Annuities

3.9 PMT of Annuities: Formula Approach

A. Payment of Ordinary Simple Annuity

In this section, we focus on calculating the periodic payment amount (PMT) required for an annuity. This payment is the amount that is either paid or received regularly throughout the term of the annuity. Depending on the scenario, whether it involves retirement plans, loans, or savings schemes, the calculation of PMT may be based on the Future Value (FV), the Present Value (PV) of the annuity, or a combination of both.

If we know the future value of an annuity, the number of payment periods, and the periodic interest rate, we can apply the future value of the ordinary simple annuity formula (Formula 3.5a) to determine the unknown periodic payment (PMT).

  [asciimath]PMT=(FV*i)/((1+i)^N-1)[/asciimath]Formula 3.5b

Likewise, when the present value of an annuity, the number of payment periods, and the periodic interest rate are known, the present value formula (Formula 3.9a) can be rearranged to solve for the unknown periodic payment PMT.

 [asciimath]PMT=(PV*i)/(1-(1+i)^(-N))[/asciimath]Formula 3.9b

 

Example 3.9.1: Compute PMT Given FV of Ordinary Simple Annuity

What deposit made at the end of each month will accumulate to $12,000 in 6 years at 4.8% p.a. compounded monthly?

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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 = 6[/asciimath] years
  • Number of payments in the term: [asciimath]N = P//Y * t = 12(6) = 72[/asciimath]
  • Nominal interest rate: [asciimath]I//Y = 4.8%[/asciimath]
  • Periodic interest rate: [asciimath]i = (I//Y)/(C//Y) = (4.8%)/12 =0.4%[/asciimath]
  • The initial balance of the account is zero, so [asciimath]PV = 0[/asciimath]
  • Future value of investments: [asciimath]FV = $12,000[/asciimath]

[asciimath]PMT=?[/asciimath]

Given the future value is known, we use Formula 3.5b to calculate the size of the payments:

  [asciimath]PMT=(FV* i)/((1+i)^N-1)[/asciimath]

  [asciimath]PMT=(12,000(0.004))/((1+0.004)^72-1)[/asciimath]

[asciimath]=144.147...[/asciimath]

[asciimath]~~$144.15[/asciimath]

The size of the deposits made must be $144.15.

 

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Example 3.9.2: Compute PMT Given PV of Ordinary Simple Annuity

What payment is required at the end of each month for 8 years to repay a $32,000 loan if the interest charged is 3.8% compounded monthly?

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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 = 8[/asciimath]   years
  • Number of payments in the term: [asciimath]N = P//Y * t = 12(8) = 96[/asciimath]
  • Nominal interest rate: [asciimath]I//Y = 3.8%[/asciimath]
  • Periodic interest rate: [asciimath]i = (I//Y)/(C//Y) = (3.8%)/12 =0.31bar(6)%[/asciimath]
  • Loan principal: [asciimath]PV = $32,000[/asciimath]
  • No remaining balance at the end of the annuity term, so [asciimath]FV=0[/asciimath]

[asciimath]PMT=?[/asciimath]

Considering the present value is known, we use Formula 3.9b to find the size of the payments:

 [asciimath]PMT=(PV* i)/(1-(1+i)^(-N))[/asciimath]

 [asciimath]PMT=(32,000(0.0031bar(6) ))/(1-(1+0.0031bar(6) )^(-96))[/asciimath]

[asciimath]=387.086...[/asciimath]

[asciimath]~~$387.09[/asciimath]

Thus, the payments of $387.09 are required to fully repay the loan.

 

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B. Payment of Ordinary General Annuity

As we did with the ordinary simple annuity, if the future value of an annuity, the number of payment periods, and the periodic interest rate are known, we can use the future value formula of an ordinary general annuity (Formula 3.6a) to calculate the unknown periodic payment, PMT.

[asciimath]PMT=(FV.i_2)/((1+i_2)^N-1)[/asciimath]Formula 3.6b

Similarly, if we have information about the present value of an annuity, along with the number of payment periods and the periodic interest rate, we can rearrange the present value formula (Formula 3.10a) to find the unknown periodic payment PMT.

