6. On May 23, 2014, the existing or current (spot) one-year, two-year, three-year, and four-year zero-coupon Treasury security rates were as follows:
1R1 = 4.55 percent, 1R2 = 4.75 percent, 1R3 = 5.25 percent, 1R4 = 5.95 percent
Using the unbiased expectations theory, calculate the one-year forward rates on zero-coupon Treasury bonds for years two, three, and four as of May 23, 2014.
To calculate the one-year forward rates on zero-coupon Treasury bonds for years two, three, and four using the unbiased expectations theory, we can use the following formula:
(1 + 1R(n))^(n) = (1 + 1R(n-1))^n * (1 + 1F(n, n+1))
Where:
1R(n) = the current zero-coupon Treasury bond rate for year n
1F(n, n+1) = the one-year forward rate for year n, as of year n
Let's calculate the one-year forward rates:
For year 2:
1R(2) = 4.75% (given)
1R(1) = 4.55% (given)
(1 + 1R(2))^2 = (1 + 1R(1))^1 * (1 + 1F(2, 3))
(1 + 0.0475)^2 = (1 + 0.0455)^1 * (1 + 1F(2, 3))
(1.0475)^2 = (1.0455) * (1 + 1F(2, 3))
1.09725625 = 1.0455 * (1 + 1F(2, 3))
1 + 1F(2, 3) = 1.09725625 / 1.0455
1 + 1F(2, 3) = 1.049
1F(2, 3) = 0.049 or 4.9%
Therefore, the one-year forward rate for year 2, as of May 23, 2014, is 4.9%.
Similarly, let's calculate the one-year forward rates for years 3 and 4.
For year 3:
1R(3) = 5.25% (given)
1R(2) = 4.75% (calculated earlier)
(1 + 1R(3))^3 = (1 + 1R(2))^2 * (1 + 1F(3, 4))
(1 + 0.0525)^3 = (1 + 0.0475)^2 * (1 + 1F(3, 4))
(1.0525)^3 = (1.0475)^2 * (1 + 1F(3, 4))
1.16639890625 = 1.09725625 * (1 + 1F(3, 4))
1 + 1F(3, 4) = 1.16639890625 / 1.09725625
1 + 1F(3, 4) = 1.0625
1F(3, 4) = 0.0625 or 6.25%
Therefore, the one-year forward rate for year 3, as of May 23, 2014, is 6.25%.
For year 4:
1R(4) = 5.95% (given)
1R(3) = 5.25% (calculated earlier)
(1 + 1R(4))^4 = (1 + 1R(3))^3 * (1 + 1F(4, 5))
(1 + 0.0595)^4 = (1 + 0.0525)^3 * (1 + 1F(4, 5))
(1.0595)^4 = (1.0525)^3 * (1 + 1F(4, 5))
1.257952003625 = 1.16639890625 * (1 + 1F(4, 5))
1 + 1F(4, 5) = 1.257952003625 / 1.16639890625
1 + 1F(4, 5) = 1.0800
1F(4, 5) = 0.0800 or 8.00%
Therefore, the one-year forward rate for year 4, as of May 23, 2014, is 8.00%.
To calculate the one-year forward rates on zero-coupon Treasury bonds for years two, three, and four using the unbiased expectations theory, we need to use the formula:
1 + 1R(n+m) = (1 + 1Rn) * (1 + 1Rn+m)^m
Where:
- 1R(n+m) is the forward rate for n+m years
- 1Rn is the spot rate for n years
- 1Rn+m is the spot rate for n+m years
- m is the number of years between the spot rates
Let's apply this formula for each forward rate:
For the one-year forward rate for year two:
- n = 1
- m = 1
- 1Rn = 4.75%
- 1Rn+m = 5.25%
Using the formula:
1 + 1R(1+1) = (1 + 1R1) * (1 + 1R1+1)^1
1 + 1R2 = (1 + 0.0475) * (1 + 0.0525)^1
1 + 1R2 = 1.0475 * 1.0525
1 + 1R2 = 1.1017
Therefore, the one-year forward rate for year two is approximately 10.17%.
Now let's calculate for year three:
- n = 1
- m = 2
- 1Rn = 4.75%
- 1Rn+m = 5.95%
Using the formula:
1 + 1R(1+2) = (1 + 1R1) * (1 + 1R1+2)^2
1 + 1R3 = (1 + 0.0475) * (1 + 0.0595)^2
1 + 1R3 = 1.0475 * 1.1246
1 + 1R3 = 1.1783
Therefore, the one-year forward rate for year three is approximately 17.83%.
Finally, let's calculate for year four:
- n = 1
- m = 3
- 1Rn = 4.75%
- 1Rn+m = None (since there's no spot rate available for year four)
Since there is no spot rate available for year four, we cannot calculate the forward rate using the unbiased expectations theory.
In summary, the approximate one-year forward rates for zero-coupon Treasury bonds as of May 23, 2014, are:
- Year 2: 10.17%
- Year 3: 17.83%
- Year 4: N/A (not available)