Comparing Spot-Rate Quotation Conventions

Why Quotation Conventions Matter

A spot rate describes the accumulation or discounting associated with a particular maturity. The same maturity-specific accumulation can be quoted as an annual effective rate, a nominal rate convertible a specified number of times per year, or a continuously compounded rate. These are different ways of expressing the same financial relationship; the numerical rate is not generally the same across conventions.

For valuation, the most useful comparison is the discount factor: the present value of one unit payable at the specified maturity. Once two quotations give the same discount factor, they produce the same present value for every cash flow paid at that maturity. This tutorial assumes a deterministic spot rate for the maturity being valued and no change in the stated cash flow. It is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: For a maturity of $t$ years, let $i_t$ be the annual effective spot rate, $i_t^{(m)}$ the nominal spot rate convertible $m$ times per year, and $\delta_t$ the continuously compounded spot rate. Each quotation implies a discount factor $v_t$, the present value at time 0 of 1 payable at time $t$.

Discount Factors Under the Three Conventions

An annual effective rate applies once per year. A nominal rate convertible $m$ times per year applies at a periodic rate of $i_t^{(m)}/m$. A continuously compounded rate uses exponential accumulation. Their corresponding discount factors are:

v_t=(1+i_t)^{-t} =\left(1+\frac{i_t^{(m)}}{m}\right)^{-mt} =e^{-\delta_t t}.

For the nominal formula, $m$ is a positive integer and $t$ is measured in years. The exponent $mt$ counts the compounding periods. When the maturity falls between scheduled compounding dates, use the contract’s stated fractional-period or day-count rule rather than assuming that this formula governs. For the annual effective and continuous formulas, $t$ is a year fraction consistent with the quote.

The equation compares quotations for the same maturity and the same discount factor. It does not say that rates at different maturities are interchangeable. A spot rate for one maturity should not be applied to another maturity simply because the quotation convention matches.

Converting Equivalent Quotations

The annual accumulation factor over one year is $1+i_t$. Under nominal conversion it is $\left(1+i_t^{(m)}/m\right)^m$; under continuous compounding it is $e^{\delta_t}$. Equating these one-year factors gives the conversion formulas:

1+i_t =\left(1+\frac{i_t^{(m)}}{m}\right)^m =e^{\delta_t}.

i_t^{(m)} =m\left((1+i_t)^{1/m}-1\right), \qquad \delta_t=\ln(1+i_t), \qquad i_t=e^{\delta_t}-1.

These conversions assume all quotations refer to the same accumulation relationship. An annual effective rate must satisfy $i_t>-1$, so its annual accumulation factor is positive. For the nominal formula, the periodic accumulation factor must be positive: $1+i_t^{(m)}/m>0$. A continuous rate $\delta_t$ can be any real number because $e^{\delta_t}$ is positive for every real $\delta_t$.

Identify the maturity and convention.
Record the term in years and, for a nominal quote, the number of conversions per year.

Find the annual accumulation factor.
Use $1+i_t$, $\left(1+i_t^{(m)}/m\right)^m$, or $e^{\delta_t}$, as appropriate.

Calculate the discount factor.
Raise the annual accumulation factor to $-t$, or use $e^{-\delta_t t}$.

Value the payment.
Multiply the discount factor by the amount payable at that maturity.

Why the Conversion Formulas Hold

Consider investing 1 unit for one year under each convention. The annual effective quote produces $1+i_t$ at the end of the year. With $m$ conversions, the periodic rate is $i_t^{(m)}/m$; compounding that rate $m$ times produces $\left(1+i_t^{(m)}/m\right)^m$. Under continuous compounding, accumulation for one year is $e^{\delta_t}$. If the quotes describe the same one-year accumulation, these three values must be equal.

For $t$ years, the corresponding accumulation factors are the one-year factors raised to the power $t$. The discount factor is the reciprocal of the accumulation factor, giving the formulas above. In particular, $v_t$ is not a different price for each convention: it is one value expressed using whichever equivalent rate quotation is convenient.

Worked Comparisons

Worked Example: Converting an Annual Effective Quote

A hypothetical zero-coupon instrument pays $10,000 in 2 years. Its 2-year spot rate is quoted as 44% annual effective. Find the equivalent nominal rate convertible semiannually, the equivalent continuous rate, and the present value. Assume the quoted rates describe the same deterministic 2-year accumulation.

Method. First convert the one-year accumulation factor, then calculate the 2-year discount factor. This avoids treating 44% as a nominal rate without conversion.

The annual effective accumulation factor is $1+0.44=1.44$. For a nominal rate convertible twice yearly, let $j=i_2^{(2)}$. Equate the annual factors:

\left(1+\frac{j}{2}\right)^2=1.44, \qquad 1+\frac{j}{2}=\sqrt{1.44}=1.20, \qquad j=2(0.20)=0.40.

The equivalent nominal rate is 40% convertible semiannually. The continuous rate is:

\delta_2=\ln(1.44)=2\ln(1.20)=0.3646431\ldots

Now calculate the discount factor for 2 years:

v_2=(1.44)^{-2} =\frac{1}{1.44^2} =\frac{1}{2.0736} =0.4822530864\ldots

Thus the present value is $10{,}000(0.4822530864\ldots)=\$4{,}822.530864\ldots$, or $4,822.53 to the nearest cent. As a check, the nominal quotation gives $(1.20)^{-4}=1/2.0736$, and the continuous quotation gives $e^{-2(0.3646431\ldots)}=1/2.0736$. Both give the same price.