[asciimath]PMT=(PV* i_2)/(1-(1+i_2)^(-N))[/asciimath]Formula 3.10b

Here [asciimath]i_2[/asciimath] is the interest rate per payment period

[asciimath]i_2=(1+i)^((C//Y)/(P//Y))-1[/asciimath]

 

Example 3.9.3: Compute PMT Given FV of Ordinary General Annuity
Dakota is planning for a renovation project that will cost $95,000. To finance this, she intends to save money in her savings account earning an interest rate of 4.12% compounded quarterly. a) Determine the amount Dakota needs to deposit at the end of each month to reach her goal of $95,000 in 5 years. b) Calculate the total amount Dakota will have deposited into the account over this 5-year period. c) Determine the total interest Dakota will earn on her savings account.
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Given information

  • Interest is compounded quarterly so [asciimath]C//Y = 4[/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 = 5[/asciimath] years
  • Number of payments in the term: [asciimath]N = P//Y * t = 12(5) = 60[/asciimath]
  • Nominal interest rate: [asciimath]I//Y = 4.12%[/asciimath]
  • Periodic interest rate: [asciimath]i = (I//Y)/(C//Y) = (4.12%)/4 =1.03%[/asciimath]
  • The initial balance of the account is zero, so [asciimath]PV = 0[/asciimath]
  • Future value of investment: [asciimath]FV = $95,000[/asciimath]

a) [asciimath]PMT=?[/asciimath]

First, we need to find the interest rate per payment period [asciimath]i_2[/asciimath]:

 [asciimath]i_2=(1+i)^((C//Y)/(P//Y) )-1[/asciimath]

 [asciimath]i_2=(1+0.0103)^(4/12)-1[/asciimath]

 [asciimath]=0.00342161...[/asciimath]

 

Given the future value is known, we then use Formula 3.6b to calculate the size of the payments:

  [asciimath]PMT=(FV* i_2)/((1+i_2)^N-1)[/asciimath]

  [asciimath]PMT=(95,000(0.00342161... ))/((1+0.00342161...)^60-1)[/asciimath]

 [asciimath]=1429.061...[/asciimath]

 [asciimath]~~$1429.06[/asciimath]

Dakota will need to make monthly deposits of $1429.06 to accumulate her desired amount.

 

b) The total amount that Dakota will have deposited over the 5-year term is equal to the number of payments (N) multiplied by the size of each payment (PMT).

[asciimath]"Total amount deposited"=N*PMT[/asciimath]

[asciimath]=60(1429.06)[/asciimath]

 [asciimath]=$85,743.60[/asciimath]

c) [asciimath]I=?[/asciimath] 

Given this is an investment scenario where [asciimath]FV[/asciimath] is known, the total interest amount earned in the savings account is given by Formula 3.2.

 [asciimath]I=FV-(N*PMT)[/asciimath]

[asciimath]=95,000-85,743.60[/asciimath]

[asciimath]=$9256.40[/asciimath]

 

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Example 3.9.4: Compute PMT Given Purchase Price

Jeff purchased a car listed for $29,900. He paid 20% of the cost as a down payment and financed the balance amount at 5.1% compounded monthly for 12 years. a) What is the size of payment made at the end of every three months to settle the loan? b) How much was the amount of interest charged?

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Given information

  • Interest is compounded monthly, so [asciimath]C//Y = 12[/asciimath]
  • Payments are made at the end of every three months so [asciimath]P//Y = 4[/asciimath]
  • [asciimath]C//Y != P//Y[/asciimath]   [asciimath]=>[/asciimath]  Ordinary General Annuity
  • Annuity term: [asciimath]t = 12[/asciimath] years
  • Number of payments in the term: [asciimath]N = P//Y * t = 4(12) = 48[/asciimath]
  • Nominal interest rate:[asciimath]I//Y = 5.1%[/asciimath]
  • Periodic interest rate: [asciimath]i = (I//Y)/(C//Y) = (5.1%)/12=0.425%[/asciimath]
  • Purchase price = $29,900
  • Down payment = 20% of the purchase price
  • The loan is fully settled, so [asciimath]FV=0[/asciimath]

a)

Finding PV of loan

First, we need to find the amount of the loan, which is the remaining balance after paying the down payment.

