If you can invest for one year at 6.00% or for two years at 6.50%, what rate for the second year makes both routes equivalent? Start with that question, then build the full connection between spot curves, forward rates and interest-rate hedges.
1Learning Roadmap
Distinguish rates agreed today from rates realised later, derive forwards from discount factors, compare equivalent quotations, and explain how an FRA hedges a future interest period.
Read the worked example first, then edit the curve and compare the results. Cream input boxes accept your own scenarios. Every rate in this guide is an illustrative teaching input from the accompanying workbook.
2Spot, Forward & Realised Rates
| Rate | Period covered | Known today? |
|---|---|---|
| Spot z(T) | Today to maturity T | Yes, for the specified curve and convention |
| Implied forward f(a,b) | Future start a to future end b | Yes, calculated from today’s curve |
| Future realised spot | A period beginning on a future date | No, observed when that date arrives |
A forward is the break-even rate embedded in today’s curve. Calculating it does not execute a hedge. A contract or a matched borrowing-and-lending strategy is needed to lock the exposure.
A zero-coupon spot rate prices a single future cash flow. A coupon bond’s yield to maturity is one internal rate applied to all its cash flows; it is not generally a spot rate for every coupon date.
3Discount Factors & No-Arbitrage
Under annual effective compounding, ₹1 invested to year T grows to (1 + z(T))T. Reverse that growth to obtain the value today of ₹1 received at T.
Two investments with identical dates, currency and risk must have the same terminal payoff under the model’s frictionless assumptions.
D(a) / D(b) is the growth factor over the future interval. Annualise that growth using the interval length b − a. Set D(0) = 1 when the interval starts today.
4The 6.00% / 6.50% Example
The workbook starts with a one-year spot of 6.00%, a two-year spot of 6.50%, and an investment of ₹1,00,000. Edit these inputs to compare the two routes.
1.065² = 1.134225. Divide by 1.06, then subtract 1: f(1,2) = 7.00235849%. Both routes finish at ₹1,13,422.50. The simple approximation 2 × 6.50% − 6.00% = 7.00% is close, but it omits compounding.
Timeline uses the original workbook inputs. The live calculation above follows your edits. The year-1 value of a direct two-year investment is not a guaranteed early-sale price.
5Explore the Spot Curve
This ten-year zero-coupon curve is the workbook’s illustrative curve. Edit a spot rate to update its discount factor, adjacent one-year forward, chart and interval calculator.
| Year T | Spot (%) | D(T) | Forward interval | 1Y forward |
|---|---|---|---|---|
| 1 | 0 → 1 | |||
| 2 | 1 → 2 | |||
| 3 | 2 → 3 | |||
| 4 | 3 → 4 | |||
| 5 | 4 → 5 | |||
| 6 | 5 → 6 | |||
| 7 | 6 → 7 | |||
| 8 | 7 → 8 | |||
| 9 | 8 → 9 | |||
| 10 | 9 → 10 |
An upward-sloping annual spot curve implies adjacent forwards above the corresponding longer spot rates. A flat curve gives equal spot and forward rates; a sufficiently inverted curve can give negative forwards.
6Forward Rate Calculator
Select integer-year nodes from the curve above. There is no interpolation or extrapolation. The calculator shows the annual effective forward rate for your selected interval.
This rate compounds annually to reproduce the growth over the selected interval: f(a,b) = G1/(b−a) − 1, where G = D(a) / D(b).
7Money Market Forwards
For this money-market example, interest is quoted on a simple basis with explicit day counts. The workbook uses Actual/365. Actual conventions depend on the instrument and contract.
| Tenor | Days | Simple spot |
|---|---|---|
| 1 month | 30 | 6.25% |
| 3 months | 91 | 6.45% |
| 6 months | 182 | 6.65% |
| 9 months | 273 | 6.80% |
| 12 months | 365 | 6.95% |
A 3 × 6 forward starts at day 91 and ends at day 182 in this example. Its covered period is 91 days.
Changing the basis reinterprets the same quoted inputs under that basis. It does not convert the original quotes while preserving cash flows. The FRA below uses the resulting contract rate.
