Forward Rates and Implied Forward Rates from Spot Rates

Understand Forward Rates and Implied Forward Rates

Key rules to remember

Implied forward rate (general)

(1 + i_{t+r})^(t+r) = (1 + i_t)^t × (1 + f_{t,r})^r

f_{t,r} is the annual effective rate from time t to t+r. Spot rates are annual effective rates.

Solving for f_{t,r}

f_{t,r} = [ (1 + i_{t+r})^(t+r) ÷ (1 + i_t)^t ]^(1/r) − 1

Take the r-th root of the ratio of accumulation factors.

One-year forward rate

1 + f_t = (1 + i_{t+1})^(t+1) ÷ (1 + i_t)^t

Here f_t is the rate for the year from t to t+1.

Spot rate from one-year forward rates

(1 + i_t)^t = (1 + f_0)(1 + f_1)…(1 + f_{t−1})

With f_0 = i_1. The spot rate is the geometric mean of the forward rates, less 1.

Discount factor from spot rate

v(t) = (1 + i_t)^(−t)

Equals the product of one-year forward discount factors (1+f_s)^(−1) for s = 0 to t−1.

How to solve Forward Rates and Implied Forward Rates questions

1. 1Write down the spot rates given and confirm they are annual effective rates. Convert if they are nominal.
2. 2Identify the forward period: start time t and length r. The end time is t+r.
3. 3Write the no-arbitrage equation: (1+i_{t+r})^(t+r) = (1+i_t)^t × (1+f_{t,r})^r.
4. 4Compute the two accumulation factors (1+i_{t+r})^(t+r) and (1+i_t)^t separately. Keep at least five decimal places.
5. 5Divide the first by the second, then take the r-th root. Subtract 1 to get f_{t,r}.
6. 6If the question asks for a value, discount each cash flow with the correct rate: spot rates for value at time 0, forward rates for value at a later time.
7. 7Check: a rising spot curve should give forward rates above the spot rates. State your answer as a percentage.
8. 8 Write the assumptions you used (annual effective rates, no arbitrage).

Quickest way: Ratio of accumulation factors

Common mistakes in Forward Rates and Implied Forward Rates

• Subtracting spot rates to get the forward rate, for example i_2 − i_1.

  It looks like a simple difference between two periods.

  Fix: Interest compounds. Use the ratio of accumulation factors, not the difference of rates.

• Forgetting to take the r-th root for multi-year forward rates.

  Students stop after dividing the accumulation factors.

  Fix: The ratio gives the r-year accumulation. Raise it to 1/r to get the annual rate.

• Using the wrong power on the spot rate, such as (1+i_2) instead of (1+i_2)^2.

  Students forget that the spot rate is annual but the term is t years.

  Fix: Always raise (1+i_t) to the power t, the full term from time 0.

• Confusing the forward rate f_{t,r} with the spot rate i_r.

  Both can cover r years, so the notation looks similar.

  Fix: A spot rate starts at time 0. A forward rate starts at time t. Check the start date before you write the equation.

• Rounding spot accumulation factors too early.

  The numbers look small and rounding seems harmless.

  Fix: Keep five or six decimals until the final step. Forward rates are sensitive to rounding because you divide two close numbers.

Worked examples