 [asciimath]"Down payment"[/asciimath] [asciimath]=20%(29,900)[/asciimath] [asciimath]=$5980[/asciimath]

Using Formula 3.5, we have

[asciimath]"Purchase Price" = "Down payment" + PV[/asciimath]

Rearranging the equation for [asciimath]PV[/asciimath]gives

 [asciimath]PV = "Purchase Price"-"Down payment"[/asciimath]

 [asciimath]PV=29,900-5980[/asciimath]

 [asciimath]=$23,920[/asciimath]

Next, we use the PV and other given values to compute the size of the payments. We first need to find the interest rate per payment period [asciimath]i_2[/asciimath]:

 [asciimath]i_2=(1+i)^((C//Y)/(P//Y) )-1[/asciimath]

 [asciimath]i_2=(1+0.00425)^(12/4)-1[/asciimath]

 [asciimath]=0.0128042...[/asciimath]

 

Substituting the values into Formula 3.10b, we obtain

 [asciimath]PMT=(PV* i_2)/(1-(1+i_2)^(-N))[/asciimath]

 [asciimath]PMT=(23,920(0.0128042... ))/(1-(1+0.0128042... )^(-48))[/asciimath]

[asciimath]=670.146...[/asciimath]

 [asciimath]~~$670.15[/asciimath]

The size of the quarterly payments is $670.15.

 

b) The amount of interest for loans is given by Formula 3.4:

 [asciimath]I=N*PMT-PV[/asciimath]

 [asciimath]=48(670.15)-23,920[/asciimath]

[asciimath]=32,167.2-23,920[/asciimath]

 [asciimath]=$8,247.20[/asciimath]

 

Example 3.9.5: Compute PMT Given Both FV and PV

Inesh has already saved $11,600 in his savings account as of today, and he plans to contribute equal deposits at the end of each month for the next 10 years. What monthly deposit is required to accumulate $105,000 in total at the end of the 10 years assuming the account earns 5.2% compounded annually?

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Given information

  • Interest is compounded quarterly 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
  • Investment term: [asciimath]t = 10[/asciimath]  years
  • Number of payments in the term: [asciimath]N_1 = P//Y * t = 12(10) = 120[/asciimath]
  • Nominal interest rate: [asciimath]I//Y = 5.2%[/asciimath]
  • Periodic interest rate: [asciimath]i = (I//Y)/(C//Y) = (5.2%)/1=5.2%[/asciimath]
  • The desired accumulated value: [asciimath]FV = $105,000[/asciimath]
  • The initial lump sum investment: [asciimath]PV = $11,600[/asciimath]

[asciimath]PMT=?[/asciimath]

The accumulated value at the end of the term consists of two components: the future values of all the periodic payments ([asciimath]FV_("PMT")[/asciimath]) and the future value of the initial lump sum investment ([asciimath]FV_("PV")[/asciimath]).

 [asciimath]FV=FV_("PMT")+FV_("PV")[/asciimath]

Timeline depicting contributions to the account with 120 monthly payments (PMTs) ending at the end of year 10, plus an initial account balance at the beginning of the investment (PV). Arrows from the payments and the initial balance direct to year 10, highlighting the requirement to calculate their future values.

To calculate the size of each periodic payment, we must first determine [asciimath]FV_("PMT")[/asciimath], the future value of these payments. Knowing that the combined future value of the periodic payments and the initial lump sum equals the target accumulation of $105,000, our first step is to calculate the future value of the initial lump sum ([asciimath]FV_("PV")[/asciimath]). This amount is then subtracted from the desired accumulated total of $105,000.

 

1. Calculating the future value of the initial lump sum ([asciimath]FV_("PV")[/asciimath]):

For compound interest, [asciimath]N_2[/asciimath]  represents the number of compounding periods in the term. So we need to calculate N before using the compound interest future value formula. Additionally, for compound interest problems, the calculation should be based on the periodic interest rate per compounding period ([asciimath]i[/asciimath]).

 [asciimath]N_2=C//Y*t=1(10)=10[/asciimath]

 [asciimath]FV_("PV")=PV(1+i)^(N_2)[/asciimath]

 [asciimath]=11,600(1+0.052)^10[/asciimath]

 [asciimath]~~19,258.186[/asciimath]    (Do not round yet)

 

2. Finding the future value of the payments ([asciimath]FV_("PMT")[/asciimath]):

 [asciimath]105,000=FV_("PMT")+FV_("PV")[/asciimath]

 [asciimath]FV_("PMT") =105,000-19,258.186[/asciimath]

 [asciimath]=85,741.814[/asciimath]

 

3. Find the size of the periodic payment given the future value:

First, we need to find the interest rate per payment period [asciimath]i_2[/asciimath].

