Actuarial Mathematics for Modelling · Interest rates over different time periods

Spot Rates and Forward Rates: How to Calculate Them

Updated 11 October 2026 · Fact-checked

A spot rate y_t is the annual rate for money invested from now for t years. A forward rate f_{t,r} is the rate agreed now for r years starting at time t. You find it from no-arbitrage: (1 + y_{t+r})^(t+r) = (1 + y_t)^t × (1 + f_{t,r})^r.

Understand Time-Varying and Spot Rates, Forward Rates

Interest rates are not the same for every term. A bank may pay a different annual rate on a 1-year deposit than on a 5-year deposit. The pattern of rates across terms is the term structure of interest rates, and a plot of it is the yield curve.

A spot rate y_t is the annual effective rate for an investment made today and held for t years. It is the yield on a t-year zero-coupon bond. If you invest ₹1 for t years you get (1 + y_t)^t. The discount factor is v(t) = (1 + y_t)^(-t).

A forward rate is a rate fixed today for a loan or investment that starts in the future. f_{t,r} is the annual effective rate for the period from time t to time t+r. The one-year forward rate starting at time t-1 is written f_{t-1,1}, often shortened to f_t in some texts. Check the notation in the question.

Forward rates follow from spot rates by no-arbitrage. Investing for t+r years at the spot rate must give the same result as investing for t years at the spot rate and then rolling into the forward contract for r years. If the two differed, you could make a risk-free profit. That equality gives the formula below.

Spot rates and forward rates are two views of the same curve. If the yield curve slopes upward, the forward rate f_{t,r} lies above the spot rate y_{t+r} for the end of the forward period. If the curve slopes downward, f_{t,r} lies below y_{t+r}. Forward rates are rates fixed now. They are not predictions of future spot rates unless a theory of the term structure says so.

Key rules to remember

Spot rate accumulation: Accumulation of ₹1 over t years = (1 + y_t)^t; y_t is the annual effective spot rate for term t, measured from time 0.
Discount factor from spot rate: v(t) = (1 + y_t)^(-t); Present value of ₹1 due at time t. Also the price of a t-year zero-coupon bond paying ₹1.
No-arbitrage relationship: (1 + y_{t+r})^(t+r) = (1 + y_t)^t × (1 + f_{t,r})^r; f_{t,r} is the annual effective forward rate for the r years starting at time t.
Forward rate from spot rates: f_{t,r} = [ (1 + y_{t+r})^(t+r) ÷ (1 + y_t)^t ]^(1/r) − 1; Rearranged from the relationship above.
One-year forward rate: f_{t,1} = (1 + y_{t+1})^(t+1) ÷ (1 + y_t)^t − 1; The case r = 1, the most common in exam questions.
Forward rate from discount factors: (1 + f_{t,r})^r = v(t) ÷ v(t+r); Useful when zero-coupon prices are given.
Spot rate from one-year forwards: (1 + y_t)^t = (1 + f_{0,1})(1 + f_{1,1}) ... (1 + f_{t-1,1}); Here f_{0,1} = y_1. Spot rate is a geometric average of the one-year forwards.

How to solve Time-Varying and Spot Rates, Forward Rates questions

Use this method for any question that gives rates for different terms and asks for a spot rate, forward rate, price or accumulation.

1Write down what you are given: spot rates, forward rates or zero-coupon prices, and note whether rates are annual effective.
2Mark the time line. Show time 0, the start of the forward period t, and the end t+r.
3Convert everything to accumulation factors, for example (1 + y_t)^t. Work with factors, not rates.
4Write the no-arbitrage equation: factor to time t+r = factor to time t × forward factor for r years.
5Solve for the unknown factor. Divide factors, do not subtract rates.
6If r is more than 1, take the r-th root, then subtract 1 to get the annual rate.
7For cash flows, discount each payment at the spot rate for its own time, or use forward rates in sequence. Both must agree.
8Check that the answer is sensible. An upward-sloping curve should give forward rates above the spot rates.

Quickest way: Factor-ratio shortcut

When to use it: Use it for MCQs or short parts where you are given spot rates and need a forward rate or a price.

Compute A = (1 + y_{t+r})^(t+r) and B = (1 + y_t)^t on your calculator and store them.
Forward factor over r years = A ÷ B.
If r = 1, the forward rate is A ÷ B − 1. Otherwise take the r-th root first.
For a price of a forward contract or a cash flow, use v(t) = (1 + y_t)^(-t) directly instead of finding the forward rate.
Sanity check: if y_{t+r} > y_t, then f_{t,r} must exceed y_{t+r}.

