Spot rates, forward rates and the yield curve

2 free worked examples

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1. Exam FMSpot rates, forward rates and the yield curveExam level

   Annual spot rates are $s_1=4\%$, $s_2=4.6\%$, $s_3=5.1\%$. Find the present value of 1,000 at the end of each of the next three years.

   1. A$2,700.00$
   2. B$2,736.89$
   3. C$2,743.79$
   4. D$2,800.00$
   5. E$3,000.00$

   Solution

   1. Each cash flow is discounted at ITS OWN spot rate - one rate per date, not one rate per bond.
   2. $1,000/1.04=961.54$; $1,000/1.046^2=913.98$; $1,000/1.051^3=861.37$.
   3. Total: $961.54+913.98+861.37$.
   4. $=2,736.89$. Using a flat 4.6% for all three would give 2,743.79 - close, and wrong.

   Trap. Applying the three-year spot rate to all three cash flows.

2. Exam FMSpot rates, forward rates and the yield curveExam level

   The two-year spot rate is 5% and the one-year forward rate from year 2 to year 3 is 7%. Find the three-year spot rate.

   1. A$5.0000\%$
   2. B$5.6625\%$
   3. C$6.0000\%$
   4. D$6.3300\%$
   5. E$7.0000\%$

   Solution

   1. Chaining: $(1+s_3{)}^3=(1+s_2{)}^2(1+f_{2,3})$.
   2. $(1.05{)}^2(1.07)=1.1025\times 1.07=1.179675$.
   3. $1+s_3=1.179675^{1/3}=1.056625$.
   4. $s_3=5.6625\%$ - between the two-year spot and the forward, as a geometric mean must be.

   Trap. Averaging 5% and 7% arithmetically to get 6%, which ignores that the 5% applies to two years and the 7% to one.