Forward rates and what they imply

Chapter 15 extracted the price of money at each date. Those prices, taken two at a time, determine the price of money between two future dates — and that is the last thing the curve contains.

The derivation

Two ways to be invested from now until year $n$:

Route A. Buy an $n$-year zero at the spot rate $s_n$, giving $(1+s_n)^n$ per rupee.

Route B. Buy an $m$-year zero at $s_m$, and today agree a rate $f$ for the period from $m$ to $n$, giving $(1+s_m)^m (1+f)^{n-m}$.

Both are certain today, so they must be equal or there is riskless profit:

$(1+s_n)^n = (1+s_m)^m\,(1+f)^{\,n-m}$

$f = \left[\frac{(1+s_n)^n}{(1+s_m)^m}\right]^{\frac{1}{n-m}} - 1$

This is the third arbitrage relation in the course with the same structure. The Option pricing subject's put-call parity and the Currency subject's covered interest parity are the same argument: two portfolios with identical certain outcomes must cost the same today. None of the three contains a forecast, and all three are routinely read as though they did.

The real curve, and a striking result

The RBI's bill yields for the week ended 2 October 2026 are spot rates, since bills are zero-coupon. Compute the forwards between them:

Period	Implied forward rate
3 months to 6 months	6.4018%
6 months to 12 months	6.4005%
3 months to 12 months	6.4009%

All three agree to within two hundredths of a basis point.

That is not a coincidence, and it is worth understanding. A curve of 5.52%, 5.96% and 6.18% at three, six and twelve months is almost exactly what you get from a flat forward rate of about 6.40% applied to a 5.52% starting point. The entire upward slope of the Indian bill curve that week is produced by one number — the market's price for money from three months out onwards.

And note how far that 6.40% sits above the near-term rates. The weighted average call money rate was 5.12% and the policy repo rate 5.25% that week. The curve was pricing money roughly 115 basis points above the overnight rate for periods starting a few months out.

Forwards from the bootstrapped curve

Extending with chapter 15's spot rates:

Spot rates		One-year forwards
$s_1$	6.1800%	—
$s_2$	6.5104%	year 1 → 2	6.8419%
$s_3$	6.8283%	year 2 → 3	7.4669%
$s_4$	7.0441%	year 3 → 4	7.6942%
$s_5$	7.2095%	year 4 → 5	7.8734%

Three features to read.

Forwards rise much faster than spots. The spot curve climbs 103 basis points from year 1 to year 5; the forward curve climbs 169 from the first year to the fifth.

Forwards sit above spots whenever the curve slopes up. A spot rate is an average of the forwards up to that date, so pulling an average upward requires each new term to exceed it.

And the forward curve exaggerates. This matters because people read forward rates as the market's expectations, and the forward curve is always the steepest version of the story the curve tells.

Why they are not forecasts

The arithmetic above used no opinion about the future. So what are forward rates?

They are breakevens. The forward rate is the future spot rate at which two strategies tie. If the actual rate in a year turns out to be above 6.8419%, rolling short bills beats holding the two-year bond; below it, the two-year wins; exactly at it, they tie.

That is a complete description and it contains no prediction. The market is quoting a price, not publishing a view.

The expectations hypothesis would make them forecasts, and it says forward rates equal expected future spot rates. If that were true, every maturity would offer the same expected return and there would be no reason to prefer any point on the curve — which is a strong claim, and the evidence against it is that long bonds have historically earned more than rolled short ones on average.

The gap is the term premium — extra compensation for bearing duration. So:

$\text{forward rate} = \text{expected future spot rate} + \text{term premium}$

And the term premium is positive most of the time, because duration is a risk somebody has to be paid to hold. Chapter 6 showed what that risk is, and chapter 13 showed how it behaves in a large move.

The consequence is specific: forward rates systematically overstate expected future rates. A forward curve implying rates 170 basis points higher in five years is not the market predicting 170 basis points of increases. Part of it is the price of taking duration risk, and nothing in the curve separates the two.

This is the same structure as the Option pricing subject's conclusion that implied probabilities overstate bad outcomes because insurance trades above expected loss — and the Portfolio theory subject's point that the skew is part distribution and part price of fear. A price that embeds a risk premium cannot be read as a belief, and that generalisation is worth more than any of the three cases.

