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Mathematics LibreTexts

9.5: Determining the Interest Rate

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    22121
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    Whether you are borrowing or investing, it is extremely important to know what compound interest rate you are being charged or earning. Unfortunately, most consumers live for today, paying attention just to the “don’t pay for one year” clause in the deals they are offered. They do not stop to consider they might be paying too much for credit.

    For example, assume you are just about to sign the purchase papers for a new 55” LED HD 3D television. The retail sales clerk turns to you, saying, “Well, I’ve run the numbers and they show you could pay $4,000 today or if you take advantage of our ‘don’t pay for one year’ offer, you will owe us $4,925.76 one year from now.” You urgently want to get your new TV home, but you had better think twice about a financial decision this important. What interest rate did the sales clerk use in determining your $4,925.76 payment? Is this interest rate fair? Could you finance the TV for less elsewhere?

    This section shows how to calculate the nominal interest rate on single payments when you know both the future value and the present value.

    The Nominal Interest Rate

    You need to calculate the nominal interest rate under many circumstances including (but not limited to) the following:

    • Determining the interest rate on a single payment loan
    • Understanding what interest rate is needed to achieve a future savings goal
    • Calculating the interest rate that generated a specific amount of interest
    • Finding a fixed interest rate that is equivalent to a variable interest rate

    The Formula

    Calculating the nominal interest rate requires you to use Formula 9.3 once again. The only difference is that the unknown variable has changed from \(FV\) to \(IY\). Note that \(IY\) is not directly part of Formula 9.3, but you are able to calculate its value after determining the periodic interest rate, or \(i\).

    clipboard_ee7a8c52fdd07da7d2312643cd4644eee.png

    How It Works

    Follow these steps to solve for the nominal interest rate on a single payment:

    Step 1: Draw a timeline to help you visualize the question. Of utmost importance is identifying the values of \(PV\) and \(FV\), the number of years involved, and the compounding for the interest rate.

    Step 2: Calculate the number of compounds, \(N\), using Formula 9.2.

    Step 3: Substitute known variables into Formula 9.3, rearrange and solve for the periodic interest rate, \(i\).

    Step 4: Substitute the periodic interest rate and the compounding frequency into Formula 9.1, rearrange, and solve for the nominal interest rate, \(IY\). Ensure that the solution is expressed with the appropriate compounding words.

    Revisiting the section opener, your $4,000 TV cost $4,925.76 one year later. What monthly compounded interest rate was being used?

    Step 1: The timeline below illustrates the scenario and identifies the values.

    clipboard_e1ac653a8e46902787b2cc23f81cfad9f.png

    The compounding is monthly, making \(CY=12\).

    Step 2: The term is one year, so \(N=1 \times 12=12\).

    Step 3: Substituting into Formula 9.3, \(\$ 4,925.76=\$ 4,000(1+i)^{12}\). Rearranging and solving for \(i\) results in \(i=0.0175\).

    Step 4: Substituting into Formula 9.1, \(0.0175=\dfrac{IY}{12}\). Rearranging and solving for \(IY\) calculates \(IY=0.21\), or 21%. Thus, the sales clerk used an interest rate of 21% compounded monthly.

    Important Notes

    Handling Decimals in Interest Rate Calculations

    When you calculate interest rates, the solution rarely works out to a terminating decimal number. Since most advertised or posted interest rates commonly involve no more than a few decimals, why is this the case? Recall that when an interest dollar amount is calculated, in most circumstances this amount must be rounded off to two decimals. In single payments, a future value is always the present value plus the rounded interest amount. This results in a future value that is an imprecise number that may be up to a half penny away from its true value. When this imprecise number is used for calculating any interest rate, the result is that nonterminating decimals show up in the solutions. To express the final solution for these nonterminating decimals, you need to apply two rounding rules:

    • Rule 1: A Clear Marginal Effect Use this rule when it is fairly obvious how to round the interest rate. The dollar amounts used in calculating the interest rate are rounded by no more than a half penny. Therefore, the calculated interest rate should be extremely close to its true value. For example, if you calculate an \(IY\) of 7.999884%, notice this value would have a marginal difference of only 0.000116% from a rounded value of 8%. Most likely the correct rate is 8% and not 7.9999%. However, if you calculate an \(IY\) of 7.920094%, rounding to 8% would produce a difference of 0.070006%, which is quite substantial. Applying marginal rounding, the most likely correct rate is 7.92% and not 7.9201%, since the marginal impact of the rounding is only 0.000094%.
    • Rule 2: An Unclear Marginal Effect Use this rule when it is not fairly obvious how to round the interest rate. For example, if the calculated \(IY\) = 7.924863%, there is no clear choice of how to round the rate. In these cases or when in doubt, apply the standard rule established for this book of rounding to four decimals. Hence, \(IY\) = 7.9249% in this example.