 [asciimath]i_2=(1+i)^((C//Y)/(P//Y) )-1[/asciimath]

 [asciimath]i_2=(1+0.052)^(1/12)-1[/asciimath]

 [asciimath]0.00423336...[/asciimath]

Plugging the values into Formula 3.6b, we get

  [asciimath]PMT=(FV* i_2)/((1+i_2)^(N_1)-1)[/asciimath]

  [asciimath]PMT=(85,741.814 (0.00423336... ))/((1+0.00423336... )^120-1)[/asciimath]

 [asciimath]=549.806...[/asciimath]

 [asciimath]~~$549.81[/asciimath]

Monthly deposits of $549.81 are needed to accumulate to the desired amount.

 

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C. Payment of Simple Annuity Due

If we know the future value of a simple annuity due, the number of payment periods, and the periodic interest rate, we can apply the future value formula (Formula 3.7a) to determine the unknown periodic payment (PMT).

[asciimath]PMT=(FV* i)/([(1+i)^N-1](1+i))[/asciimath]Formula 3.7b

Similarly, when the present value of a simple annuity due, the number of payments per year, and the periodic interest rate are known, the present value formula (Formula 3.11a) can be rearranged to solve for the unknown periodic payment PMT.

[asciimath]PMT=(PV* i)/([1-(1+i)^(-N)](1+i))[/asciimath]Formula 3.11b

 

Example 3.9.6: Compute PMT Given PV of Simple Annuity Due

What monthly rent payment at the beginning of each month for three years is needed to fulfill a lease contract that is worth $10,000 if money is worth 5% compounded monthly?

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Given information

  • Interest is compounded monthly so [asciimath]C//Y = 12[/asciimath]
  • Payments are made 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 = 3[/asciimath] years
  • Number of payments in the term: [asciimath]N = P//Y * t = 12(3) = 36[/asciimath]
  • Nominal interest rate: [asciimath]I//Y = 5%[/asciimath]
  • Periodic interest rate: [asciimath]i = (I//Y)/(C//Y) = (5%)/12=0.41bar(6)%[/asciimath]
  • The present value: [asciimath]PV = $10,000[/asciimath]
  • [asciimath]FV=0[/asciimath]

[asciimath]PMT=?[/asciimath]

Considering the present value is known, we use Formula 3.11b to find the size of the payments:

 [asciimath]PMT=(PV* i)/([1-(1+i)^(-N)](1+i))[/asciimath]

 [asciimath]PMT=(10,000(0.0041bar(6) ))/([1-(1+0.0041bar(6) )^(-36)](1+0.0041bar(6) ))[/asciimath]

[asciimath]=298.465...[/asciimath]

[asciimath]~~$298.47[/asciimath]

Monthly rents of $298.47 are required to fulfill the lease contract.

 

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D. Payment of General Annuity Due

If the future value of a general annuity due, along with the number of payment periods and the periodic interest rate, are known, we can use the future value formula (Formula 3.8a) to calculate the unknown periodic payment (PMT).

  [asciimath]PMT=(FV* i_2)/([(1+i_2)^N-1](1+i_2))[/asciimath]    Formula 3.8b

Similarly, if we have information about the present value of a general annuity due, as well as the number of payment periods and the periodic interest rate, we can use the present value formula (Formula 3.12a) to find the unknown periodic payment PMT.

 [asciimath]PMT=(PV* i_2)/([1-(1+i_2)^(-N)](1+i_2))[/asciimath]    Formula 3.12b

 

Example 3.9.7: Compute PMT Given Both PV and FV of general Annuity Due

Colleen entered into a lease agreement for a car with a list price of $42,000, which is expected to have a residual value of $21,450 after four years. She agreed to make a down payment of 10% of the car’s list price. Given an interest rate of 2.8% compounded monthly, calculate the amount of Colleen’s lease payment that is due at the beginning of each week.

Note: The residual value of a leased vehicle represents an estimate of how much the car will be worth at the end of the lease term.

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Given information

  • Interest is compounded monthly so [asciimath]C//Y = 12[/asciimath]
  • Payments are made at the beginning of every week so [asciimath]P//Y = 52[/asciimath]
  • [asciimath]C//Y != P//Y[/asciimath]   [asciimath]=>[/asciimath]  General Annuity Due
  • Annuity term: [asciimath]t = 4[/asciimath] years
  • Number of payments in the term: [asciimath]N_1 = P//Y * t = 52(4) = 208[/asciimath]
  • Nominal interest rate: [asciimath]I//Y = 2.8%[/asciimath]
  • Periodic interest rate: [asciimath]i = (I//Y)/(C//Y) = (2.8%)/12=0.2bar(3)%[/asciimath]
  • List price (cash value) [asciimath]= $42,000[/asciimath]
  • Down payment [asciimath]= 10%[/asciimath] of the list price
  • Residual value [asciimath]= $21,450[/asciimath]

