Converting Rates for Refinancing Calculations

Why Rate Conversion Matters in Refinancing

A refinancing calculation may involve an old loan quoted at a nominal annual rate compounded monthly and a replacement loan quoted at an effective annual rate, while payments on the replacement loan are quarterly. Those quoted rates do not apply directly to the same time interval. Before using a rate in a loan calculation, convert it to the effective rate for the interval between the relevant payments.

The central discipline is to match the rate period to the cash-flow period. A monthly payment calculation needs a monthly effective rate; a quarterly payment calculation needs a quarterly effective rate. The annual labels on two quotes do not make their quoted rates directly comparable.

Definition: An effective rate for a specified period is the proportion by which an amount grows over that period, after compounding within the period. The effective annual rate is \(i\), so an amount \(1\) accumulates to \(1+i\) in one year. A nominal annual rate \(i^{(m)}\), convertible \(m\) times per year, means a rate of \(i^{(m)}/m\) for each conversion period, with annual accumulation \(\left(1+i^{(m)}/m\right)^m\).

Convert the Quote Before Using It

Assume a quoted rate describes a constant accumulation at the stated compounding frequency, and that the loan uses the same rate convention throughout the period being modeled. If the nominal annual rate is \(i^{(m)}\), the effective annual rate equivalent to that quote is

i=\left(1+\frac{i^{(m)}}{m}\right)^m-1.

If payments occur \(p\) times per year, let \(j_p\) denote the effective rate per payment period. Equivalence requires that accumulating through \(p\) payment periods produces the same annual accumulation as \(i\). Thus \((1+j_p)^p=1+i\), giving

j_p=(1+i)^{1/p}-1.

Combining these expressions gives a direct conversion from a nominal rate to an equivalent payment-period rate:

j_p=\left(1+\frac{i^{(m)}}{m}\right)^{m/p}-1.

This conversion is an equivalence of annual accumulations. It does not mean that the nominal rate itself is the effective rate per payment period. If the quote and payment frequencies differ, the period rate will generally not be \(i^{(m)}/p\).

Result: For every rate used in a refinancing calculation, identify the payment interval first. Convert each quoted rate to the effective rate over that interval, using the quote's compounding convention. Do not apply an annual rate directly to a monthly or quarterly payment interval.

Why the Conversion Formula Holds

The conversion follows from matching the accumulation over a year, not from matching rate labels. Under a nominal annual rate \(i^{(m)}\), one conversion period has rate \(i^{(m)}/m\). Applying that rate \(m\) times turns \(1\) into \(\left(1+i^{(m)}/m\right)^m\). The corresponding effective annual rate is therefore that accumulation minus \(1\).

Now suppose the payment interval is one of \(p\) equal intervals in a year. If the effective rate per interval is \(j_p\), then \(p\) such intervals turn \(1\) into \((1+j_p)^p\). To make the periodic rate equivalent to the annual quote, set the two annual accumulations equal:

(1+j_p)^p=1+i \quad\Longrightarrow\quad j_p=(1+i)^{1/p}-1.

When the original quote is nominal, substitute its effective annual accumulation to obtain \(j_p=\left(1+i^{(m)}/m\right)^{m/p}-1\). The derivation assumes a constant rate and equivalent annual accumulation. If a loan instead specifies different rates at different times, a single annual-to-period conversion does not describe that changing rate pattern.

Worked Refinancing Conversions

Worked Example: Monthly Payments With Different Rate Quotes

A borrower is considering replacing a loan that charges a nominal annual rate of \(9.6\%\), compounded monthly. The replacement loan quotes an effective annual rate of \(11.2\%\), and both loans have monthly payments. Find the effective monthly rate for each loan and the effective annual rate equivalent to the old loan quote. Use rates to four decimal places as percentages.

Method. Convert each quote to an effective rate for the payment interval. The old loan's nominal monthly quote directly determines its monthly rate; the replacement's effective annual quote must be converted to a monthly equivalent.

Old loan, monthly rate. Since the nominal rate is convertible monthly, divide by 12:

j_{12,\text{old}}=\frac{0.096}{12}=0.008=0.8000\%.

Old loan, effective annual rate. Compound the monthly rate for 12 months:

i_{\text{old}}=(1.008)^{12}-1 \approx 0.1003387=10.0339\%.

Replacement loan, monthly rate. Convert its effective annual accumulation to an equivalent monthly accumulation:

j_{12,\text{new}}=(1.112)^{1/12}-1 \approx 0.008885931=0.8886\%.

The old and replacement monthly rates are therefore \(0.8000\%\) and \(0.8886\%\), respectively. The old quote's nominal annual rate of \(9.6\%\) is not its effective annual rate: its effective annual rate is \(10.0339\%\). Common mistake avoided: using \(11.2\%/12\) for the replacement loan. The \(11.2\%\) is effective annually, not a nominal rate convertible monthly, so it must be converted by taking a twelfth root.

Check. The replacement monthly rate gives \((1+0.008885931)^{12}\approx 1.112\), as required by its effective annual quote. The old monthly rate gives \((1.008)^{12}\approx 1.1003387\), matching the old loan's effective annual accumulation.

Worked Example: Quarterly Payments and Different Compounding Frequencies

An existing loan has quarterly payments and a nominal annual rate of \(8.4\%\), compounded monthly. A proposed replacement loan also has quarterly payments but quotes a nominal annual rate of \(8.0\%\), compounded semiannually. Find the effective quarterly rate under each quote and the effective annual rate for each loan. Give rates to four decimal places as percentages.

Method. Convert each loan's quote to an effective annual rate, then take a fourth root to get the rate per quarterly payment period. Equivalently, convert the compounding-period accumulation directly to three months.

