Final answer:
To calculate the r₁ₗ in a binomial tree, we need to estimate the possible interest rate movements based on the given annual volatility. Assuming the interest rate can increase or decrease by one standard deviation, the r₁ₗ is either 6.325% or 3.825%.
Step-by-step explanation:
To calculate the r₁ₗ in a binomial tree, we need to estimate the possible interest rate movements. Based on the given information, the annual volatility is 15%. Let's assume that the interest rate can either increase or decrease by one standard deviation, which is 15%.
The YTM of 1-year and 2-year treasury bonds are 4.5% and 5.5%, respectively. The 1-year bond is zero coupon and the 2-year bond trades at par. If the annual volatility is 15%, calculate the r₁ₗ (i.e., the lower branch of the binomial tree at T=1 ). Report the result in the percentage format (e.g., if the answer is 0.0321 , report it as 3.21) Assume "annual" coupon payments.
ChatGPT
To calculate the lower branch rate (
�
1
�
r
1L
) for the binomial tree at
�
=
1
T=1, we can use the following relationship between the spot rate (
�
0
r
0
), the lower branch rate (
�
1
�
r
1L
), and the upper branch rate (
�
1
�
r
1H
):
�
1
�
=
(
1
+
�
0
)
×
(
1
−
1
−
2
×
YTM
1
1
+
YTM
2
)
−
1
r
1L
=(1+r
0
)×(1−
1−
1+YTM
2
2×YTM
1
)−1
Here,
�
0
r
0
is the current spot rate.
YTM
1
YTM
1
is the yield to maturity of the 1-year bond.
YTM
2
YTM
2
is the yield to maturity of the 2-year bond.
Given:
�
0
r
0
is not explicitly provided, but we'll assume it's the rate for a risk-free bond, often considered equivalent to the yield of a Treasury bond. Let's assume
�
0
=
4.5
r
0
=4.5.
YTM
1
=
4.5
%
YTM
1
=4.5%
YTM
2
=
5.5
%
YTM
2
=5.5%
Now we can substitute these values into the formula:
�
1
�
=
(
1
+
0.045
)
×
(
1
−
1
−
2
×
0.045
1
+
0.055
)
−
1
r
1L
=(1+0.045)×(1−
1−
1+0.055
2×0.045
)−1
�
1
�
=
0.045
×
(
1
−
1
−
0.09
1.055
)
−
1
r
1L
=0.045×(1−
1−
1.055
0.09
)−1
�
1
�
=
0.045
×
(
1
−
1
−
0.085308
)
−
1
r
1L
=0.045×(1−
1−0.085308
)−1
�
1
�
=
0.045
×
(
1
−
0.914692
)
−
1
r
1L
=0.045×(1−
0.914692
)−1
�
1
�
=
0.045
×
(
1
−
0.9565
)
−
1
r
1L
=0.045×(1−0.9565)−1
�
1
�
=
0.045
×
(
−
0.9565
)
−
1
r
1L
=0.045×(−0.9565)−1
�
1
�
=
−
0.043
r
1L
=−0.043
Convert to percentage:
−
0.043
×
100
=
−
4.3
%
−0.043×100=−4.3%
Therefore, the lower branch rate (
�
1
�
r
1L
) is approximately 4.3%.