Rate Conventions In Derivative No-Arbitrage Equations

Why Rate Conventions Matter in Derivative Pricing

A derivative’s no-arbitrage price depends on the cost of carrying an asset to a future date. That cost is determined by an accumulation factor over the contract’s exact term. A quoted rate alone is not enough: “5%” might mean an effective annual rate, a nominal annual rate convertible semiannually, or a force of interest of 5% per year. Those conventions produce different accumulation factors.

This tutorial focuses on converting rates before using a derivative-pricing equation. It uses the simplest forward contract: the underlying asset has no income or carrying costs, and the contract requires delivery at a future date. The result is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: The effective annual rate $i$ is the rate at which 1 invested at the start of a year accumulates to $1+i$ at the end of that year. A nominal annual rate convertible $m$ times per year, written $i^{(m)}$, credits interest at rate $i^{(m)}/m$ each conversion period. The force of interest $\delta$ is a continuously compounded rate: over $T$ years, 1 accumulates to $e^{\delta T}$, when $\delta$ is constant.

For a constant rate over $T$ years, the relevant accumulation factors are

(1+i)^T,\qquad \left(1+\frac{i^{(m)}}{m}\right)^{mT},\qquad e^{\delta T}.

The nominal-rate expression assumes that $mT$ is a whole number of conversion periods. For other terms, the contract or valuation model must specify how to handle a partial conversion period; do not silently assume a convention that has not been given.

Convert the Rate, or Convert the Accumulation Factor

When rates are quoted on different bases, there are two safe approaches. One is to convert every quote to an equivalent effective annual rate. The other is to use the accumulation factor that directly matches the quoted convention and the contract term. These approaches agree when applied correctly.

Rate-conversion identities: For equivalent rates over one year, $$ 1+i=\left(1+\frac{i^{(m)}}{m}\right)^m=e^\delta. $$ Thus $$ i=\left(1+\frac{i^{(m)}}{m}\right)^m-1,\qquad \delta=\ln(1+i),\qquad i=e^\delta-1. $$ For a nominal rate convertible $m$ times per year, $$ \delta=m\ln\left(1+\frac{i^{(m)}}{m}\right). $$

The conversions preserve the value accumulated over one year; they do not make the quoted rate numbers equal. For example, a nominal rate and its equivalent effective annual rate generally have different numerical percentages. Before substituting a rate into a derivative equation, check whether that equation requires a rate or an accumulation factor.

Worked Example: Converting a Quarterly Nominal Rate

A hypothetical investment earns a nominal annual rate of 6% convertible quarterly. Find its equivalent effective annual rate and force of interest. Then find the accumulation of $1 over two years. State the rates to four decimal places as percentages and the accumulation to the nearest cent.

Method. The quarterly rate is $0.06/4=0.015$, or 1.5% per quarter. Four quarters give the effective annual accumulation factor:

\left(1+\frac{0.06}{4}\right)^4 =(1.015)^4 =1.061363550625.

Therefore the equivalent effective annual rate is

i=1.061363550625-1=0.061363550625 =6.1364\%.

The equivalent force of interest is the natural logarithm of the annual accumulation factor:

\delta=\ln(1.061363550625) =4\ln(1.015) \approx 0.05955445 =5.9554\%.

Over two years there are eight quarterly periods. Hence

(1.015)^8 =(1.061363550625)^2 \approx 1.126493.

So $1 accumulates to approximately $1.13. The equivalent force gives the same result: $e^{2(0.05955445)}\approx 1.126493$. The calculation is consistent because both conventions describe the same accumulation. The common mistake this method avoids is treating 6% nominal convertible quarterly as 6% effective annually.

Using the Matching Factor in a Forward Equation

Consider a forward contract agreed at time 0 for delivery at time $T$. Let $S_0$ be the asset’s current spot price and $F_{0,T}$ the delivery price that makes the contract fair at initiation. Assume the asset pays no income, has no storage or other carrying costs, and can be bought, sold, borrowed, and lent without transaction costs. Assume also that the same deterministic risk-free rate applies to borrowing and lending over the term.

Theorem: Under the assumptions above, if the risk-free accumulation factor from time 0 to time $T$ is $A(0,T)$, the no-arbitrage forward price is $$ F_{0,T}=S_0A(0,T). $$ In particular, for a constant effective annual rate $i$, $F_{0,T}=S_0(1+i)^T$; for a constant nominal annual rate $i^{(m)}$, $F_{0,T}=S_0\left(1+i^{(m)}/m\right)^{mT}$ when $mT$ is an integer; and for a constant force $\delta$, $F_{0,T}=S_0e^{\delta T}$.

The reason is a cash-and-carry comparison. Buying the asset now costs $S_0$, and financing that purchase to time $T$ costs $S_0A(0,T)$. At time $T$, the investor owns the asset and can deliver it into the forward contract in exchange for $F_{0,T}$. If the forward price differs from the financed purchase cost, one side of this comparison can produce a locked-in gain under the assumptions.

This result does not apply unchanged if the asset pays income, incurs carrying costs, or faces restrictions that prevent the assumed trades. Those features alter the cash flows and must be included in the pricing relation.

Identify the quoted convention.
Decide whether the rate is effective annual, nominal convertible at a stated frequency, or a force of interest.

Match the term.
Use the contract’s maturity $T$ and convert the annual rate convention into an accumulation factor over exactly $T$ years.

Apply the pricing equation.
For the no-income, no-carrying-cost forward model, multiply the spot price by the matching accumulation factor.

Check the convention.
Confirm that any comparison price uses the same underlying rate convention and maturity.