Common mistakes in Time-Varying and Spot Rates, Forward Rates

Subtracting rates, for example f = 2 × y_2 − y_1 for the second-year forward rate.
It looks like the average idea 'y_2 is the average of y_1 and f'. That is only approximately true.
Fix: Use accumulation factors: 1 + f_{1,1} = (1 + y_2)^2 ÷ (1 + y_1).
Forgetting the power t on the spot rate, using (1 + y_t) instead of (1 + y_t)^t.
Students copy the one-year pattern and drop the exponent.
Fix: Always write the exponent. Spot rates are annual, so a t-year accumulation needs the power t.
Not taking the r-th root when the forward period is longer than one year.
The ratio of factors gives the total growth over r years, not an annual rate.
Fix: Take the power 1/r of the ratio and then subtract 1.
Treating forward rates as forecasts of future spot rates.
The names sound alike.
Fix: A forward rate is the rate fixed today by no-arbitrage. It equals the expected future spot rate only under specific theories, such as pure expectations.
Discounting every cash flow at one rate, such as the yield on the longest bond.
Habit from constant-rate questions.
Fix: Discount the payment at time t with y_t. Each time has its own spot rate.
Mixing up the notation f_{t,r} with the year number, so the forward period starts at the wrong time.
Different texts label one-year forwards differently.
Fix: Draw the time line. Start and end points decide which spot rates you use.

Worked examples

Example 1

The annual effective spot rates are y_1 = 4%, y_2 = 5% and y_3 = 5.5%. Calculate the one-year forward rates f_{1,1} and f_{2,1}, correct to 3 decimal places in percent.

Show the solution

f_{1,1}: 1 + f_{1,1} = (1.05)^2 ÷ 1.04.
(1.05)^2 = 1.1025. Dividing by 1.04 gives 1.060096.
So f_{1,1} = 6.010%.
f_{2,1}: 1 + f_{2,1} = (1.055)^3 ÷ (1.05)^2.
(1.055)^3 = 1.055 × 1.113025 = 1.174241 (to 6 decimals).
Divide by 1.1025: 1.174241 ÷ 1.1025 = 1.065072.
So f_{2,1} = 6.507%.

Answer: f_{1,1} ≈ 6.010% and f_{2,1} ≈ 6.507%.

Example 2

The annual effective spot rates are y_2 = 5% and y_5 = 6%. An investor agrees today to lend ₹10,00,000 at time 2 for 3 years at the forward rate. Find the annual forward rate f_{2,3} and the amount repaid at time 5.

Show the solution

Use (1 + y_5)^5 = (1 + y_2)^2 × (1 + f_{2,3})^3.
(1.06)^5 = 1.338226 and (1.05)^2 = 1.1025.
(1 + f_{2,3})^3 = 1.338226 ÷ 1.1025 = 1.213810.
Take the cube root: 1.213810^(1/3) ≈ 1.06672.
So f_{2,3} ≈ 6.67% per year.
Amount repaid = ₹10,00,000 × 1.213810 ≈ ₹12,13,810.

Answer: f_{2,3} ≈ 6.67% per year, and the amount repaid at time 5 is about ₹12,13,810.

Exam tips

Write the no-arbitrage equation in full before you substitute numbers. Examiners give method marks for it.
Keep at least six decimal places in factors. Rounding early distorts forward rates because you divide two close numbers.
If the question gives zero-coupon prices, use (1 + f)^r = v(t) ÷ v(t+r) and skip converting to spot rates.
In written questions, state your assumptions: annual effective rates, no arbitrage, no transaction costs.
Be ready to explain in words why an upward-sloping yield curve means forward rates exceed spot rates.

Practice questions from Interest rates over different time periods

Time-Varying and Spot Rates, Forward Rates in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Time-Varying and Spot Rates, Forward Rates: frequently asked questions

What is the difference between a spot rate and a forward rate?

A spot rate applies to money invested from today for a given term. A forward rate applies to a period that starts in the future, and is fixed today. Both are read from the same yield curve.

How do I calculate a forward rate from spot rates?

Divide the accumulation factor for the longer term by the factor for the shorter term. For a one-year forward, f_{t,1} = (1 + y_{t+1})^(t+1) ÷ (1 + y_t)^t − 1. For r years, take the r-th root before subtracting 1.

Is the spot rate the same as the yield to maturity?

No, not in general. A spot rate is the yield on a zero-coupon bond. The yield to maturity on a coupon bond is a single rate that equates its price to all its payments, so it blends several spot rates.

Why are forward rates higher than spot rates when the yield curve slopes upward?

The spot rate is a geometric average of the one-year forward rates up to that term. If spot rates keep rising with term, each new forward period must be above the earlier average to pull it up.