Extract a forward rate from spot discounting

Learning goals

• Calculate the one-year forward rate starting in one year.
• Explain why a rate consistent with the current spot curve applies to this case.
• Identify the case-specific assumption or failure mode before taking action.

Draft · approximately 70 minutes, excluding further revision.

Shared case and decision context

Annual-compounded one- and two-year spot rates are 3% and 4%. A strategist interprets the implied second-year forward rate as a certain prediction. The curve may contain term and liquidity premiums.

Economic or professional mechanism

Spot rates discount single dated cash flows. A forward rate equates investing directly to a later date with rolling through an earlier date, using the same compounding convention. Expectations theories and term-premium models explain curve shape differently; an implied forward is not automatically an unbiased or certain future short rate.

Apply, challenge and communicate

The one-year forward starting in one year is 5.0097%. Separate parallel level shifts from slope and curvature changes when evaluating bond exposure. A single duration can miss nonparallel risk, so key-rate analysis and scenario repricing are useful complements.

Bootstrap the second maturity

Bootstrapping solves discount factors sequentially from liquid benchmark prices. If a one-year zero price is.97 and a two-year par bond pays4 annually on100, then100=4×.97+104×d2, giving d2=.9242308. Earlier cash flows use already known factors. Apply the same settlement, day count and quotation basis to all instruments. Interpolation choices between benchmark maturities introduce model risk.

Identify the curve-evolution assumption

An active bond strategy takes a view on how future spot rates will compare with rates already implied by the curve. A forecast of unchanged spot rates is not the same as assuming future spot rates equal current forwards. Expected return reflects coupon income, reinvestment and the future sale price under the forecast. Calculate the price at the shortened maturity using the forecast curve, not the original yield mechanically.

Understand rolling down an unchanged curve

On an unchanged upward-sloping spot curve a bond becomes a shorter-maturity instrument over the holding period. Its future required yield can be lower, creating a price benefit in addition to coupon income. This roll-down is conditional: a curve shift or credit-spread widening can offset it. Include transaction costs and reinvestment assumptions before treating a roll-down estimate as an implementable expected return.

Use the swap curve for its actual market convention

A swap curve contains par fixed rates on swaps across maturities. Its instruments can be liquid and useful valuation benchmarks, but counterparty, collateral, projection and discount conventions matter. A collateralized swap framework may use an overnight-index curve for discounting and another curve for projected floating payments. Do not label every swap rate sovereign-risk-free or use a par swap rate as if it were the matching zero rate.

Keep swap-spread units consistent

A conventional maturity-matched swap spread is the par swap rate minus the government bond yield. If the swap rate is4.20% and the selected government yield3.80%, the spread is.40 percentage points, or40 basis points. This spread can reflect relative funding, liquidity, supply and balance-sheet conditions; it is not a pure probability of a single issuer’s default. State the benchmark and maturity.

Read short-rate spreads as joint credit and liquidity signals

A short-term unsecured funding rate can exceed a comparable overnight-index benchmark because lenders demand credit and liquidity compensation. Widening is a warning of funding stress, not proof of a particular bank’s default. Match currency and tenor, understand current benchmark definitions and avoid blindly extending obsolete reference-rate relationships. Collateral, market segmentation and central-bank operations can affect the observed difference.

Separate yield volatility and price sensitivity

Price volatility depends on both yield volatility and the bond’s price sensitivity to the relevant yield changes. A long-duration bond can have substantial price risk even if its yield volatility is lower than that of short rates. Correlated shifts across maturities and convexity matter beyond a single-duration approximation. Compare horizon and yield-change units before multiplying duration by a volatility estimate.

Form a rate view from a coherent macro scenario

A rate view combines inflation expectations, real growth, monetary-policy expectations, term premiums and bond supply-demand conditions. An inflation surprise may raise nominal yields without proportionally raising real yields. Credit spreads can widen while government yields fall during risk aversion. State which curve segment and spread the scenario moves, then reprice the portfolio rather than calling every yield a single interest rate.

Formula and assumptions

(1+s2)^2=(1+s1)*(1+f1,1)

Case analysis with explicit assumptions

1. Annual-compounded one- and two-year spot rates are 3% and 4%. A strategist interprets the implied second-year forward rate as a certain prediction. The curve may contain term and liquidity premiums.
2. Method: (1+s2)^2=(1+s1)*(1+f1,1)
3. Case result: 5.01%. The one-year forward starting in one year is 5.0097%. Separate parallel level shifts from slope and curvature changes when evaluating bond exposure. A single duration can miss nonparallel risk, so key-rate analysis and scenario repricing are useful complements.

Errors to catch

• Do not replace the stated assumptions with an unstated shortcut. Reconcile units, timing and the claim being valued or duty being assessed.
• Always equal realized spot rates is not a defensible conclusion for these case facts.

Original knowledge check

Answer every question before checking. Review explanations after submitting; retry starts a new attempt.

Sources and coverage status

These original lessons are drafts. Public outlines are used to check scope; the official learning-outcome text and proprietary question banks are not reproduced here. Follow the source for exam-year requirements and updates.

• Reference · www.cfainstitute.org
• Reference · www.bis.org