    It is important to stress that the above recommendations for rounding apply to final solutions. If the calculated interest rate is to be used in further calculations, then you should carry forward the unrounded interest rate.

    Your BAII Plus Calculator

    Solving for the nominal interest rate requires the computation of \(I/Y\) on the BAII Plus calculator. This requires you to enter all six of the other variables, including \(N\), \(PV\), \(PMT\) (which is zero), \(FV\), and both values in the \(P/Y\) window (\(P/Y\) and \(C/Y\)) following the procedures established in Section 9.2. Ensure proper application of the cash flow sign convention to \(PV\) and \(FV\), according to which one number must be negative while the other is positive.

    Things To Watch Out For

    Pay careful attention to what the situation requires you to calculate—the nominal interest rate or the periodic interest rate. The most common mistake when calculating the interest rate is to confuse these two rates. If you need the periodic interest rate, then you must rearrange and solve Formula 9.3 for \(i\), and no further calculations are required after step 3. If you need the nominal interest rate, first calculate the periodic interest rate (\(i\)) but then substitute it into Formula 9.1 and rearrange to solve for \(IY\), thus completing step 4 of the process.

    Exercise \(\PageIndex{1}\): Give It Some Thought

    Round off the following calculated values of \(IY\) to the appropriate decimals.

    1. 4.5679998%
    2. 12.000138%
    3. 6.8499984%
    4. 8.0200121%
    5. 7.1224998%
    Answer
    1. 4.568%
    2. 12%
    3. 6.85%
    4. 8.02%
    5. 7.1225%

    When Sandra borrowed $7,100 from Sanchez, she agreed to reimburse him $8,615.19 three years from now including interest compounded quarterly. What interest rate is being charged?

    Solution

    Find the nominal quarterly compounded rate of interest (\(IY\)).

    What You Already Know

    Step 1:

    The present value, future value, term, and compounding are known, as illustrated in the timeline.

    clipboard_eff20402325fbe092aa57cc5c3b72cfa3.png

    \(CY\) = quarterly = 4 Term = 3 years

    How You Will Get There

    Step 2:

    Calculate \(N\) using Formula 9.2.

    Step 3:

    Substitute into Formula 9.3 and rearrange for \(i\).

    Step 4:

    Substitute into Formula 9.1 and rearrange for \(IY\).

    Perform

    Step 2:

    \(N = 4 × 3 = 12\)

    Step 3:

    \[\begin{aligned} \$ 8,615.10 &=\$ 7,100(1+i)^{12} \\ 1.213394 &=(1+i)^{12} \\ 1.213394 ^{\frac{1}{12}} &=1+i \\ 1.016249 &=1+i \\ 0.016249 &=i \end{aligned} \nonumber \]

    Step 4:

    \[\begin{aligned}
    0.016249 &=\dfrac{IY}{4} \\
    IY &=0.064996=0.065 \text { or } 6.5 \%
    \end{aligned} \nonumber \]

    Calculator Instructions

    N I/Y PV PMT FV P/Y C/Y
    12 Answer: 6.499709 -7100 0 8615.10 4 4

    Sanchez is charging an interest rate of 6.5% compounded quarterly on the loan to Sandra.

    Paths To Success

    When a series of calculations involving the nominal interest rate must be performed, many people find it helpful first to rearrange Formula 9.3 algebraically for \(i\), thus bypassing a long series of manipulations. The rearranged formula appears as follows:

    \[i=\left[\left(\dfrac{FV}{PV}\right)^{\frac{1}{N}}-1\right]\nonumber\]

    This rearrangement calculates the periodic interest rate. If the nominal interest rate is required, you can combine Formula 9.3 and Formula 9.1 together:

    \[IY=\left[\left(\dfrac{FV}{PV}\right)^{\frac{1}{N}}-1\right] \times CY\nonumber\]

    Example \(\PageIndex{2}\): Known Interest Amount

    Five years ago, Taryn placed $15,000 into an RRSP that earned $6,799.42 of interest compounded monthly. What was the nominal interest rate for the investment?

    Solution

    Find the nominal monthly compounded rate of interest (\(IY\)).

