will want to know how the security being hedged as well as the hedging tool will respond to interest rate changes.

## DV01 of a Fixed Income Security

DV01 (dollar value of an 01) measures the dollar change in the value of a security for a basis point change in interest rates. DV01 is the change in dollars, not a percentage. It gives the dollar change in bond price for every basis point change in yield. The “01” refers to one basis point (that is, 0.0001).

DV01 is computed as follows:

$$DV01=-\frac { \Delta BV }{ 10000\times \Delta y }$$

Where:

$$\Delta BV$$=change in bond value

$$\Delta y$$=change in yield

Note that yield here implies the yield to maturity.

#### Example of DV01:

Suppose the yield on a zero-coupon bond declines from 5.00% to 4.95%, and the price of the zero increases from $22.45 to$22.87. Compute the DV01.

#### Solution:

$$DV01=-\frac { \Delta BV }{ 10000\times \Delta y }$$

$$=-\frac { 22.87-22.45 }{ 10,000\times \left( -0.0005 \right) } =\frac { 0.42 }{ 5 } =0.84$$

Exam tip: Since the DV01 formula is preceded by a negative sign, when rates decline and prices increase, DV01 will be positive.

## Calculating the Face Amount of Bonds Required to Hedge an Option Position Given the DV01 of Each

As illustrated below, as yields fall and the price approaches $105, the price-yield curve rises more slowly than that of an identical but non-callable bond. At the point where 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, exhibiting negative convexity. As can be seen, so long as the yield remains below level $${ k }^{ \prime }$$, callable bonds will exhibit negative convexity. For yields above $${ k }^{ \prime }$$, however, the bonds will exhibit positive convexity. The underlying concept is quite straightforward: At higher yields, the value of the call becomes very small such that the “collables” behave more or less like the “non-collables”. When convexity is positive, bond returns will increase provided interest rates move. When convexity is negative, movement in either direction will reduce bond returns. Thus, if you, as an investor, wish to be long volatility, you should choose a security whose convexity is positive. If you wish to be short volatility, choose a security whose convexity is negative. ## Barbell Portfolio vs. Bullet Portfolio A barbell is an investment strategy applicable to fixed-income securities whereby half the portfolio is made up of long-term bonds and the other half of short-term bonds. A bullet strategy, on the other hand, is an investment strategy where the investor buys bonds concentrated only in the intermediate maturity range. Given the prices, coupon rates, maturities, yields, durations, and convexities of a set of bonds, it is possible to construct a barbell portfolio that has the same cost and duration as the bullet portfolio. This involves determining the proportion of each security in the barbell that should be bought such that their total value equals that of the bullet. ### Why would we do this? For the same amount of duration risk, the barbell portfolio has greater convexity, which means that its value will increase more than the value of the bullet when rates rise or fall. ### Advantages and Disadvantages of Barbell Portfolios #### Advantages: • Can potentially achieve higher yields than would be possible under a bulleted approach • If rates rise, the investor has the opportunity to reinvest the proceeds of the shorter term securities at the higher rate • Less risk that the investor will be forced to reinvest their funds at lower rates when their bonds mature. • Offers greater diversification than a bullet strategy • Presence of short-term securities provides the investor with liquidity and flexibility to deal with potential emergencies. #### Disadvantages: • Presence of long-term securities increases volatility and the potential for capital losses if rates rise • A steepening yield curve may force the investor to invest short-term proceeds in low-yielding bonds. ### Advantages and Disadvantages of Bullet Portfolios #### Advantages: • May be more liquid compared to a barbell portfolio • Spreading out bond purchases ensures higher yields when rates are rising • The investor need not have a “war chest” at the onset since the portfolio is built gradually. #### Disadvantages: • When the yield curve flattens (short rates go up; long rates go down), the barbell OUTPERFORMS the bullet. Note: Sometimes the bullet and barbell have the same DURATION but they will have different CONVEXITIES. ## Questions ### Question 1 A portfolio manager controls CAD 100,000,000 par value of zero coupon bonds expiring in 5 years. The bonds’ current yield is 6%, and, upon scrutinizing market intelligence, the portfolio manager has reason to believe interest rates will rise. To hedge the exposure, the manager wishes to sell part of the 5-year bond position. He will then use the proceeds from the sale to finance the purchase of zero-coupon bonds maturing in 3 years and yielding 4%. If the manager intends to reduce the duration of the combined position to 3.5 years, determine the market value of the 3-year bonds the manager should purchase: 1.$22 million
2. $55.56 million 3.$55.0 million
4. \$45 million

First, compute the current market value of the portfolio:

$${ market\quad value }_{ portfolio }=P=A\times { exp }^{ -r }=100\times { exp }^{ -0.06\times 5 }=74.0818 \quad million$$

Now, let $$X$$ represent the market value of the zero coupon bond with a maturity of 3 years

$$5\times \frac { P-X }{ P } +3\times \frac { X }{ P } =3.5$$

$$5\left( P-X \right) +3X=3.5P$$

$$5P-5X+3X=3.5P$$

$$1.5P=2X$$

$$X=0.75P=55.5614\quad million$$

### Question 2

Susan Mendoza, FRM, has been asked to evaluate the sensitivity of an investment-grade callable bond using her employer’s valuation system. Information on the bond, including the embedded option, is presented in the table below. The prevailing interest rate environment is flat at 10%.

Value in USD per USD 100 face value

$$\begin{array}{|l|l|l|} \hline Level\quad of\quad interest\quad rate & Callable\quad bond & Call\quad option \\ \hline 9.98\% & 106.33472 & 4.0617 \\ \hline 10.00\% & 105.52241 & 4.0421 \\ \hline 10.02\% & 104.25832 & 4.0152 \\ \hline \end{array}$$

Required:

Determine the DV01 of a comparable option-free bond that has the same maturity and coupon rate as the callable bond.

1. 0.53
2. 0.56
3. 0.005
4. 0.05

Any existing call option on a bond reduces the bond’s price. Therefore,

$$Price\quad of\quad comparable\quad option-free\quad bond=price\quad of\quad callable\quad bond+price\quad of\quad call\quad option.$$

At a rate of 10.02%,

$$Price\quad of\quad comparable\quad option-free\quad bond=104.25832+4.0152=108.27352$$

At a rate of 9.98%,

$$Price\quad of\quad comparable\quad option-free\quad bond=106.33472+4.0617=110.39642$$

$$DV01=-\frac { \Delta BV }{ 10000\times \Delta y }$$

Where:

$$\Delta BV$$=change in bond value

$$\Delta y$$=change in yield

$$DV01=-\frac { 108.27352-110.39642 }{ 10,000\times \left( 10.02\%-9.98\% \right) }$$

$$=-\left( -\frac { 2.1229 }{ 4 \quad basis \quad points } \right) =0.5307$$