Nominally Convertible Spot Rates

Why Conversion Frequency Matters

A spot rate applies to a particular maturity. It can be expressed as an annual effective rate or as a nominal annual rate convertible a specified number of times per year. The conversion frequency matters: a nominal rate convertible quarterly and one convertible semiannually are different quoted rates even when each produces the same accumulation over the maturity.

The discount factor for a maturity records the present value of one unit payable at that maturity. Once the maturity and discount factor are fixed, a nominal spot rate can be found for any stated conversion frequency, provided the assumed compounding periods fit the maturity. The result is an equivalent rate convention, not a new discount factor.

Definition: For a maturity of \(t\) years, let \(D(t)\) be the discount factor for one unit payable at time \(t\). A nominal spot rate \(i_t^{(m)}\), convertible \(m\) times per year, is an annualized rate whose periodic rate is \(i_t^{(m)}/m\) and whose accumulation over the \(mt\) conversion periods is the reciprocal of \(D(t)\). This convention assumes \(mt\) is an integer and the same periodic rate applies throughout the term.

The subscript \(t\) identifies the maturity to which the spot rate applies. The superscript \(m\) identifies the conversion frequency. Both are needed: a quoted nominal rate without its frequency does not specify a periodic rate. The annual effective spot rate for the same maturity is a different convention, even though it must produce the same maturity-specific discount factor.

Deriving the Conversion Formula

Under the nominal-rate convention, each conversion period accumulates at the factor \(1+i_t^{(m)}/m\). There are \(mt\) periods in \(t\) years. Thus the accumulation over the full term is \((1+i_t^{(m)}/m)^{mt}\). Since the discount factor is the reciprocal of that accumulation, equating the two gives the conversion formula.

D(t)=\left(1+\frac{i_t^{(m)}}{m}\right)^{-mt}

Solving for the nominal spot rate gives

i_t^{(m)}=m\left(D(t)^{-1/(mt)}-1\right)

The annual effective spot rate \(i_t\) satisfies \(D(t)=(1+i_t)^{-t}\). Therefore, when both rates refer to the same maturity and discount factor,

1+i_t=\left(1+\frac{i_t^{(m)}}{m}\right)^m \qquad\text{and}\qquad i_t^{(m)}=m\left((1+i_t)^{1/m}-1\right)

These equations express equivalence: each quoted rate convention implies the same accumulation over one year, and therefore the same accumulation over \(t\) years when its convention is applied consistently. They do not mean that the numerical values of the rates are equal.

Condition: Use these formulas when the nominal rate is compounded at a constant rate every \(1/m\) year and the maturity contains an integer number \(mt\) of conversion periods. If the maturity does not align with the stated conversion periods, a fractional-period convention must be specified; do not apply the integer-period formula without that convention.

Worked Conversions

Worked Example: Recover a Quarterly Nominal Rate and an Annual Effective Rate

A three-year discount factor is \(D(3)=(1.015)^{-12}\). Find the equivalent nominal spot rate convertible quarterly and the equivalent annual effective spot rate. Check both against the stated discount factor.

Method. There are \(3(4)=12\) quarterly periods. Apply the nominal-rate formula with \(m=4\), then use the annual effective accumulation over one year. The common mistake to avoid is treating a quarterly periodic rate as if it were already an annual rate.

For quarterly conversion,

i_3^{(4)} =4\left(D(3)^{-1/12}-1\right) =4\left(\left((1.015)^{-12}\right)^{-1/12}-1\right) =4(1.015-1) =0.0600

The nominal quarterly-convertible rate is \(6.0000\%\). Its quarterly periodic rate is \(0.0600/4=0.0150\), or \(1.5000\%\). The discount factor is approximately \(D(3)=0.836387\), to six decimal places.

For the annual effective spot rate, the one-year accumulation is the accumulation over four quarterly periods:

1+i_3=(1.015)^4=1.061363550625, \qquad i_3=0.061363550625

Thus the annual effective rate is \(6.1364\%\), to four decimal places. Checking the full three-year accumulation gives \((1.061363550625)^{-3}=(1.015)^{-12}=D(3)\). Both rates reproduce the given discount factor.

Worked Example: Convert a Quarterly Rate to a Semiannual Quote

A two-year discount factor is \(D(2)=(1.02)^{-8}\). Find the equivalent nominal spot rate convertible semiannually and the annual effective spot rate. Explain why the semiannual nominal rate is not simply the quarterly nominal rate multiplied by a frequency ratio.

Method. First identify the accumulation represented by the discount factor. It corresponds to eight quarterly periods at \(2.0000\%\) per quarter. Two quarterly periods make one half-year, so the equivalent half-year accumulation factor is \((1.02)^2\).