8Forward Rate Agreements
A forward rate agreement (FRA) exchanges the interest difference on a notional amount. The principal is not exchanged. A buyer pays fixed and receives floating, so a fixing above the contract rate produces a receipt that offsets higher borrowing interest.
| Stage | What happens |
|---|---|
| Trade date | Agree the contract rate K today. |
| Fixing date | Observe the contract’s reference rate L at its specified fixing time. |
| Settlement date a | In this conventional FRA example, settle the discounted interest difference at the beginning of the period. |
| Maturity date b | The underlying borrowing ends. No further FRA payment in this example. |
Here δ = 91/B. Discounting is necessary because the interest difference belongs to the end of the interval but the cash payment occurs at the start.
Carry the settlement to the end at the same reference rate. Then N × Lδ − Settlement × (1 + Lδ) = N × Kδ. This is the workbook’s matched borrowing and FRA model, assuming equal notionals, dates and conventions and no spreads or costs.
From FRAs to other derivatives
A swap contains a series of future floating-rate exposures. Forward rates project these payments; discount factors bring them back to today. A par swap rate is a discount-weighted fixed rate balancing the legs. Caps and floors additionally require an option model and volatility.
Contract details matter: modern overnight-rate products can fix in arrears and use different settlement mechanics. The start-settled term-rate formula here should not be applied automatically to every benchmark or derivative.
9Sensitivity to Spot Rates
Rows vary the one-year spot; columns vary the two-year spot. The highlighted cell is the workbook’s 6.00% / 6.50% example. All rates use annual effective compounding.
| z(1) ↓ / z(2) → | 5.50% | 6.00% | 6.50% | 7.00% | 7.50% |
|---|---|---|---|---|---|
| 5.00% | 6.0024% | 7.0095% | 8.0214% | 9.0381% | 10.0595% |
| 5.50% | 5.5000% | 6.5024% | 7.5095% | 8.5213% | 9.5379% |
| 6.00% | 5.0024% | 6.0000% | 7.0024% | 8.0094% | 9.0212% |
| 6.50% | 4.5094% | 5.5023% | 6.5000% | 7.5023% | 8.5094% |
| 7.00% | 4.0210% | 5.0093% | 6.0023% | 7.0000% | 8.0023% |
Increasing the two-year spot raises the forward. Increasing the one-year spot lowers it. A negative forward occurs when (1 + z(2))² < (1 + z(1)); under the stated model it is a possible result, not a calculation error.
Use the calculator in section 4 to explore your own combinations. The scenario table above remains fixed to the workbook’s original sensitivity grid.
10Practice Problems
Enter each answer as a percentage, then check the calculated answer and your signed error in basis points. One basis point is 0.01 percentage point.
1. Upward curve: z(1) = 4%, z(2) = 5%. Find f(1,2).
2. Flat curve: z(1) = z(2) = 5%. Find f(1,2).
3. Inverted curve: z(1) = 7%, z(2) = 4%. Find f(1,2).
4. Recover z(2) when z(1) = 5% and f(1,2) = 7%.
Discussion: Why do equal spots imply an equal forward? Why does a plan to reinvest later leave interest-rate risk unless that future rate is contracted today?
11Questions & Answers
What exactly is a spot rate?
The interest rate agreed today on money that starts today and is repaid in a single amount at maturity T. It is a zero-coupon rate: one cash flow in, one cash flow out, no coupons in between.
What is a forward rate?
The interest rate agreed today for a period that BEGINS on a future date. It is a rate for the interval from a to b, not from today to b.
Is the forward rate known today, or is it a guess?
It is known today. Once the spot curve and the compounding convention are fixed, the forward rate is arithmetic. What is unknown today is the future realised spot rate, which is a different thing entirely.
Where does the forward rate formula come from?
From no-arbitrage: (1 + z(b))^b = (1 + z(a))^a × (1 + f(a,b))^(b − a). Growing money to b in one step must equal growing to a and then over the forward interval, or a riskless profit exists.
Why is f(1,2) equal to 7.0024% and not 6.91%?
With annual effective compounding, 1.065² / 1.06 − 1 = 7.00235849%. A result of 6.91% does not follow from these stated inputs and convention.
Why does an upward sloping curve give a forward above the spot rate?
