FRM Exam Part I · Interest Rates

Spot, Forward and Par Rates for FRM Part I

Updated 11 October 2026 · Fact-checked

A spot (zero) rate is the yield on a single cash flow paid at one future date. A forward rate is the rate implied between two future dates by two spot rates. A par rate is the coupon that prices a bond at par. Bootstrap par yields into spot rates, then derive forwards from them.

Understand Spot, Forward and Par Rates

A spot rate (or zero rate) is the annual rate earned on a zero-coupon investment from today to date T. It prices one cash flow only. The discount factor is d(T) = 1 ÷ (1 + z)^T with annual compounding, or e^(−zT) with continuous compounding.

A forward rate is the rate for a future period, such as year 1 to year 2, that you can lock in today. It comes from no-arbitrage. Investing for 2 years at the 2-year spot rate must give the same result as investing for 1 year at the 1-year spot rate and then rolling at the forward rate. If it did not, you could make a risk-free profit.

A par rate (par yield) is the coupon rate at which a bond with that maturity prices exactly at its face value. Most observed market data are par yields or swap rates, not zero rates, because coupon bonds are what trade. A coupon bond is a bundle of zero-coupon cash flows, so its par yield is a weighted average of the discount factors. It is not the same as the zero rate for that maturity.

Bootstrapping turns par yields into zero rates one maturity at a time. Start with the shortest maturity, where par yield equals the zero rate. For each longer maturity, discount all coupons except the last using the zero rates you already have. Then solve for the one unknown zero rate that makes the bond price equal par.

When the curve slopes upward, zero rates sit above par yields at long maturities, and forward rates sit above zero rates. When the curve slopes downward, the order reverses. This ordering is a frequent conceptual question.

Key formulas to remember

Discount factor (annual compounding): d(T) = 1 ÷ (1 + z_T)^T; z_T is the T-year spot rate. For m compounding periods per year use (1 + z/m)^(mT).
Discount factor (continuous compounding): d(T) = e^(−z_T × T); Use when the question states continuous compounding.
Forward rate (annual compounding): (1 + z_2)^T2 = (1 + z_1)^T1 × (1 + f)^(T2 − T1), so f = [(1 + z_2)^T2 ÷ (1 + z_1)^T1]^(1 ÷ (T2 − T1)) − 1; f is the forward rate between T1 and T2. Equivalent to d(T1) ÷ d(T2) = (1 + f)^(T2 − T1).
Forward rate (continuous compounding): f = (z_2 × T2 − z_1 × T1) ÷ (T2 − T1); The forward rate is a time-weighted difference of the spot rates. Its compounding is also continuous.
Par yield: c = m × (1 − d(T)) ÷ Σ d(t_i); m is coupons per year. The sum runs over all coupon dates up to T. This gives the annual coupon rate per 1 of face value.
Bootstrapping a par bond (annual coupons): 100 = c × Σ d(t_i) for earlier dates + (100 + c) × d(T); Solve for d(T), then convert to z_T.
FRA value (receive fixed R_K, annual accrual, per unit notional): Value = N × (R_K − R_F) × τ × d(T2); R_F is the current forward rate for T1 to T2, τ is the accrual length and d(T2) discounts from the payment date. Use d(T1) instead if the FRA settles at T1 on a discounted basis. Reverse the sign for the pay-fixed side.

How to solve Spot, Forward and Par Rates questions

Use this order for any question on spot, forward or par rates. Check the compounding convention before you calculate anything.

1Read the data type. Decide whether the given rates are spot (zero) rates, forward rates or par yields, and note the compounding (annual, semiannual or continuous).
2Write down the target: a zero rate, a forward rate, a par yield or a FRA value.
3If you are given par yields, bootstrap. Set the 1-period zero rate equal to the 1-period par yield. For each longer maturity, discount the earlier coupons with known discount factors and solve the par-bond equation for the last discount factor.
4Convert the discount factor to a zero rate when needed: z = d^(−1/T) − 1 for annual compounding, or z = −ln(d) ÷ T for continuous.
5For forwards, use the no-arbitrage relation. Compute d(T1) ÷ d(T2) or the continuous formula, then adjust for the accrual period.
6For a par yield from zero rates, compute all discount factors up to T and apply c = m × (1 − d(T)) ÷ Σ d(t_i).
7For a FRA, compare the contract rate with the forward rate, multiply by notional and accrual length, and discount from the payment date.
8Sense-check. With an upward-sloping curve, forwards should be above spots. Rates should look reasonable.

Quickest way: Discount-factor shortcut

When to use it: Use it whenever the question gives several maturities or asks for forwards and par yields. Convert everything to discount factors first. Every other quantity then comes from ratios and sums.

Convert each rate to a discount factor and store it in the calculator memory.
Forward rate for T1 to T2 (annual): d(T1) ÷ d(T2) − 1 for a one-year gap. For longer gaps raise the ratio to 1 ÷ (T2 − T1) before subtracting 1.
Par yield: (1 − d(T)) ÷ sum of discount factors, times m.
Bootstrapping: d(T) = (1 − c × sum of earlier d) ÷ (1 + c), with c as a decimal and annual coupons.
With continuous compounding, skip discount factors for forwards and use (z_2 T2 − z_1 T1) ÷ (T2 − T1) directly.

