Annuities
3.6 Nominal Interest Rate (I/Y): Calculator Approach
A. Introduction
In this section, we learn how to calculate both nominal and periodic interest rates. Calculating the periodic interest ([asciimath]i[/asciimath]) algebraically (formula approach) is difficult as [asciimath]i[/asciimath] cannot be easily isolated in the PV or FV formula. Therefore, we only discuss the calculator approach to find the nominal interest rate ([asciimath]I//Y[/asciimath]). Once [asciimath]I//Y[/asciimath] is determined, we can easily calculate the periodic interest rate with Formula 2.1a:
[asciimath]i=(I//Y)/(C//Y)[/asciimath]
We use the time-value-of-money (TVM) worksheet in financial calculators. It’s also important to follow the cash flow sign convention (see Table 2.2.1 in Section 2.2) when inputting monetary values into the calculator. This ensures consistency and accuracy in our calculations.
Depending on the scenario, whether it involves retirement plans, loans, or savings schemes, the calculation of [asciimath]I//Y[/asciimath] may be based on the Periodic Payment (PMT) and either the Future Value (FV), the Present Value (PV) of the annuity, or a combination of both. The following examples demonstrate determining the interest rate in different scenarios.
B. Nominal and Periodic Interest Rates of Ordinary Annuities
Reza has $500,000 in his registered retirement income fund (RRIF). He wants to withdraw $4000 at the end of each month for 20 years. a) What nominal annual interest rate compounded monthly is required? b) What is the periodic interest rate per month? Give your answers in percent rounded to two decimal places.
Show/Hide Solution
Given Information
- Interest is compounded monthly so [asciimath]C//Y = 12[/asciimath]
- Payments are made at the end of each month so [asciimath]P//Y = 12[/asciimath]
[asciimath]C//Y = P//Y[/asciimath] [asciimath]=>[/asciimath] Ordinary Simple Annuity
- Annuity term: [asciimath]t = 20[/asciimath] years
- Number of payments in the term: [asciimath]N = P//Y* t = 12(20) = 240[/asciimath]
- Payments are received, so they are cash inflow: [asciimath]PMT = $4000[/asciimath]
- Present value is invested, so it’s cash outflow: [asciimath]PV = -$500,000[/asciimath]
- The balance at the end of the term will be zero: [asciimath]FV=0[/asciimath]
a) [asciimath]I//Y=?[/asciimath]
The nominal interest rate, [asciimath]I//Y[/asciimath], should be 7.41% compounded monthly.
b) [asciimath]i=?[/asciimath]
The periodic interest rate then is given by Formula 2.1a:
[asciimath]i=(I//Y)/(C//Y)[/asciimath]
[asciimath]i=(7.41%)/12[/asciimath]
[asciimath]=0.6175%[/asciimath]
[asciimath]~~0.62%[/asciimath] per month
Try an Example
Sydney currently has $12,500 in her investment account. She intends to deposit an additional $100 at the end of each month for the next 20 years. To meet her target of $48,000 at the end of this 20-year period, what effective interest rate must her investment earn?
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Given Information
- Effective interest rate is compounded annually so [asciimath]C//Y = 1[/asciimath]
- Payments are made at the end of each month so [asciimath]P//Y = 12[/asciimath]
[asciimath]C//Y != P//Y[/asciimath] [asciimath]=>[/asciimath] Ordinary General Annuity
- Annuity term: [asciimath]t = 20[/asciimath] years
- Number of payments in the term: [asciimath]N = P//Y * t = 12(20) = 240[/asciimath]
- Payments are made, so they are cash outflow: [asciimath]PMT = -$100[/asciimath]
- The initial investment is the present value and it is cash outflow: [asciimath]PV = -$12,500[/asciimath]
- The future value is received, so it is a cash inflow: [asciimath]FV = $48,000[/asciimath]
a) [asciimath]I//Y=?[/asciimath]
The effective interest rate, I/Y, is 2.00% compounded annually.
b) [asciimath]i=?[/asciimath]
Given the interest is compounded annually, the periodic interest rate is equal to the nominal interest rate.
[asciimath]i=2.00%[/asciimath]
C. Nominal and Periodic Interest Rates of Annuities Due
Mona won a lottery prize of $475,000 and invested it in a savings account. She intends to make monthly withdrawals of $3,500 at the beginning of each month for 15 years. Determine the nominal interest rate compounded monthly, needed to sustain these withdrawals. Provide your answer as a percentage rounded to two decimal places.
