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

A hedge ratio determines the amount of par of the hedge position that needs to be bought or sold for every $1 par value of the original position. The goal of hedging is to lock in the value of a position even in the face of small changes in yield. The hedge ratio is given by:

$$ HR=\frac { DV01_{initial \quad position} }{ DV01_{hedge \quad position} } $$

**Example of hedging a position with DV01**

A 20-year semiannual coupon bond has a DV01 of 0.18125. An investor wishes to hedge his position in this bond with another 10-year semiannual coupon bond whose DV01 is equal to 0.11369. Calculate the hedge ratio:

#### Solution

$$ HR=\frac { 0.18125 }{ 0.11369 } =1.59 $$

Interpretation: For every $100 par value of the 20-year bond, short $159 of par of the 10-year bond.

## Effective Duration of a Fixed Income Security

Duration is a measure of the sensitivity of the price of a fixed-income security to a change in interest rates. It’s a rather complex computation that incorporates the security’s present value, yield, coupon, final maturity, and call characteristics. Duration is measured in years.

Bond prices have an inverse relationship with interest rates: if rates rise, a bond’s price is likely to fall. If rates decline, a bond’s price is likely to rise. The bigger the duration, the greater the interest rate risk, (and hence the reward) for bonds.

### How is duration interpreted?

If a fixed-income security has a high duration, this implies investors would need to wait a long period to get their hands on coupon payments and the principal. Even more crucially, the higher the duration, the more the security’s price would fall in the event of a rise in interest rates. The lower the duration, the lesser the security’s price will fall when interest rates rise.

$$ duration=-\frac { 1 }{ P } \left( \frac { \partial P }{ \partial y } \right) $$

## DV01 vs. Dollar Duration

The difference between DV01 and dollar duration is quite clear. While DV01 measures the change in **dollar value** of a security for every basis point change in rates, dollar duration gives the **percentage change** in a security’s value for every unit change in rates. The two have the following relationship:

$$ DV01=duration\times 0.0001\times bond\quad value $$

## Convexity

Duration of a bond is the linear relationship between the bond price and interest rates where, as interest rates increase bond price decreases. It’s a good measure of sensitivity to interest rates when for small/sudden changes in yield. However, for much larger changes in yield, duration is not so desirable since the relationship between price and interest rates tends to be non-linear.

Therefore, convexity measures the degree of the non-linear relationship between price and yield of a bond. Mathematically speaking, convexity is the second derivative of the formula for change in bond prices with a change in interest rates.

$$ Convexity=\frac { 1 }{ BV } \times \frac { { d }^{ 2 }BV }{ { dy }^{ 2 } } $$

The actual calculation of convexity is complex and makes use of tenets learnt in calculus. The formula for convexity approximation is as follows:

$$ convexity=\frac { { BV }_{ -\Delta y }+{ BV }_{ +\Delta y }-2{ BV }_{ 0 } }{ { BV }_{ 0 }\times \Delta { y }^{ 2 } } $$

### Price Change Using Both Duration and Convexity

When used together, duration and convexity offer a better estimation of the percentage price change resulting from a particular percentage change in yield than using duration alone. Convexity is also very helpful when comparing bonds with the same duration.

$$ percentage\quad price\quad change\approx duration\quad effect+convexity\quad effect $$

$$ =\left( -duration\times \Delta y\times 100 \right) +\left( \left( \frac { 1 }{ 2 } \right) \times convexity\times { \left( \Delta y \right) }^{ 2 }\times 100 \right) $$

## Calculating the Effective Duration and Convexity of a Portfolio

Portfolio duration can be calculated as the **weighted sum** of the individual durations. The weight attached to each security is equal to its value as a percentage of total portfolio value.

$$ Portfolio\quad duration=\sum _{ j=1 }^{ K }{ { w }_{ j }{ D }_{ j } } $$

Where:

\({ D }_{ j }\)=duration of the bond \(j\)

\({ w }_{ j }\)=market value of the bond \(j\) divided by total portfolio market value

\(k\)=number of bonds in the portfolio

Portfolio convexity is computed using a similar approach. It’s defined as the value-weighted average of the individual bond convexities making up the portfolio:

$$ Portfolio\quad convexity=\sum _{ j=1 }^{ K }{ { w }_{ j }{ C }_{ j } } $$

## The Impact of Negative Convexity on the Hedging of Fixed Income Securities

When interest rates fall, bond prices generally rise. However, a bond with negative convexity loses value when interest rates fall. Bonds have negative convexity when the yield increases as the duration decreases. In other words, there is a negative correlation between duration and yield, resulting in a downward sloping yield curve. Securities that can have negative convexity include callable bonds, mortgage-backed securities, and bonds that have a repayment option.

An investor in a callable bond has given the issuing company the right to repurchase the bond at a fixed price. Suppose that rates fall, in which case a non-callable bond would see its price increase from $100 to, say, $107. If the issuer has the right to call the bond at, say, $105, will most likely exercise this right, simply because they will be buying the bond cheaply. That effectively generates negative convexity in the bond.

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:

- $22 million
- $55.56 million
- $55.0 million
- $45 million

The correct answer is **B**.

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.

- 0.53
- 0.56
- 0.005
- 0.05

The correct answer is **A**.

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

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

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