Existing loan. The monthly rate is \(0.084/12=0.007\). Its annual accumulation and effective annual rate are

(1.007)^{12}\approx 1.0873107, \qquad i_{\text{old}}\approx 0.0873107=8.7311\%.

A quarter contains three months, so its effective rate is

j_{4,\text{old}}=(1.007)^3-1 =1.021147343-1 =0.021147343 \approx 2.1147\%.

Replacement loan. The semiannual rate is \(0.080/2=0.04\). Two semiannual periods make a year, so

i_{\text{new}}=(1.04)^2-1=0.0816=8.1600\%.

A quarter is half of a semiannual period. The equivalent quarterly rate is

j_{4,\text{new}}=(1.04)^{1/2}-1 \approx 0.019803903 =1.9804\%.

The quarterly rates are \(2.1147\%\) for the existing loan and \(1.9804\%\) for the replacement loan. Common mistake avoided: dividing each nominal annual rate by four just because payments are quarterly. The nominal conversion frequencies are monthly and semiannual, not quarterly, so neither quoted rate can be divided by four to obtain its quarterly equivalent.

Check. Four quarters at the existing quarterly rate accumulate by \((1.021147343)^4=(1.007)^{12}\), and four quarters at the replacement rate accumulate by \((1.019803903)^4=(1.04)^2\). Each agrees with its respective effective annual accumulation.

Worked Example: Payments Every Two Months

A replacement loan makes payments every two months. Its rate is quoted as a nominal annual rate of \(9.0\%\), compounded quarterly. Find the effective rate per payment period and verify that six payment periods reproduce the annual accumulation. Give the periodic rate to four decimal places as a percentage.

Method. First find the quarterly compounding rate, then convert its annual accumulation to six equal payment periods per year.

The quarterly rate is \(0.09/4=0.0225\), so the effective annual accumulation is \((1.0225)^4\). Since payments occur six times per year, the effective rate per two-month period is

j_6=(1.0225)^{4/6}-1 =(1.0225)^{2/3}-1 \approx 0.014944302 =1.4944\%.

The annual effective rate is

i=(1.0225)^4-1 \approx 0.0930831=9.3083\%.

Check. Six payment periods reproduce the annual accumulation: \((1+j_6)^6=(1.0225)^4\). Common mistake avoided: using \(0.0225\) as the two-month payment rate. That is the rate for a three-month quarter, not a two-month interval.

Other Rate Conventions

The same accumulation-equivalence approach applies when a loan's rate is given as a force of interest or an effective discount rate. These conventions must first be identified correctly; a force of interest and an effective discount rate are not interchangeable with an effective annual interest rate.

Conversion results: If \(\delta\) is a constant force of interest, the effective annual rate is \(i=e^\delta-1\), and the effective rate for a period of \(1/p\) year is \(e^{\delta/p}-1\). If \(d\) is an effective annual discount rate, then \(v=1-d=1/(1+i)\), so \(i=d/(1-d)\). Its effective rate over \(1/p\) year is \((1-d)^{-1/p}-1\).

These results assume the stated force or discount rate is constant over the relevant periods. Once an effective rate per payment period has been found, it can be used with the corresponding payment schedule. Do not combine one loan's monthly rate with another loan's annual rate in a calculation unless both rates have first been converted to a common time basis.

Common Rate-Conversion Traps

Dividing every annual quote by the number of payments. Direct division gives a periodic rate only when the quote is nominal and its conversion frequency matches the payment frequency. Otherwise, use accumulation equivalence.
Treating nominal and effective annual rates as the same. A nominal rate \(i^{(m)}\) has annual accumulation \(\left(1+i^{(m)}/m\right)^m\), not \(1+i^{(m)}\).
Using a rate for the wrong interval. A quarterly rate applies over three months. It is not automatically the rate for a two-month or monthly payment period.
Rounding too early. Keep full calculator precision through the conversion and round only the reported rate. Small rounding differences can otherwise affect later loan calculations.
Comparing rates without first matching periods. For a meaningful comparison, convert both rates to the same effective period, such as the interval between payments.

For refinancing work, write down the quote convention, compounding frequency, and payment frequency before calculating. Then convert the quoted rates and check that compounding the periodic rate over a year reproduces the quoted annual accumulation. This check catches many unit and convention errors before they enter a longer calculation.

Check Your Understanding

Convert each quote as requested. Retain unrounded values during the calculations, and round the final rates to four decimal places as percentages.

A loan quotes a nominal annual rate of \(6.6\%\), compounded monthly. Find its effective monthly rate and effective annual rate. (Answer: Monthly \(0.5500\%\); effective annual \(6.8034\%\).)
A replacement loan quotes an effective annual rate of \(9.5\%\) and has quarterly payments. Find the effective quarterly rate and verify the conversion using the annual accumulation. (Answer: \(2.2947\%\) per quarter; \((1.022947)^4\approx1.095\).)
A loan quotes a nominal annual rate of \(10.0\%\), compounded semiannually, and makes monthly payments. Find the equivalent effective monthly rate. (Answer: \((1.05)^{1/6}-1\approx0.8165\%\) per month.)
An existing loan quotes \(7.2\%\) nominal compounded quarterly; a proposed replacement quotes \(7.1\%\) nominal compounded monthly. Both have quarterly payments. Convert both quotes to effective quarterly rates and identify which rate is lower. (Answer: Existing \(1.8000\%\); replacement \((1+0.071/12)^3-1\approx1.7855\%\); the replacement rate is lower.)
A loan has a constant force of interest of \(6.0\%\) per year and semiannual payments. Find the effective rate per payment period and the effective annual rate. (Answer: Semiannual \(e^{0.03}-1\approx3.0455\%\); effective annual \(e^{0.06}-1\approx6.1837\%\).)

What You'll Learn