What this does to riding the curve

Chapter 9 described rolling down the curve and called it "a bet on stability, not an arbitrage." Forward rates say precisely what the bet is.

Riding the curve profits if rates in a year are below today's forward rates, and loses if they are above. The forward rate is the exact breakeven — so the strategy is a position against the forward curve, taken deliberately or not.

And because forwards contain a term premium, that bet has historically paid more often than not. If forwards overstate expected rates, then betting that rates come in below the forwards is betting with the premium. That is the honest case for riding the curve: not that rates will fall, but that the forwards embed compensation you collect by holding duration.

With the honest counterweight. The premium is compensation for a real risk, and it is paid out in the episodes where rates rise sharply — chapter 13's 300 basis point column shows a ten-year bond losing nearly a fifth of its value. You are being paid to bear that, and sometimes you bear it.

The limits

Forwards past the liquid maturities are extrapolation. Chapter 15 noted that real curves have gaps filled by interpolation; forwards amplify whatever that interpolation assumed, because they are built from differences between neighbouring points. A one-year forward starting in year 23, derived from a curve with liquid points at 20 and 30 years, is mostly an artefact of the fitting method.

Small errors in spot rates become large errors in forwards. The forward depends on a ratio of two compounded numbers, so a basis point of noise at each end can be several basis points in the forward.

And the instruments to transact at the forward rate may not exist. The arbitrage argument assumes you can contract today for a future period. Where that market is thin, the forward is a computed number rather than a tradeable rate — true as arithmetic, unavailable as a trade.

Working the problem

One-year spot 6.18%; two-year spot 6.5104%.

Step 1 — equate the two routes.

$(1 + 0.065104)^2 = (1 + 0.0618)(1 + f)$

Step 2 — compute each side.

$1.065104^2 = 1.134447$

$1 + f = \frac{1.134447}{1.0618} = 1.068419$

$f = 6.8419\%$

So the market is pricing one-year money, a year from now, at 6.8419%.

Step 3 — what has to happen for rolling to win. Buy a one-year bill at 6.18%, and in a year buy another at whatever rate $r$ then prevails. Over two years that gives $(1.0618)(1+r)$ against the two-year bond's $1.065104^2 = 1.134447$.

Rolling wins when

$(1.0618)(1 + r) > 1.134447 \quad\Longrightarrow\quad r > 6.8419\%$

The one-year rate must be above 6.8419% in a year's time — which is 66 basis points above where one-year money is priced today.

Step 4 — the part that is usually got wrong. It is tempting to conclude the market expects a 66 basis point rise, and to roll if you disagree.

That is not what the number says. The 6.8419% is the breakeven, and it equals the expected future rate plus a term premium. If the term premium over this horizon is, say, 30 basis points, the market's actual expectation is nearer 6.54% — and rates could rise by 36 basis points, vindicating the "expectation", while the two-year bond still wins.

So the correct framing is the one chapter 9 reached by a different route. Do not ask whether you can out-forecast the market. Ask what you are being paid:

You are being offered 6.5104% for two years against 6.18% for one. The extra 33 basis points a year is the compensation for committing for a second year — part expected rate rise, part premium for duration risk, in proportions nobody can observe.

Decide on that term sheet. If 33 basis points is not enough to make the second year of duration risk worth taking, roll the bills — and that reasoning holds whether or not rates do what the forwards imply, which is exactly what makes it more robust than a forecast.

The point

Spot rates at two dates fix the rate between them by arbitrage, $f = [(1+s_n)^n/(1+s_m)^m]^{1/(n-m)} - 1$ — the same structure as put-call parity and covered interest parity, and containing no forecast. The Indian bill curve for the week ended 2 October 2026 implies forwards of about 6.40% between every pair of its points, so a single number generates its whole slope, sitting about 115 basis points above the 5.12% call rate. Forwards rise faster than spots and exceed them on an upward-sloping curve. They are breakevens, not expectations: a forward equals the expected future spot plus a term premium, so forwards systematically overstate expected rates — which is the honest case for riding the curve, and the reason a forward curve implying sharp rises is not a prediction of them.