Reprice cash flows without arbitrage

Learning goals

• Calculate the bond value from the given discount factors.
• Explain why the 105 principal-plus-coupon cash flow applies to this case.
• Identify the case-specific assumption or failure mode before taking action.

Draft · approximately 60 minutes, excluding further revision.

Shared case and decision context

A two-year bond pays coupon 5 each year and principal 100 at year two. Discount factors are d1=0.97 and d2=0.93. Ignore default and transaction costs. A dealer quotes 104 for the same contractual cash flows.

Economic or professional mechanism

No-arbitrage valuation discounts each dated cash flow with the matching discount factor rather than using one coupon rate as a discount rate. Identical cash flows should have identical value under the same risk and trading assumptions. A recombining interest-rate tree must be calibrated so its model prices reproduce the selected market benchmarks.

Apply, challenge and communicate

The cash-flow value is 102.5. A quote of 104 is above that benchmark, but an executable arbitrage also requires the ability to buy the replicating cash flows, short the bond and finance positions at assumed terms. Risk-neutral probabilities are pricing devices, not claims about physical frequency.

Make rate-tree probabilities and node rates explicit

A binomial interest-rate tree gives two future rate states at each step and specifies transition pricing probabilities and discount rates. Recombination can reduce computational complexity but does not ensure calibration. Bond values at a node depend on the next cash flow and discounted continuation values. The tree must represent its compounding and time-step conventions consistently; rates quoted annually require conversion for shorter steps.

Perform one backward-induction step

Suppose next-period bond values including the relevant next payment are102 and98, risk-neutral weights.5/.5 and the current one-period rate4%. The current value is(.5×102+.5×98)/1.04=96.1538. Include a coupon either in the next value or as an explicit added cash flow, never twice. Continue backward to the valuation date and check that known benchmark prices are reproduced.

Reconcile zero-curve and tree prices

For an option-free bond with deterministic cash flows, an appropriately calibrated arbitrage-free rate tree and the same zero-coupon discount curve should produce the same price within numerical precision. A discrepancy can reveal coupon timing, compounding, calibration or cash-flow errors. A tree becomes particularly useful when future cash flows depend on rates; equality for a straight bond does not imply that embedded options can be valued without a rate model.

Discount each path before averaging

Pathwise valuation discounts cash flows along each realized sequence of short rates and averages using the appropriate pricing weights. Path dependence matters for instruments whose cash flows depend on prior rates or exercise decisions. Averaging rates first and then discounting generally differs from averaging discounted values because discounting is nonlinear. Ensure path probabilities sum to one and each path’s cash flow and discount factors use matching dates.

Simulate future curves without confusing measures

Monte Carlo rate valuation simulates rate paths under a specified model and pricing measure, evaluates cash flows and discounts them consistently. More paths reduce simulation sampling error but do not cure misspecified dynamics or wrong volatility inputs. Standard errors and convergence diagnostics help distinguish numerical noise from model differences. Early-exercise claims need an appropriate exercise method; a simple terminal-payoff average may not capture optimal exercise.

Select a rate model for the decision

Equilibrium-style short-rate models describe dynamics using economically motivated parameters; no-arbitrage models are fitted to reproduce current market prices. Mean reversion, volatility structure, positivity constraints and the ability to accommodate current curves differ across models. A model fitting today’s curve can still give poor option values if volatility dynamics are wrong. Select the model against the instrument, calibration data and risk horizon, and disclose material limitations.

Formula and assumptions

P=sum(CF_t*d_t)

Case analysis with explicit assumptions

1. A two-year bond pays coupon 5 each year and principal 100 at year two. Discount factors are d1=0.97 and d2=0.93. Ignore default and transaction costs. A dealer quotes 104 for the same contractual cash flows.
2. Method: P=sum(CF_t*d_t)
3. Case result: 102.50. The cash-flow value is 102.5. A quote of 104 is above that benchmark, but an executable arbitrage also requires the ability to buy the replicating cash flows, short the bond and finance positions at assumed terms. Risk-neutral probabilities are pricing devices, not claims about physical frequency.

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.
• Guarantee physical outcomes 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
• Reference · www.cfainstitute.org