Forward rate

You have money to invest for two years and two ways to do it. Buy a two-year bond. Or buy a one-year bond and, when it matures, reinvest for a second year at whatever the one-year rate is then.

You know today's one-year and two-year rates. You do not know next year's one-year rate. But the market does price it, implicitly: there is exactly one value for next year's rate that makes the two strategies end with the same amount of money. Any other value and somebody borrows through one route and lends through the other for a riskless profit.

That implied value is the forward rate. It is not a forecast and nobody claims it will come true. It is the rate at which no arbitrage exists today.

The mechanism is simply compounding equality. Invest for the longer term at the longer spot rate, or invest for the shorter term at the shorter spot rate and then for the remaining period at the forward rate — the two totals must match.

Because a longer spot rate is a blend of the shorter rate and the forward rate, an upward-sloping term structure forces the forward rate above both spot rates. The 7.01% in the example below is higher than the 2-year 6% precisely because it has to pull the average up from the first year's 5%.

This is the machinery behind the 91-day T-Bill futures contract. On 1 October, with expiry on 27 October, the market needs the 91-day rate that will apply on 27 October — a rate 26 days forward. It takes the 26-day and 117-day yields off the FBIL Treasury bill curve, interpolating or extrapolating where a tenor is not published, and solves for the 91-day forward rate between them. The futures quote is then 100 minus that yield.

A forward rate agreement locks the same rate contractually instead of implying it.

          ⎡ (1 + S₁)ⁿ¹ ⎤ 1/(n₁ − n₂)
F  =  ⎢ ────────── ⎥            − 1
          ⎣ (1 + S₂)ⁿ² ⎦

   S₁ = spot rate to the further date,  n₁ = years to it
   S₂ = spot rate to the nearer date,   n₂ = years to it

The compounding frequency must be carried through. For semi-annual rates, divide each rate by 2, use 2 × years as the exponent, and multiply the answer by 2 to annualise.

The two-route argument in rupees. An investor has Rs 1,000 for two years. The 2-year spot rate is 6%; the 1-year spot rate is 5%.

Route 1 — two-year bond:
   1,000 × (1.06)²                     = Rs 1,123.60

Route 2 — one year, then reinvest at F:
   1,000 × (1.05) × (1 + F)            = Rs 1,050 × (1 + F)

Set them equal:

1,050 × (1 + F) = 1,123.60
              F = 7.0095%

The market is implying a one-year rate of 7.01%, one year from now. If a bank offered to lock next year's money at 6.50%, a dealer would borrow for two years at 6%, lend for one at 5%, and lock the second year at 6.50% — and lose money. At 7.50% the trade runs the other way. Only at 7.0095% is there nothing to do.

Semi-annual, the way the T-Bill curve actually comes. The 6-month spot rate is 5.60% and the 1-year spot rate is 5.85%:

⎡ (1 + 0.0585/2)^(2×1) ⎤
⎢ ──────────────────── ⎥ − 1, then × 2  =  6.10%
⎣ (1 + 0.0560/2)^(2×0.5) ⎦

The 6-month rate, 6 months forward, is 6.10% — 25 basis points above the 1-year spot and 50 above the 6-month, exactly as an upward-sloping curve demands.

Turning it into a futures price. If the 91-day forward yield implied for 27 October works out at 5.25%, the 91-day T-Bill futures contract theoretically quotes:

100 − 5.25 = Rs 94.7500
Contract value = 2,000 × (100 − 0.25 × 5.25) = Rs 1,97,375 per lot

Chapter 3, section 3.7.1.1 (Forward Rate), derives it as the first of the concepts needed before futures can be priced, and section 3.7.5 applies it to computing the price of the 91-day T-Bill futures contract off the FBIL yield curve. Chapter 2, section 2.2.1, gives the contractual version — the forward rate agreement — and Chapter 1, section 1.9, supplies the spot rates it is built from.

Expect to be given two spot rates and asked for the implied forward rate, or to be asked what a forward rate represents — the answer being a no-arbitrage implication, not a prediction.

• Put the longer maturity on top. S₁ and n₁ belong to the further date; inverting the fraction inverts the answer.
• The exponent is 1 ÷ (n₁ − n₂), the length of the forward period — not 1 ÷ n₁.
• A forward rate is not a forecast. It is what today's prices imply; whether the rate materialises is a separate matter entirely.
• On an upward-sloping curve the forward rate exceeds both spot rates. A forward rate sitting between the two spot rates is arithmetically impossible and signals an inverted fraction.
• Carry the compounding through. Semi-annual inputs need the rate halved, the exponent doubled and the result annualised by 2 — three adjustments, and candidates typically make one.
• Where a tenor is missing from the curve, interpolate. The workbook expects the 26-day and 117-day yields to be interpolated or extrapolated from the published FBIL curve, not guessed.