    What You Already Know

    Step 1:

    The present value, interest earned, term, and compounding are known, as illustrated in the timeline.

    clipboard_ec4c6fdc24c2f9bb6e9dc910afdab57c0.png

    Use Formula 9.3 to arrive at the \(FV\) in the figure. \(CY\) = monthly = 12 Term = 5 years

    How You Will Get There

    Step 2:

    Calculate \(N\) using Formula 9.2.

    Step 3:

    Substitute into Formula 9.3 and rearrange for \(i\).

    Step 4:

    Substitute into Formula 9.1 and rearrange for \(IY\).

    Perform

    Step 2:

    \(N = 12 × 5 = 60\)

    Step 3:

    \[\begin{aligned} \$ 21,799.42 &=\$ 15,000(1+i)^{60} \\ 1.453294 &=(1+i)^{60} \\ 1.453294^{\frac{1}{60}} &=1+i \\ 1.00625 &=1+i \\ 0.00625 &=i \end{aligned} \nonumber \]

    Step 4:

    \[\begin{aligned}
    0.00625 &=\dfrac{IY}{12} \\
    IY &=0.075 \text { or } 7.5 \%
    \end{aligned} \nonumber \]

    Calculator Instructions

    N I/Y PV PMT FV P/Y C/Y
    60 Answer: 7.500003 -15000 0 21799.42 12 12

    Taryn’s investment in his RRSP earned 7.5% compounded monthly over the five years.

    Converting Variable Interest Rates to a Fixed Interest Rate

    When you deal with a series of variable interest rates it is extremely difficult to determine their overall effect. This also makes it hard to choose wisely between different series. For example, assume that you could place your money into an investment earning interest rates of 2%, 2.5%, 3%, 3.5%, and 4.5% over the course of five years, or alternatively you could invest in a plan earning 1%, 1.5%, 1.75%, 3.5%, and 7% (all rates compounded semi-annually). Which plan is better? The decision is unclear. But you can make it clear by converting the variable rates on each investment option into an equivalent fixed interest rate.

    How It Works

    Follow these steps to convert variable interest rates to their equivalent fixed interest rates:

    Step 1: Draw a timeline for the variable interest rate. Identify key elements including any known \(PV\) or \(FV\), interest rates, compounding, and terms.

    Step 2: For each time segment, calculate \(i\) and \(N\) using Formula 9.1 and Formula 9.2, respectively.

    Step 3: One of three situations will occur, depending on what variables are known:

    1. \(PV\) Is Known Calculate the future value at the end of the transaction using Formula 9.3 and solving for \(FV\) in each time segment, working left to right across the timeline.
    2. \(FV\) Is Known Calculate the present value at the beginning of the transaction using Formula 9.3 and rearranging to solve for \(PV\) in each time segment, working right to left across the timeline.
    3. Neither \(PV\) nor \(FV\) Is Known Pick an arbitrary number for \(PV\) ($10,000 is recommended) and use Formula 9.3 in each time segment to solve for the future value at the end of the transaction, working left to right across the timeline.

    Step 4: Determine the compounding required on the fixed interest rate (\(CY\)) and use Formula 9.2 to calculate a new value for \(N\) to reflect the entire term of the transaction.

    Step 5: Rearrange and solve Formula 9.3 for i using the \(N\) from step 4 along with the starting \(PV\) and ending \(FV\) for the entire timeline.

    Step 6: Rearrange and solve Formula 9.1 for \(IY\).

    Example \(\PageIndex{3}\): Interest Rate under Variable Rate Conditions

    Continue working with the two investment options mentioned previously. The choices are to place your money into a fiveyear investment earning semi-annually compounded interest rates of either:

    1. 2%, 2.5%, 3%, 3.5%, and 4.5%
    2. %, 1.5%, 1.75%, 3.5%, and 7%

    Calculate the equivalent semi-annual fixed interest rate for each plan and recommend an investment.

    Solution

    What You Already Know

    Step 1:

    Draw a timeline for each investment option, as illustrated below.

    clipboard_e573d982269871b5b5314b18a96bfbdef.png

    How You Will Get There

    Step 2:

    For each time segment, calculate the \(i\) and \(N\) using Formula 9.1 and Formula 9.2. All interest rates are semi-annually compounded with \(CY\) = 2.

