A 6% six-year bond yields 10.5% and a 10% six-year bond yields 8.5%. Calculate the six-year spot rate. Assume annual coupon payments.
To calculate the six-year spot rate, we need to use a process called bootstrapping. Bootstrapping is a method used to derive spot rates from the yields of bonds with different maturities.
Step 1: Calculate the present value (PV) of the future cash flows of each bond using its yield.
For the 6% bond with a yield of 10.5%, we can use the formula:
PV = Coupon / (1 + Yield)^Year + Face Value / (1 + Yield)^Year
PV = (0.06 / (1 + 0.105)^6) + (100 / (1 + 0.105)^6)
PV = 0.02865 + 0.52597
PV ≈ 0.55462
For the 10% bond with a yield of 8.5%, we use the same formula:
PV = (0.1 / (1 + 0.085)^6) + (100 / (1 + 0.085)^6)
PV = 0.05225 + 0.62092
PV ≈ 0.67317
Step 2: Calculate the present value of the future cash flows of the difference between these two bonds.
Difference = PV of 10% Bond - PV of 6% Bond
Difference = 0.67317 - 0.55462
Difference ≈ 0.11855
Step 3: Calculate the spot rate using the formula:
Spot Rate = (Coupon / (1 + r)^Year + PV of Difference / (1 + r)^Year) / (Face Value / (1 + r)^Year)
0.06 / (1 + r)^6 + 0.11855 / (1 + r)^6 ≈ 0.55462 / (1 + r)^6
We will need to find the value of 'r' (the spot rate) that makes the equation balance.
This equation is simplified as:
0.06 + 0.11855 = 0.55462 * (1 + r)^6
Rearranging the equation:
(1 + r)^6 = (0.06 + 0.11855) / 0.55462
Solving for (1 + r)^6:
(1 + r)^6 ≈ 0.3125
Taking the sixth root of both sides:
1 + r ≈ 0.8016
Subtracting 1 from both sides:
r ≈ 0.8016 - 1
The six-year spot rate is approximately -0.1984 or -19.84%. Note that a negative rate indicates an downward-sloping yield curve.