The two-year growth factor combines the first-year spot and the second-year forward geometrically. When the two-year annual spot exceeds the one-year spot, the second-year forward must exceed both.
What happens on a perfectly flat curve?
Every forward equals the spot rate. Averaging equal numbers changes nothing, which is why a flat curve is the sanity check for any forward calculation you build.
What does an inverted curve imply?
Forwards fall below the short spot rate, and if the inversion is steep enough the implied forward can be negative. A negative forward is not an error; it simply means the discount factor ratio is below one.
Is the forward rate a forecast of the future spot rate?
No. It is a curve-implied break-even rate. Expectations and risk premia can affect the curve, and the realised future rate can differ.
What does the forward rate actually tell me then?
It is the future reinvestment rate at which rolling short investments would match the terminal value of today’s longer investment. A decision to lock or roll also depends on funding, risk and actual market terms.
How is the money market view different from the bond view?
The workbook’s money-market example uses simple interest and explicit days over a 365-day basis. Its annual bond-curve example uses effective annual compounding. Instrument-specific conventions must be checked before comparing quotes.
Does calculating a forward rate lock in my return?
No. Only a contract locks it: an FRA, a forward deposit, or a matched borrow-and-lend position. Simply planning to reinvest at the future spot rate leaves you fully exposed to whatever that rate turns out to be.
How does an FRA use the forward rate?
In this simplified single-curve model the money-market forward is the fair FRA contract rate at inception. Actual dealer quotes can include bid-offer spreads and adjustments. Settlement compares the contract rate with the specified reference fixing.
What does the 3 × 6 notation mean?
The contract covers the interval starting in three months and ending in six months. This workbook represents it with day 91 to day 182; actual dates and business-day adjustments come from the contract.
Why is FRA settlement discounted?
Because the interest difference accrues over the contract period but is paid on the settlement date at the START of that period. Paying it early means paying its present value, discounted at the reference rate that fixed.
Why is an interest rate swap called a strip of forwards?
Each floating payment of a swap is priced off the forward rate for its own period. The swap rate is simply the single fixed rate that makes that whole strip of forward exposures worth zero today.
What breaks these clean equalities in practice?
Bid-offer spreads, counterparty credit and capital charges, day-count mismatches between the hedge and the exposure, collateral conventions that force OIS discounting, and taxes. The arithmetic is the floor, not the quote.
12Formula Sheet & Excel
| Quantity | Formula |
|---|---|
| Discount factor | |
| Interval growth | |
| Annual effective forward | |
| Simple annualised forward | |
| Continuous forward | |
| Semiannual nominal forward | |
| Recover two-year spot |
Trace the workbook
| Workbook location | Excel formula / purpose |
|---|---|
| Worked Example!E13 | =E12-1 |
| Worked Example!F13 | =(1+E7)^2/(1+E6)^1-1 |
| Spot Curve!D7 | =1/(1+C7)^B7 |
| Spot Curve!F8 | =D7/D8-1 |
| Forward Calculator!E14 | =IF(E12="","",E12^(1/E11)-1) |
| FRA in Derivatives!E45 | =E43*E44 — discounted settlement |
Enter 6% as 0.06 in the spreadsheet’s rate cells. In the webpage’s boxes labelled “(%)”, enter 6. Displayed values are rounded; calculations retain full precision.
Download the source workbook13Sources & Teaching Assumptions
Primary teaching source: 1 A Spot-Rates-to-Forward-Rates.xlsx, supplied by Prof. V. Ravichandran. The lesson follows its Concepts, Worked Example, Spot Curve, Forward Calculator, Money Markets, FRA in Derivatives, Sensitivity, Practice, Q and A, and Sources sheets.
Assumptions: one currency, consistent risk and discounting basis, no transaction costs. All curve inputs are illustrative, not current market quotations. Calculator inputs are independent except that the curve feeds the interval calculator and the money-market forward feeds the FRA.
Additional reference: Bank of England — Yield curve terminology and concepts. Spot rates discount dated cash flows; forward rates describe future intervals embedded in today’s curve. The Bank’s instantaneous forward convention differs from the discrete interval forwards used here.
A spot rate begins today. A forward rate covers a specified future interval. A future realised rate is unknown today. Keep the dates and the quotation convention explicit in every calculation.