Using the discount factors in the table below:

Discount Factor
Term (years) Discount Factor
.5 .998752
1 .996758
1.5 .994529

what must be the forward rate from 1 to 1.5 years? Assume semiannual compounding.

Question

Using the discount factors in the table below:

Discount Factor
Term (years)	Discount Factor
.5	.998752
1	.996758
1.5	.994529

what must be the forward rate from 1 to 1.5 years? Assume semiannual compounding.

BlackTom AI · Accepted Answer

We’re given discount factors for 0.5, 1, and 1.5 years under semiannual compounding: DF(0.5) = 0.998752, DF(1) = 0.996758, DF(1.5) = 0.994529. The goal is the forward rate from 1 to 1.5 years with semiannual compounding.

First, recall the relation between discount factors and a forward rate over a period: for a half-year (0.5 year) interval under semiannual compounding, the discount factor for 1.5 years can be written as DF(1.5) = DF(1) × 1/(1 + f/2), where f is the annualized forward rate for the 1 to 1.5 year period. This arises because, over a 0.5-year horizon, there is one semiannual compounding period.

Rearranging gives: 1 + f/2 = DF(1) / DF(1.5).

Compute the ratio: DF(1) / DF(1.5) = 0.996758 / 0.994529 ≈ 1.002241.

Thus f/2 ≈ 0.002241, so f ≈ 0.004482, which is about 0.4482% per year.

Now evaluate the answer choices by checking how closely each matches the computed forward rate:
- Option A: ".45%" — near 0.448%, very close to the calculated value. This is a plausible match.
- Option B: ".65%" — significantly higher than the computed ~0.45%; would require a larger ratio DF(1)/DF(1.5) than observed.
- Option C: ".25%" — substantially lower than the computed rate; would imply a much smaller DF(1)/DF(1.5) ratio than we have.
- Option D: ".53%" — somewhat higher than 0.45%; while not extremely far, it does not align as neatly with the ratio 0.996758/0.994529.
- Option E: ".50%" — also close to 0.45% but still a bit higher than the precise calculation; would correspond to a slightly larger forward discount factor gap than observed.

To decide, the key step is applying the forward-rate relationship for a 0.5-year interval under semiannual compounding and using the given discount factors. The ratio DF(1)/DF(1.5) directly gives the required (1 + f/2). The computed value yields f ≈ 0.448%, which aligns best with the 0.45% option and is less consistent with the other choices due to the sensitivity of plugging the numbers into the (1 + f/2) relation.

If you plug the numbers back into the formula to verify: 1 + f/2 ≈ 1.002241 → f ≈ 0.004482 ≈ 0.4482%, which rounds to 0.45%.

Using the discount factors in the table below: Discount Factor Term (years) Discount Factor .5 .998752 1 .996758 1.5 .994529 what must be the forward rate from 1 to 1.5 years? Assume semiannual compounding.

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