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Given Information
- Annuity Due: Set your calculator to BGN mode
- Interest is compounded monthly so [asciimath]C//Y = 12[/asciimath]
- Payments are withdrawn at the beginning of each month so [asciimath]P//Y = 12[/asciimath]
[asciimath]C//Y = P//Y[/asciimath] [asciimath]=>[/asciimath] Simple Annuity Due
- Annuity term: [asciimath]t = 15[/asciimath] years
- Number of payments in the term: [asciimath]N = P//Y* t = 12(15) = 180[/asciimath]
- Payments are received, so they are cash inflow: [asciimath]PMT = $3,500[/asciimath]
- Present value is invested in a bank, so it is a cash outflow: [asciimath]PV = -$475,000[/asciimath]
- The balance at the end of the term will be zero: [asciimath]FV=0[/asciimath]
[asciimath]I//Y=?[/asciimath]
The nominal interest rate, I/Y, should be 3.99% compounded monthly.
Try an Example
Sandeep leases a new car by making a $5600 down payment and payments of $520 at the beginning of every month for three years. The lease amount (list price) of the car is $51,000, and the residual value of the car at the end of the lease term will be $32,000. a) What nominal annual interest rate compounded quarterly is he being charged? b) What is the periodic interest rate per quarter? Give your answers in percent rounded to two decimal places.
Show/Hide Solution
Given Information
- Interest is compounded quarterly so [asciimath]C//Y = 4[/asciimath]
- Payments are made at the beginning of every month so [asciimath]P//Y = 12[/asciimath]
[asciimath]C//Y != P//Y[/asciimath] [asciimath]=>[/asciimath] General Annuity Due
- Lease term: [asciimath]t = 3[/asciimath] years
- Number of payments in the term: [asciimath]N = P//Y* t = 12(3) = 36[/asciimath]
- Payments are made, so they are cash outflow: [asciimath]PMT = -$520[/asciimath]
- List price [asciimath]=$51,000[/asciimath]
- Down payment [asciimath]= $5600[/asciimath]
- Residual value (FV) is a cash outflow [asciimath]= -$21,450[/asciimath]
a) [asciimath]I//Y=?[/asciimath]
First, we need to find the amount of PV (the loan on the lease), which is the remaining balance after paying the down payment.
[asciimath]"Lease Amount" = "Down payment" + PV[/asciimath]
Rearranging the equation for [asciimath]PV[/asciimath]gives
[asciimath]PV = "Lease Amount"-"Down payment"[/asciimath]
[asciimath]PV=51,000-5600[/asciimath]
[asciimath]=$45,400[/asciimath]
Next, we compute the nominal interest rate using the above PV, residual as FV, PMT, and N. We use the calculator approach. Note that PV is entered as a positive value since it is considered a cash inflow (Sandeep receives the lease loan), and FV is entered as a negative value since Sandeep must either return the car (equivalent to the residual value) or pay the residual value to own the car (if such an option is offered) at the end of the lease term (cash outflow). The periodic payments are also cash outflow.
The nominal interest rate, I/Y, being charged is 4.62% compounded quarterly.
b) [asciimath]i=?[/asciimath]
The periodic interest rate is then given by Formula 2.1a.
[asciimath]i=(I//Y)/(C//Y)[/asciimath]
[asciimath]i=(4.62%)/4[/asciimath]
[asciimath]=1.155%[/asciimath]
[asciimath]~~1.16%[/asciimath] per quarter
Try an Example
Section 3.6 Exercises
- Blake has saved $79,500 in his registered retirement income fund (RRIF). He plans to make withdrawals of $4,222.32 every six months for a period of 12 years. a) Determine the nominal annual interest rate required for his plan, assuming it is compounded semi-annually. b) Calculate the periodic interest rate for each six-month interval. Give your answers in percent rounded to two decimal places.
Show/Hide Answer
a) I/Y = 4.08% compounded semi-annually
b) i = 2.04% per six months
- Bisa won a lottery prize of $29,000 and invested it in a savings account. She intends to make monthly withdrawals of $1,597.64 at the beginning of each quarter for 5 years. What nominal annual interest rate compounded quarterly is required to sustain these withdrawals? Give your answer in percent rounded to two decimal places.
Show/Hide Answer
I/Y = 4.18% compounded quarterly
- Tom leases a new car by making a $2,444 down payment and payments of $452 at the beginning of every month for 6 years. The lease amount (list price) of the car is $48,870, and the residual value of the car at the end of the lease term will be $20,525. What nominal annual interest rate compounded semi-annually is Tom being charged? Give your answer in percent rounded to two decimal places.
Show/Hide Answer
I/Y = 3.31% compounded semi-annually