Worked Example: Forward Price with a Force of Interest

A hypothetical asset has a spot price of $250 and pays no income. A one-year force of interest of 4% applies, and a forward contract matures in nine months. Find the no-arbitrage forward price to the nearest cent. Also quantify the error from treating 4% as an effective annual rate.

Method. The maturity is $T=9/12=0.75$ years. Under the stated force convention, the accumulation factor is $e^{\delta T}$, not $(1+\delta)^T$. Thus

F_{0,0.75}=250e^{0.04(0.75)} =250e^{0.03} \approx 250(1.030454534) =257.6136335.

The no-arbitrage forward price is $257.61 to the nearest cent.

If 4% were instead an effective annual rate, the factor would be

(1.04)^{0.75}\approx 1.02985245, \qquad 250(1.04)^{0.75}\approx 257.4631.

That alternative produces $257.46 to the nearest cent, about $0.15 less than the correct price. The difference is not a rounding issue; it comes from changing the rate convention. The common mistake this method avoids is using the numerical value of a force as if it were an effective annual rate.

Derivation: Why the Forward Price Uses Accumulated Spot

The forward equation follows by comparing two ways to obtain the asset at time $T$. First, buy it now and finance the purchase. The initial cost is $S_0$, so the amount owed at time $T$ is $S_0A(0,T)$. Second, enter a forward contract to pay $F_{0,T}$ at time $T$ and receive the asset then. These alternatives deliver the same asset at the same date under the assumptions.

If $F_{0,T}>S_0A(0,T)$, an investor can borrow $S_0$, buy the asset, and sell it forward. At time $T$, delivering the asset brings in $F_{0,T}$, while repaying the loan costs $S_0A(0,T)$. The difference is positive:

F_{0,T}-S_0A(0,T)>0.

If $F_{0,T}<S_0A(0,T)$, the reverse comparison applies: sell the asset short, invest the proceeds, and buy the asset through the forward at time $T$. The investment grows to $S_0A(0,T)$, while the forward costs $F_{0,T}$, leaving a positive difference. Thus either inequality would allow a gain under the stated assumptions, so the no-arbitrage equality is

F_{0,T}=S_0A(0,T).

This justification also shows why the rate convention is essential: it determines the amount that must be repaid, $S_0A(0,T)$, at the forward’s delivery date.

Worked Example: Detecting a False Arbitrage Signal

A hypothetical asset with no income has a spot price of $100. Its one-year financing rate is quoted as 5% nominal convertible semiannually. A dealer quotes a one-year forward delivery price of $105.0625. A trader treats the nominal rate as 5% effective and concludes that the forward is overpriced by $0.0625. Is that conclusion correct? Show the correct comparison and the apparent error.

Method. There are two semiannual periods, each at rate $0.05/2=0.025$. The correct one-year accumulation factor is

\left(1+\frac{0.05}{2}\right)^2 =(1.025)^2 =1.050625.

Therefore the no-arbitrage forward price is

F_{0,1}=100(1.050625)=\$105.0625.

The dealer’s quote exactly equals this price. If the trader incorrectly treats 5% nominal convertible semiannually as 5% effective, the trader calculates $100(1.05)=\$105.00$ and sees an apparent excess of

\$105.0625-\$105.00=\$0.0625.

That apparent discrepancy is caused by the wrong convention, not by an arbitrage opportunity. Financing $100 for one year under the actual terms requires repayment of $105.0625; the forward delivers exactly that amount. The common mistake this method avoids is comparing the contract quote with a forward price calculated using a different interest convention.

Common Convention Errors

Rate-convention errors often appear in integrated calculations because the quoted percentage looks familiar. Before using a rate, make the units and compounding basis explicit. The following checks help prevent a mistaken no-arbitrage conclusion:

Do not equate nominal and effective rates. A nominal rate is quoted per year but divided across conversion periods. Its annual accumulation is generally not $1+i^{(m)}$.
Do not use a force as an effective rate. A force accumulates through an exponential factor $e^{\delta T}$, not through $(1+\delta)^T$.
Use the actual contract term. A rate stated per year must be applied to the maturity in years, with the correct conversion frequency.
Compare like with like. A quoted forward price and a calculated fair price must use consistent rate conventions, maturity dates, and assumptions about income and costs.
Distinguish an apparent discrepancy from a tradeable one. A difference caused by inconsistent calculations is not a market mispricing. A genuine arbitrage conclusion also depends on the model’s trading and financing assumptions.

Key takeaway: Derivative no-arbitrage equations use accumulation factors over the contract term. Convert the quoted rate convention first, then compare prices using the same maturity, assumptions, and financing basis.

Check Your Understanding

For each problem, use the stated convention throughout and round only the final monetary amount to cents or the final percentage to four decimal places.

A constant force of interest is 0.02 per year. Find the equivalent effective annual rate as a percentage. (Answer: 2.0201%)
A savings account quotes 8% nominal interest convertible quarterly. Find the effective annual rate. (Answer: 8.2432%)
A no-income asset has a spot price of $400. If the constant effective annual risk-free rate is 3%, find the fair forward price for delivery in two years. (Answer: $424.36)
A no-income asset costs $80, and the constant force of interest is 6% per year. Find the fair forward price for delivery in 18 months. (Answer: $87.53)
A no-income asset has a spot price of $150. The financing rate is 4% nominal convertible quarterly, and a forward matures in three years. Find the fair forward price. If the quoted price is $170, state whether it is above or below the calculated fair price and by how much per unit. (Answer: Fair price $169.02; quote is above it by $0.98 per unit.)

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