Common mistakes in Spot, Forward and Par Rates

Treating the 2-year par yield as the 2-year zero rate.
Both are quoted as a rate for the same maturity, so they look interchangeable.
Fix: The par yield reflects the coupon paid at year 1 too, so the 2-year zero rate differs from it (it is higher when the curve slopes up). Bootstrap to get the zero rate.
Taking the forward rate as the simple difference of spot rates, such as 4% − 3% = 1%.
It feels natural to subtract. The shortcut is only roughly right for small differences.
Fix: With annual compounding, divide the compounded growth factors: (1 + z_2)^2 ÷ (1 + z_1) − 1. With continuous compounding, use (z_2 T2 − z_1 T1) ÷ (T2 − T1). Both are time-weighted.
Mixing compounding conventions in one calculation.
Questions switch between annual, semiannual and continuous rates, and students apply one formula throughout.
Fix: Match the formula to the stated compounding. Convert first if needed: z_annual = e^(z_cont) − 1.
Forgetting to discount the earlier coupons when bootstrapping, or using the wrong discount factors.
Students discount every cash flow at the unknown rate.
Fix: Earlier coupons use the zero rates already found. Only the final cash flow (coupon plus principal) uses the unknown rate.
Valuing a FRA without discounting, or discounting from the wrong date.
The payoff is (R_K − R_F) × N × τ, and students stop there.
Fix: Discount from the payment date at the matching zero rate. State whether the FRA settles at T1 or T2 and use the right discount factor.
Assuming forwards are always above spot rates.
Most textbook curves slope upward.
Fix: The forward rate exceeds the spot rate only when the curve is upward sloping. In an inverted curve, forwards sit below spots.

Worked examples

Example 1

Annual-coupon par yields are 3% for 1 year and 4% for 2 years. Annual compounding. Find the 2-year zero rate and the 1-year forward rate starting in 1 year.

Show the solution

The 1-year zero rate equals the 1-year par yield: z_1 = 3%, so d(1) = 1 ÷ 1.03 = 0.970874.
Set up the 2-year par bond (face 100, coupon 4): 100 = 4 × d(1) + 104 × d(2).
4 × 0.970874 = 3.883495, so 104 × d(2) = 100 − 3.883495 = 96.116505.
d(2) = 96.116505 ÷ 104 = 0.924197.
z_2 = d(2)^(−1/2) − 1 = (1 ÷ 0.924197)^0.5 − 1 = 1.082020^0.5 − 1 = 1.040202 − 1 = 4.02%.
Forward rate for year 1 to 2: f = d(1) ÷ d(2) − 1 = 0.970874 ÷ 0.924197 − 1 = 1.050504 − 1 = 5.05%.

Answer: The 2-year zero rate is about 4.02% (above the 4% par yield). The 1-year forward rate, 1 year from now, is about 5.05%.

Example 2

Annual spot rates are 3% for 1 year and 4% for 2 years. You receive a fixed 5.5% on a ₹ notional equivalent of USD 10,000,000 for the year from t = 1 to t = 2, against the forward rate, with settlement at t = 2. Find the FRA value today.

Show the solution

Discount factors: d(1) = 1 ÷ 1.03 = 0.970874 and d(2) = 1 ÷ 1.0816 = 0.924556.
Forward rate: R_F = d(1) ÷ d(2) − 1 = 1.0816 ÷ 1.03 − 1 = 5.0097%.
Rate difference: R_K − R_F = 5.5% − 5.0097% = 0.4903%.
Settlement amount at t = 2: 10,000,000 × 0.004903 × 1 = USD 49,029.
Discount from t = 2: 49,029 × 0.924556 = USD 45,330.
Check using legs: fixed leg PV = 550,000 × 0.924556 = 508,506. Floating leg PV = 10,000,000 × (0.970874 − 0.924556) = 463,176. Difference = 45,330.

Answer: The FRA is worth about USD 45,330 to the party receiving fixed. The notional is in USD, so ignore the rupee mention.

Exam tips

Check the compounding convention first. It is the most common trap, and each option usually corresponds to a different convention error.
Expect the curve-shape logic in conceptual questions: upward curve means forwards above zeros above par yields at long maturities, downward curve reverses it.
Bootstrapping needs only the formula for the last discount factor. Practise the calculator memory routine so a two- or three-period bootstrap takes under two minutes.
For FRA questions, identify the discounting date and the sign (receive fixed or pay fixed) before calculating.
Round only at the end. Small rounding in discount factors can move the answer into the neighbouring option.

Practice questions from Interest Rates

Spot, Forward and Par Rates in other exams

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

Spot, Forward and Par Rates: frequently asked questions

What is the difference between a spot rate and a par yield?

A spot rate applies to a single cash flow at one date, as in a zero-coupon bond. A par yield is the coupon that makes a coupon bond price at par, so it blends several spot rates. On an upward-sloping curve, the spot rate is higher than the par yield at the same maturity.

How do I calculate a forward rate from spot rates?

With annual compounding, divide the compounded growth factors: (1 + z_2)^T2 ÷ (1 + z_1)^T1, raise to the power 1 ÷ (T2 − T1), and subtract 1. With continuous compounding, use (z_2 T2 − z_1 T1) ÷ (T2 − T1).

How does bootstrapping a zero curve work?

Start with the shortest maturity, where the par yield equals the zero rate. For each next maturity, discount the earlier coupons with known zero rates. Solve the par-bond price equation for the final discount factor, then convert it to a zero rate. Repeat for each maturity.

How is a forward rate agreement valued?

Value the difference between the contract rate and the current forward rate for the same period. Multiply by the notional and accrual length, then discount from the payment date. The sign depends on whether you receive or pay fixed.