Worked Example: Starting From Continuous Compounding

A hypothetical instrument pays $5,000 in 3 years. Its 3-year spot rate is quoted as a continuously compounded rate of 8% per year. Find the equivalent annual effective rate, the equivalent nominal rate convertible monthly, and the present value. Assume the monthly quote compounds for exactly 36 periods.

Method. Convert the continuous rate to the annual accumulation factor. Then use that factor for the rate conversions and the stated 3-year maturity for discounting.

The one-year accumulation factor is $e^{0.08}=1.0832870677\ldots$. Therefore the annual effective rate is:

i_3=e^{0.08}-1=0.0832870677\ldots

That is 8.3287% to four decimal places. For the equivalent nominal monthly rate $j=i_3^{(12)}$, equate annual accumulation factors:

\left(1+\frac{j}{12}\right)^{12}=e^{0.08}, \qquad j=12\left(e^{0.08/12}-1\right) =0.0802672603\ldots

The nominal rate is 8.0267% convertible monthly to four decimal places. The discount factor at 3 years is:

v_3=e^{-0.08(3)}=e^{-0.24}=0.7866278611\ldots

The present value is $5{,}000(0.7866278611\ldots)=\$3{,}933.139305\ldots$, or $3,933.14 to the nearest cent. Checking with the annual effective rate, $(1.0832870677\ldots)^{-3}=e^{-0.24}$, as required.

Worked Example: Reading a Nominal Quote at Its Stated Frequency

A hypothetical 2-year zero-coupon payment of $12,000 is valued using a nominal spot rate of 10% convertible semiannually. Find the equivalent annual effective and continuous rates, then find the present value. Assume the 2-year term contains four complete half-year periods.

Method. Use the stated semiannual periodic rate, $0.10/2=0.05$, and count the four half-year periods. Do not apply 10% as though it were an annual effective rate.

The annual accumulation factor is:

\left(1+\frac{0.10}{2}\right)^2 =1.05^2 =1.1025.

Thus the equivalent annual effective rate is $1.1025-1=0.1025$, or 10.25%. The equivalent continuous rate is:

\delta_2=\ln(1.1025)=2\ln(1.05)=0.0975803283\ldots

For two years, there are $2(2)=4$ half-year periods. The discount factor is:

v_2=(1.05)^{-4} =\frac{1}{1.05^4} =\frac{1}{1.21550625} =0.8227024748\ldots

The present value is $12{,}000(0.8227024748\ldots)=\$9{,}872.429697\ldots$, or $9,872.43 to the nearest cent. The annual effective check gives $(1.1025)^{-2}=1/(1.05^4)$, the same discount factor.

Common Mistake: Treating the Rate Number as the Accumulation

In the first example, a common incorrect calculation would treat the 44% annual effective rate as a nominal rate convertible semiannually. That gives a periodic rate of $0.44/2=0.22$, and an accumulation factor of $1.22^2=1.4884$ per year. But the stated annual effective accumulation factor is only 1.44. The incorrect calculation therefore changes the financial terms rather than translating the quotation.

The correction is to equate accumulation factors: $(1+j/2)^2=1.44$, which gives $j=40\%$, not 44%. Another version of the same error is to discount continuously using $(1-\delta_t)^t$. Continuous discounting uses $e^{-\delta_t t}$; the exponential expression follows from continuous compounding and is not interchangeable with subtracting the rate from 1.

Key takeaway: Compare quotations through the maturity-specific discount factor, not by comparing the numerical rate values. Convert using equal accumulation factors, account for the nominal compounding frequency, and then apply the factor for the correct maturity.

Check Your Understanding

For each problem, treat the stated quotation as a deterministic spot rate for the stated maturity. Round dollar amounts to the nearest cent and rates to four decimal places where needed.

A hypothetical payment of $8,000 is due in 3 years. The spot rate is 3.5% annual effective. Find its discount factor and present value. (Answer: $v_3=(1.035)^{-3}=0.9019427\ldots$; present value = $7,215.54.)
A hypothetical payment of $3,000 is due in 4 years at a continuously compounded spot rate of 6%. Find the equivalent annual effective rate and present value. (Answer: annual effective rate $=e^{0.06}-1=6.1837\%$; present value $=3{,}000e^{-0.24}=\$2,359.88$.)
A hypothetical payment of $5,000 is due in 1.5 years. The spot rate is 8% nominal convertible quarterly, and the term contains six quarters. Find the discount factor and present value. (Answer: $v_{1.5}=(1.02)^{-6}=0.8879714\ldots$; present value = $4,439.86.)
Convert a 10.25% annual effective spot rate into an equivalent nominal rate convertible semiannually and an equivalent continuously compounded rate. (Answer: nominal rate $=10\%$ convertible semiannually; continuous rate $=\ln(1.1025)=0.0975803\ldots$, or 9.7580%.)
A hypothetical payment of $5,000 is due in 2 years under a 6% nominal spot rate convertible monthly. Find the equivalent annual effective rate, equivalent continuous rate, and present value. (Answer: annual effective rate $=(1.005)^{12}-1=6.1678\%$; continuous rate $=12\ln(1.005)=0.0598505\ldots$, or 5.9851%; present value $=5{,}000(1.005)^{-24}=\$4,435.93$.)

What You'll Learn