    Step 3:

    There is no value for \(PV\) or \(FV\). Choose an arbitrary value of \(PV\) = $10,000 and solve for \(FV\) using Formula 9.3. Since only the interest rate fluctuates, solve in one calculation:

    \[FV_{5}=PV \times\left(1+i_{1}\right)^{N_{1}} \times\left(1+i_{2}\right)^{N_{2}} \times \ldots \times\left(1+i_{5}\right)^{N_{5}} \nonumber \]

    Step 4:

    For each investment, calculate a new value of \(N\) to reflect the entire five-year term using Formula 9.2.

    Step 5:

    For each investment, substitute into Formula 9.3 and rearrange for \(i\)

    Step 6:

    For each investment, substitute into Formula 9.1 and rearrange for \(IY\).

    Perform

    Step 2:

    All periodic interest rate (\(i\)) and \(N\) calculations are found in the timeline figure above.

    Step 3:

    First Investment: \(FV_{5}=\$ 10,000(1+0.01)^{2}(1+0.0125)^{2}(1+0.015)^{2}(1+0.0175)^{2}(1+0.0225)^{2}=\$ 11,661.65972 \)

    Second Investment: \(FV_{5}=\$10,000(1+0.005)^{2}(1+0.0075)^{2}(1+0.00875)^{2}(1+0.0175)^{2}(1+0.035)^{2} = \$ 11,570.14666\)

    Step 4:

    First Investment: \(N = 2\times 5=10 \)

    Second Investment: \(N = 2\times 5=10 \)

    Step 5:

    First Investment:

    \[\begin{aligned}
    &\$ 11,661.65972=\$ 10,000(1+i)^{10}\\
    &1.166165=(1+i)^{10}\\
    &1.166165^{\frac{1}{10}}=1+i\\
    &1.001549=1+i\\
    &0.001549=i
    \end{aligned} \nonumber \]

    Second Investment:

    \[\begin{aligned}
    &\$ 11,570.14666=\$ 10,000(1+i)^{10}\\
    &1.157014=(1+i)^{10}\\
    &1.157015^{\frac{1}{10}}=1+i\\
    &1.014691=1+i\\
    &0.014691=i
    \end{aligned} \nonumber \]

    Step 6:

    First Investment:

    \[\begin{aligned}
    &0.001549=\dfrac{IY}{2}\\
    &IY =0.030982 \text { or } 3.0982 \%
    \end{aligned} \nonumber \]

    Second Investment:

    \[\begin{aligned}
    &0.014691=\dfrac{IY}{2}\\
    &\text { IY }=0.029382 \text { or } 2.9382 \%
    \end{aligned} \nonumber \]

    Calculator Instructions

    Investment #1

    Time segment N I/Y PV PMT FV P/Y C/Y
    1 2 2 -10000 0 Answer: 10,201 2 2
    2 \(\surd\) 2.5 -10201 \(\surd\) Answer: 10,457.61891 \(\surd\) \(\surd\)
    3 \(\surd\) 3 -10457.61891 \(\surd\) Answer: 10,773.70044 \(\surd\) \(\surd\)
    4 \(\surd\) 3.5 -10773.70044 \(\surd\) Answer: 11,154.0794 \(\surd\) \(\surd\)
    5 \(\surd\) 4.5 -11154.0794 \(\surd\) Answer: 11661.65972 \(\surd\) \(\surd\)
    All 10 Answer: 0.030982 -10000 \(\surd\) 11661.66 \(\surd\) \(\surd\)

    Investment #2

    Time segment N I/Y PV PMT FV P/Y C/Y
    1 60 1 -10000 0 Answer: 10,100.25 2 2
    2 \(\surd\) 1.5 -10100.25 \(\surd\) Answer: 10,252.32189 \(\surd\) \(\surd\)
    3 \(\surd\) 1.75 -10252.32189 \(\surd\) Answer: 10,432.52247 \(\surd\) \(\surd\)
    4 \(\surd\) 3.5 -10432.52247 \(\surd\) Answer: 10,800.85571 \(\surd\) \(\surd\)
    5 \(\surd\) 7 -10800.85571 \(\surd\) Answer: 11,570.14666 \(\surd\) \(\surd\)
    All 10 Answer: 0.029382 -10000 \(\surd\) 11570.15 \(\surd\) \(\surd\)

    The variable interest rates on the first investment option are equivalent to a fixed interest rate of 3.0982% compounded semi-annually. For the second option, the rates are equivalent to 2.9382% compounded semi-annually. Therefore, recommend the first investment since its rate is higher by 3.0982% − 2.9382% = 0.16% compounded semi-annually.

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