The half-year periodic rate is

$$ (1.02)^2-1=1.0404-1=0.0404 $$

A semiannual nominal rate is twice its half-year periodic rate. Therefore,

i_2^{(2)}=2(0.0404)=0.0808

The equivalent semiannual-convertible nominal spot rate is \(8.0800\%\). The annual effective rate follows from four quarterly periods in one year:

$$ i_2=(1.02)^4-1=1.08243216-1=0.08243216 $$

So the annual effective rate is \(8.2432\%\), to four decimal places. The check over two years is \((1.0404)^{-4}=(1.02)^{-8}=D(2)\): there are four half-year periods, each accumulating by \(1.0404\). Multiplying a quoted nominal rate by a frequency ratio would not preserve this accumulation; equivalent periodic accumulation factors must be matched instead.

Worked Example: Compare Three Equivalent Rate Conventions

For a two-year maturity, let \(D(2)=(1.01)^{-8}\). Find the equivalent nominal rates convertible quarterly and semiannually, and the annual effective spot rate. Use the results to verify that the three quoted rates describe the same maturity-specific discount factor.

Method. The given factor represents eight quarterly periods with a \(1.0000\%\) periodic rate. The quarterly nominal rate is four times that periodic rate:

i_2^{(4)}=4(0.01)=0.0400

Thus the nominal quarterly-convertible rate is \(4.0000\%\). For a half-year, there are two quarterly periods, so the equivalent half-year accumulation factor is \((1.01)^2=1.0201\). The half-year periodic rate is \(1.0201-1=0.0201\), and

i_2^{(2)}=2(0.0201)=0.0402

The nominal semiannually convertible rate is \(4.0200\%\). For annual effective interest, four quarterly periods produce

$$ i_2=(1.01)^4-1=1.04060401-1=0.04060401 $$

The annual effective rate is \(4.0604\%\), to four decimal places. For the check, the quarterly quote gives \((1+0.04/4)^{-8}=(1.01)^{-8}\). The semiannual quote gives \((1+0.0402/2)^{-4}=(1.0201)^{-4}=(1.01)^{-8}\). The annual effective quote gives \((1.04060401)^{-2}=(1.01)^{-8}\). All three are equivalent at this maturity, even though their quoted rates differ.

Conversion Procedure and Exam Traps

A reliable conversion begins with the discount factor, not with a comparison of rate percentages. Write down the maturity, the number of conversion periods, and the periodic accumulation factor. Then annualize the periodic rate using the requested frequency. This separates the mathematical equivalence from the reporting convention.

Fix the maturity and convention.
Record \(t\) in years and the requested number \(m\) of conversions per year. Confirm that \(mt\) is an integer under the stated convention.

Find the periodic rate.
Use \(D(t)^{-1/(mt)}-1\). This is the rate per conversion period, not the nominal annual rate.

Annualize the quote.
Multiply the periodic rate by \(m\) to obtain \(i_t^{(m)}\). For an annual effective rate, compound the periodic factor through \(m\) periods and subtract one.

Check the result.
Substitute the quoted rate into \((1+i_t^{(m)}/m)^{-mt}\). It must reproduce the original \(D(t)\), apart from any rounding.

A frequent error is to call an annual effective rate a nominal rate with the same numerical value. For example, the annual effective rate in the first worked example is \(6.1364\%\), but its equivalent nominal quarterly rate is \(6.0000\%\). If \(6.1364\%\) were incorrectly treated as nominal convertible quarterly, its quarterly rate would be \(0.061364/4\), approximately \(1.5341\%\), rather than the required \(1.5000\%\). The resulting discount factor would not be \((1.015)^{-12}\).

Another error is to omit the maturity from the exponent. For a three-year maturity and quarterly conversion, the exponent is \(mt=12\), not \(m=4\). Using four periods would convert the rate as if the discount factor referred to one year. A final check using the full number of periods catches this mistake.

Key takeaway: A discount factor and maturity determine the total accumulation. A conversion frequency determines how that accumulation is reported as a nominal rate. Preserve the periodic accumulation factor when changing frequencies, and verify every quoted rate by rebuilding the same discount factor.

Check Your Understanding

For each problem, keep the maturity and conversion frequency explicit. Answers are given so you can check both the rate and the convention.

A two-year discount factor is \(D(2)=(1.0125)^{-8}\). Find the equivalent nominal spot rate convertible quarterly. (Answer: \(5.0000\%\).)
A three-year discount factor is \(D(3)=(1.02)^{-12}\). Find the equivalent annual effective spot rate. (Answer: \(8.2432\%\).)
A five-year discount factor is \(D(5)=(1.015)^{-20}\). Find the equivalent nominal spot rate convertible quarterly. (Answer: \(6.0000\%\).)
A three-year spot rate is \(7.2000\%\) nominal convertible semiannually. Find the discount factor for the three-year maturity. (Answer: \((1.036)^{-6}\), approximately \(0.8088\).)
A three-year discount factor is \(D(3)=(1.0125)^{-12}\). Find the equivalent annual effective spot rate and nominal spot rate convertible semiannually. (Answer: annual effective \(5.0945\%\); nominal semiannually convertible \(5.0313\%\).)

What You'll Learn