Bond Duration

The duration of a level perpetuity is

(1+y) / y • For example, at a 10% yield, the duration of a perpetuity that pays $100 once a year forever will equal 1.10/.10=11 years

So How Would Duration Help?

- If the portfolio duration is chosen appropriately, these two effects will cancel out exactly - When the portfolio's duration is set equal to the investor's horizon date, the accumulated value of the investment fund at the horizon date will be unaffected by interest rate fluctuations

Example: Insurance Company Suppose an insurance company issues a Guaranteed Investment Contract for $10,000. It has a 5-year maturity and a guaranteed interest rate of 8%...

...hence the insurance company has an obligation of $10,000 x (1.08)5 = $14,693.28 in 5 years time

How can a perpetuity, which has an infinite maturity, have a duration as short as 10 or 20 years?

Answer: Because the weight of future cash flows is very small since the PV of future cash flows is small

Immunization - Motivation

Banks and financial institutions have a natural mismatch between the asset side and the liability side - Think about a pension fund, where liabilities are long term... - Or think about a bank that provides long-term loans...

Immunization

Immunization techniques refer to strategies used by investors to shield their overall financial position from exposure to interest rate fluctuations - If this is not taken into consideration, a change in interest rates could wreak havoc with the institution's balance sheet, threatening bankruptcy

Duration - Its Importance

Duration is a key concept in fixed income portfolio management • It is a simple summary statistic of the effective average maturity of the portfolio; • It is a measure of the interest rate sensitivity of the portfolio; • It is an essential tool in immunizing portfolio from interest rate risk

Offsetting Risks

Fixed income investors face two offsetting types of interest rate risks: • Price risk; and • Reinvestment rate risk - Increases in interest rates cause capital losses but at the same time increase the rate at which reinvested income will grow

Rebalancing

Matching durations of assets and liabilities immunizes the portfolio from small changes in interest rates • As interest rates change, durations change • Even if interest rates don't change, just the passage of time changes durations > It is necessary for the portfolio manager to rebalance the portfolio in response to changes in interest rates. That is, the manager must realign the proportions of assets in a portfolio as needed

Example: Rebalancing Continuing with our pension fund that has to make a payment of $19,487. Suppose a year has passed but the interest rate remained at 10%. Hence the present value of this obligation is now $11,000 • The zero-coupon bond has increased in value from $5000 to $5500 with the passage of time, and the duration is now 2 years. The perpetuity has paid a $500 coupon and remains worth $5000 and still has a duration of 11 years. Note: the obligation is still fully funded by the asset position

Rebalancing the weights to immunize the liability with its new duration, we solve the following: [w x 2 years] + [(1-w) x 11 years] = 6 years w = 0.556 We will invest $6111 = 0.556*$11000 in the zero coupon bond, and $4889 in the perpetuity. To do this, we invest the entire $500 coupon payment from the perpetuity into the zero-coupon bond, plus sell an additional $111 of the perpetuity and invest it in the zero

Interest Rate Sensitivity

Recall: 1. As maturity increases, price sensitivity to yield changes increases. 2. Interest rate risk is inversely related to the bond's coupon rate 3. Bond prices are more sensitive to changes in yields when the bond is selling at a lower initial yield to maturity

Example: Obtaining Immunization Consider a pension fund that has to make a payment of $19,487 in 7 years. The market interest rate is 10%. Hence the present value of this obligation is $10,000 • The manager decides to fund this obligation using 3-year zero-coupon bonds and perpetuities paying annual coupons • How can we construct an immunized portfolio?

Step 1: Calculate the liability side's duration. In this case, it is a single-payment obligation with duration of 7 years Step 2: Calculate the duration of the asset side. In this case, we have some zero-coupon bonds...hence their duration is 3 years...and some perpetuities yielding 10%...hence their duration is obtained as 1.10/.10 = 11 years. Say we invest w% in zeros and (1-w)% in perpetuities Step 3: Find the asset mix that sets the duration of assets equal to the 7-year duration of liabilities. Hence, solve the following: [w x 3 years] + [(1-w) x 11 years] = 7 years => w = 0.5 Hence, the manager should half the portfolio ($5000) in the zero-coupon bonds and half ($5000) in perpetuities

Summary of Interest Rate Sensitivity

The concept: • Any security that gives an investor more money back sooner (as a % of your investment) will have lower price volatility when interest rates change • Maturity is a major determinant of bond price sensitivity to interest rate changes, but... • it is not the only factor; in particular the coupon rate and the current YTM are also major determinants Duration is a way to estimate the interest rate sensitivity of a bond by taking into account the timing and amount of each payment to the investor

Bond Price Changes: Example

Using our previous example 1: - The coupon-paying bond sells for $964.54 at the initial semiannual interest rate of 5% - If the bond's semiannual yield increases by 1 basis point (i.e. 0.01%) to 5.01%, how much will the price change? • ΔP = -(D* Δy) × P • ΔP = (1.8853/1.05) * (2*0.01%) * 964.54 •ΔP = 0.35, or the new P = $964.19

Consider the following 8 year 9% coupon annual payment corporate bond:

• Because the bond pays cash prior to maturity it has an "effective" maturity less than 8 years • We can think of this bond as a portfolio of 8 zero coupon bonds with the given maturities • The average maturity of the 8 zeros would be the coupon bond's effective maturity • We need a way to calculate the effective maturity

Duration - Bond Price Changes

• Bond price volatility is proportional to the bond's duration • Duration becomes a natural measure of interest rate exposure

Duration

• Duration is the term for the effective maturity of a bond • Time value of money tells us we must calculate the present value of each of the eight zero coupon bonds to construct an average • We then need to take the present value of each zero and divide it by the price of the coupon bond. This tells us what percentage of our money we get back each year • We can now construct the weighted average of the times until each payment is received. This is the Duration

Key Points to Remember

• Duration measures an "effective" maturity of a bond by weighting the timing of each payment by that payments contribution to the price of the bond • Know how to calculate durations • Use duration to estimate bond price changes to interest rate changes • Using duration, we can immunize our fixed income portfolios from interest rate risk

Bond Price Changes

• Long-term bonds are more sensitive to interest rate movements than are short-term bonds • The duration measure enables us to quantify this relationship • When interest rates change, the proportional change in a bond's price can be related to the change in its yield to maturity

Duration Calculation

• Start by calculating the present value of each payment • Then find the percentage of the present value of each payment to the total price of the bond. This gives you the weight assigned to each cash flow • The weight is then multiplied with the time until that payment is made • The results are then summed up

Duration - Basic Rules

• The duration of a zero-coupon bond equals its time to maturity; • Holding maturity constant, a bond's duration is higher when the coupon rate is lower; • Holding the coupon rate constant, a bond's duration generally increases with its time to maturity. Duration always increases with maturity for bonds selling at par or at a premium to par; • Holding other factors constant, the duration of a coupon bond is higher when the bond's yield to maturity is lower

Example: Insurance Company Now, suppose that this insurance company decides to fund this obligation with a $10,000 of 8% annual coupon bonds, selling at par, and having 6- year maturity. What do the asset side's duration and the liability side's duration look like?

• The duration of the GIC (essentially a zero-coupon bond) is: 5 years • The duration of the coupon bond at 8% is 4.99 years

The weight, w, associated with the cash flow (CF) made at time t is given by

• Where y is the bond's yield to maturity