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Question

You currently hold a 7-year fixed rate bond 5% annually. You would like to hedge against changes in the level and the slope of the yield curve and you plan to use a 1-year zero coupon bond and a 7-year zero coupon bond. Use the following table to compute the adequate positions in the hedging instruments. (10 points)

Maturity        β1            β2            Z(t, 1)
1.00          1.1150      -0.2540       0.9800
2.00          0.9940      -0.3010       0.9600
3.00          0.9640      -0.1470       0.9300
4.00          0.9330       0.0080       0.8900
5.00          0.9300       0.1620       0.8500
6.00          0.9260       0.3160       0.8100
7.00          0.9270       0.4230       0.7700
8.00          0.9270       0.5300       0.7300

Answer 1

Hi Brantley!

This is an interesting fixed-income problem! The key is recognizing that we're trying to hedge two different types of interest-rate risk: changes in the overall level of the yield curve and changes in its slope.

Because we have two risk factors, we'll need two hedging instruments: the 1-year and 7-year zero-coupon bonds.

Let's work through it!

Step 1: Identify the cash flows of the bond.

Assuming a $100 face value and annual coupon payments, the 7-year bond pays:

Years 1–6: $5 annually

Year 7: $105 ($5 coupon + $100 principal)

The present value of each cash flow is calculated using the discount factors Z(t,1) provided in the table.

The bond's price is:

P = 5(0.98 + 0.96 + 0.93 + 0.89 + 0.85 + 0.81) + 105(0.77)

P = $107.95

Step 2: Calculate exposure to each yield-curve factor.

For each factor, we calculate the bond's sensitivity using:

Exposure = Σ [t × CF(t) × Z(t,1) × β(t)]

Here:

1. t = time to maturity
2. CF(t) = cash flow at time t
3. Z(t,1) = discount factor
4. β(t) = sensitivity to the corresponding yield-curve factor

For the first factor (level):

E₁ = Σ [t × CF(t) × Z(t,1) × β₁(t)]

E₁ = 611.9611

For the second factor (slope):

E₂ = Σ [t × CF(t) × Z(t,1) × β₂(t)]

E₂ = 244.4757

These represent the bond's exposures to the two interest-rate risk factors, before applying the common negative sign associated with price sensitivity to yields.

Step 3: Calculate the exposures of the hedging instruments.

For the 1-year zero-coupon bond:

Level exposure:

1 × 0.98 × 1.115 = 1.0927

Slope exposure:

1 × 0.98 × (−0.254) = −0.24892

For the 7-year zero-coupon bond:

Level exposure:

7 × 0.77 × 0.927 = 4.99653

Slope exposure:

7 × 0.77 × 0.423 = 2.27997

Step 4: Construct the hedge.

Let:

x = number of 1-year zero-coupon bonds held

y = number of 7-year zero-coupon bonds held

To eliminate both risk exposures, the combined sensitivities must equal zero.

Level hedge:

611.96105 + 1.0927x + 4.99653y = 0

Slope hedge:

244.47570 − 0.24892x + 2.27997y = 0

Solving these equations simultaneously gives:

x = −46.51

y = −112.31

Step 5: Interpret the results.

The negative signs indicate short positions.

Therefore, to hedge one 7-year, 5% annual coupon bond with a $100 face value, you should:

Short 46.51 units of the 1-year zero-coupon bond.

Short 112.31 units of the 7-year zero-coupon bond.

These positions offset the original bond's first-order exposure to both changes in the level and slope of the yield curve, under the two-factor model provided.

Key takeaway: A single duration hedge generally protects against parallel interest-rate movements. By using two bonds with different maturities, we can construct a hedge against two independent yield-curve risk factors.

One important distinction: this is a first-order hedge, so it does not eliminate convexity risk or guarantee protection against large, nonlinear yield-curve movements.

Hope this helps clarify the process!

Michael