No-Arbitrage Between Locking And Rolling

Why Locking and Rolling Must Match

An investor who wants funds at a future date may invest directly to that date at a spot rate. Alternatively, the investor may invest to an earlier date and then reinvest for the remaining period at a forward rate. If the forward rate is the one implied by the spot rates, these two strategies produce the same accumulation at the target date.

This equality is a basic consistency condition for interest rates across maturities. The comparison is meaningful only when both strategies use the same initial amount, end at the same time, and use rates with compatible compounding conventions. It does not require the spot and forward rates themselves to be equal; it requires their accumulated values over the relevant periods to agree.

The exponent is the length of the investment period in years. Thus, for a forward period lasting \(1.5\) years, the annual effective forward rate \(f\) gives an accumulation factor of \((1+f)^{1.5}\), not \(1+1.5f\). The latter is simple-interest accumulation and is a different model.

The No-Arbitrage Equality

Suppose the spot and forward investments are comparable: they have the same currency, credit quality, payment certainty, and relevant contractual terms, and there are no taxes, transaction costs, or other frictions that change the comparison. Assume that the stated rates are annual effective rates and that fractional periods use annual-effective compounding. Under these assumptions, define the two-year-horizon strategies (more generally, the strategies ending at \(t\)) as follows:

• Lock: invest one unit at time 0 to time \(t\) at the spot rate \(i_t\).
• Roll: invest one unit at time 0 to time \(s\) at the spot rate \(i_s\), then reinvest the proceeds from \(s\) to \(t\) at the forward rate \(f_{s,t}\).

The locked investment accumulates to \((1+i_t)^t\). The rolled investment first accumulates to \((1+i_s)^s\), then that amount accumulates by \((1+f_{s,t})^{t-s}\). Equating their terminal values gives the no-arbitrage relation:

This equation compares terminal amounts, not quoted rates. The exponents encode how long each rate applies. If the relation fails for rates that are otherwise comparable, the direct investment and the two-stage investment do not have the same terminal value.

Why the Relation Holds

The justification is a same-cash-flow comparison. Starting with one unit at time 0, the direct investment produces a definite amount at time \(t\). The staged investment also produces a definite amount at time \(t\), provided the forward rate is contractually specified. If the two amounts differ, an investor could choose the strategy with the larger terminal amount for the same initial outlay, while taking the opposite position in the other strategy where available. In a frictionless market with comparable investments, persistent unequal values would therefore be inconsistent with no-arbitrage pricing.

Solving the equality for the implied forward rate makes the role of the two spot rates explicit. Divide by the accumulation to time \(s\), then take the power \(1/(t-s)\):

The numerator is the total accumulation to the later date, and the denominator removes the accumulation already earned by time \(s\). The remaining factor covers only the interval from \(s\) to \(t\). Taking the appropriate root converts that period factor into an annual effective forward rate.

Fractional Forward Periods

The same equality applies when either maturity is fractional. The rate convention must remain consistent: if a spot rate is quoted as an annual effective rate to maturity \(t\), its accumulation factor is \((1+i_t)^t\). If a forward rate applies for \(t-s\) years, its factor is \((1+f_{s,t})^{t-s}\). The duration of the period, rather than the number of rate quotes, determines the exponent.

For example, a forward rate from year 1 to year 2.5 applies for \(2.5-1=1.5\) years. Its accumulation factor is \((1+f_{1,2.5})^{1.5}\). This must be multiplied by the one-year spot accumulation factor to obtain the total accumulation to year 2.5.

Compare Terminal Values Before Comparing Rates

A reliable way to use the relation is to put both strategies on the same timeline and calculate their terminal accumulation factors. Only after confirming the time intervals should you solve for a missing rate or compare quoted rates. This prevents a common error: comparing annual rates directly even though they apply for different lengths of time.

The following example uses a two-year forward period. The later spot rate is the annual effective rate that makes a four-year locked investment equivalent to rolling through the two-year forward rate.

Common Mistakes and Exam Checks

A no-arbitrage equation can look correct algebraically and still be wrong if the intervals or conventions do not match. Before accepting a calculation, identify the initial date, the intermediate date, and the final date. Then label each factor by the interval over which it accumulates.

• Using a rate for the wrong interval. A spot rate to year \(t\) applies from time 0 through year \(t\); it is not automatically the rate for the final year alone.
• Forgetting the forward-period length. From year 1 to year 2.5 is \(1.5\) years, so an annual effective forward rate is raised to the power \(1.5\).
• Adding rates instead of multiplying factors. The accumulated factors chain multiplicatively: first \((1+i_s)^s\), then \((1+f_{s,t})^{t-s}\).
• Mixing conventions. The formulas here assume annual effective rates. A nominal rate or a force of interest must be converted or used with its own accumulation formula before making the comparison.
• Rounding too early. Keep the implied factor at full precision through the calculation, then round the requested rate or money amount at the end.

A useful check is to substitute the computed forward rate back into the original equality. If the two accumulation factors do not agree, check the exponents first, then the rate convention, and finally the arithmetic.