This Exam FM sample reference tests Spot Rate from Forward Rates. Taking the cube root gives s₃ = 0.05987, which rounds to 6%. The result agrees with the published answer key, choice C.
How to solve this Spot Rate from Forward Rates question
Setup
Setup
The first three one-year forward rates are 4%, 6%, and 8%. A three-year zero-coupon accumulation must equal the product of these three annual forward accumulations.
s1=0.04,(1+s2)2=(1.04)(1.06)
Model
Model
Thus (1 + s₃)³ = 1.04 × 1.06 × 1.08. This no-arbitrage product, not an arithmetic average of rates, determines the spot rate.
(1+s3)3=(1+s2)2(1.08)
Compute
Compute
Taking the cube root gives s₃ = 0.05987, which rounds to 6%.
s3=[1.04(1.06)(1.08)]1/3−1=0.05987
Answer
Answer
The calculation gives 6% for spot rate from forward rates, matching published choice C.
s3=6%(C)
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These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AChoice A (4%) does not match the checked spot rate from forward rates result (6%); no distinct standard one-step error is identifiable.
BChoice B (5%) does not match the checked spot rate from forward rates result (6%); no distinct standard one-step error is identifiable.
DChoice D (7%) does not match the checked spot rate from forward rates result (6%); no distinct standard one-step error is identifiable.
EChoice E (8%) does not match the checked spot rate from forward rates result (6%); no distinct standard one-step error is identifiable.
Original practice · fully worked
Original variant: missing fourth-year forward rate from a spot quote
A four-year annual effective spot rate is 4.5%. The one-year forward rates for years 1, 2, and 3 are 3%, 4%, and 5%. Calculate the one-year forward rate for year 4.
A 5.42%
B 5.72%
C 6.02%
D 6.33%
E 6.63%
Variant answer in brief
The residual factor gives fourth-year forward rate 6.02%. The missing forward rate is 6.02%, selecting choice C.
Setup
Setup
The spot accumulation and the first three forward factors are known; isolate the missing final-year factor.
(1.045)4=(1.03)(1.04)(1.05)(1+f4)
Model
Model
Divide the four-year spot accumulation by the product of the first three forward accumulations.
1+f4=(1.045)4/[(1.03)(1.04)(1.05)]
Compute
Compute
The residual factor gives fourth-year forward rate 6.02%.
f4=0.06024272
Answer
Answer
The missing forward rate is 6.02%, selecting choice C.
The 2210-page Financial Mathematics Proof Manual reorganizes 461 verified Exam FM solutions by syllabus skill and adds formula proofs, error patterns, and original worked practice.
