Within a timeline of 10 years, and assuming that all of the following rates are annualized, which one of the following statements is false?

Question

BlackTom AI · Accepted Answer

Knowing each 2-year forward rate nf2_t gives the relation (1 + nf2_t)^2 = D_t / D_{t+2}, where D_t are discount factors. Rearranged: D_{t+2} = D_t / (1 + nf2_t)^2. With D_0 = 1 this determines D_2, D_4, D_6, D_8, D_10 uniquely, but it never yields D_1, D_3, D_5, D_7, D_9 because those are linked only to each other via the same recurrence starting from D_1. Thus nf2 data leave an unspecified degree of freedom (the odd-year anchor D_1), so one-year short rates r1..r10 cannot be uniquely recovered.

1) Relevant concepts:
- Discount factors D_t relate to spot (zero) rates z_t by D_t = 1/(1+z_t)^t.
- One-year short rates r_t satisfy D_t = D_{t-1}/(1 + r_t) or r_t = D_{t-1}/D_t - 1.
- An n-year forward rate starting at t, F_{t}^{(n)}, satisfies (1+F_{t}^{(n)})^n = D_t / D_{t+n}.

2) Step-by-step reasoning:
- Given nf2_t for t = 0,1,...,8, we have (1+nf2_t)^2 = D_t / D_{t+2}.
- Start with D_0 = 1 ⇒ D_2 = 1 / (1+nf2_0)^2. Then D_4 = D_2 / (1+nf2_2)^2, etc. All even D's follow.
- For odd indices: D_3 = D_1 / (1+nf2_1)^2, D_5 = D_3 / (1+nf2_3)^2, ... so all odd D's are proportional to the unknown D_1.
- Short rates require both D_{t-1} and D_t; for example r1 = D_0/D_1 - 1 depends on D_1, r2 = D_1/D_2 - 1 depends on D_1 and D_2. Since D_1 is not determined by nf2 data, r1, r2, r3, ... cannot be uniquely obtained.

3) Calculations (illustrative):
- D_2 = D_0/(1+nf2_0)^2 = 1/(1+nf2_0)^2.
- r2 = D_1/D_2 - 1 → needs D_1 which is unknown.

4) Why the correct answer is correct:
Because nf2 provides only relations between D_t and D_{t+2}, it yields two disjoint chains (even and odd years). With only nf2 one chain can be anchored at D_0 but the other remains free, so short rates are not uniquely determined.

5) Why other choices are true:
- a) Given short rates r1..r10, compute D_t = ∏_{i=1}^t 1/(1+r_i) and then z_t = (1/D_t)^{1/t} − 1.
- b) Given spot rates z1..z10, get D_t = 1/(1+z_t)^t and then r_t = D_{t-1}/D_t − 1.
- c) Given short rates, forward rates (multi-year) follow from D_t: F_{t}^{(n)} = (D_t/D_{t+n})^{1/n} − 1.

Conclusion: The false statement is "Given 2-year Forward Rates for each future year, i.e. nf2, you can always derive Short Rates." This matches the provided correct answer.

Within a timeline of 10 years, and assuming that all of the following rates are annualized, which one of the following statements is false?单项选择题

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