To determine the size of the payment, we need to have the present value (PV) of the lease payments. The car’s cash value is equal to the combined present value of the lease payment and the residual plus the down payment:

 [asciimath]"Cash value" =[/asciimath]  [asciimath]"Down"+PV_("PMT") +PV_("Residual")[/asciimath]

To find the present value of the payments ([asciimath]PV_("PMT")[/asciimath]), we first need to calculate the present value of the residual ([asciimath]PV_("Residual")[/asciimath]).

 

1. Calculating the present value of the residual ([asciimath]PV_("Residual")[/asciimath]):

For compound interest, we need to calculate [asciimath]N_2[/asciimath] before using the compound interest future value formula. Additionally, for compound interest problems, the calculation should be based on the periodic interest rate per compounding period ([asciimath]i[/asciimath]).

 [asciimath]N_2=C//Y*t=12(4)=48[/asciimath]

 [asciimath]PV_("Residual")=FV(1+i)^(-N_2)[/asciimath]

 [asciimath]=21,450(1+0.002bar(3) )^-48[/asciimath]

 [asciimath]~~19,179.751...[/asciimath]  (Do not round yet)

 

2. Finding the present value (loan amount) of the payments ([asciimath]PV_("PMT")[/asciimath]):

 [asciimath]"Cash value" =[/asciimath]  [asciimath]"Down"+PV_("PMT") +PV_("Residual")[/asciimath]

Rearranging the equation for [asciimath]PV_("PMT")[/asciimath] gives

[asciimath]PV_("PMT")="Cash value"-[/asciimath] [asciimath]"Down"-PV_("Residual")[/asciimath]

 [asciimath]=42,000-10%(42,000)-19,179.751...[/asciimath]

 [asciimath]=18,620.249...[/asciimath]  (Do not round yet)

 

3. Find the size of the periodic payment given the present value:

We should first the interest rate per payment period [asciimath]i_2[/asciimath].

 [asciimath]i_2=(1+i)^((C//Y)/(P//Y) )-1[/asciimath]

 [asciimath]i_2=(1+0.002bar(3) )^(12/52)-1[/asciimath]

 [asciimath]0.000537978...[/asciimath]

Plugging the values into Formula 3.12b, we get

 [asciimath]PMT=(PV* i_2)/([1-(1+i_2)^(-N_1)](1+i_2))[/asciimath]

 [asciimath]PMT=((18,620.249...) (0.000537978... ))/([1-(1+0.000537978... )^(-208)](1+0.000537978... ))[/asciimath]

 [asciimath]=94.595...[/asciimath]

 [asciimath]~~$94.60[/asciimath]

The size of the weekly payments is $94.60.

 

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E. Payments in Scenarios with Multiple Annuities

The following example delves into calculating payments for a scenario involving a Registered Retirement Savings Plan (RRSP) and a Registered Retirement Income Fund (RRIF). Before we begin, it is beneficial to introduce these concepts.

RRSP and RRIF are fundamental components of Canada’s retirement savings system, designed to help individuals save for their future. The RRSP is a savings account with tax advantages to encourage saving for retirement. Contributions to an RRSP are tax-deductible, meaning they can reduce the amount of income tax you pay in the year you make the contribution. Once you retire, RRSP is converted into a RRIF which provides you with a regular income during retirement. Withdrawals from a RRIF are taxed as income at your marginal tax rate. The frequency of these withdrawals can be monthly, quarterly, semi-annually, or annually, offering flexibility in managing retirement income.

 

Example 3.9.8: Compute PMT of First Term Given PMT of the Second Term

Darius is planning for his retirement, which is 30 years away, by making end-of-quarter contributions to his Registered Retirement Savings Plan (RRSP), set to later convert into a Registered Retirement Income Fund (RRIF). Upon retirement, he intends to withdraw $3,400 at the end of each month for 20 years. An annual interest rate of 4.8%, compounded monthly, applies to both the RRSP during the accumulation phase and the RRIF during the distribution phase.

a) What is the total amount Darius must have in his RRIF at the start of retirement to support his goal of monthly withdrawals of $3,400?

b) How much must Darius deposit into his RRSP at the end of each quarter to ensure he accumulates the required amount in his RRIF by his retirement in 30 years?

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Timeline illustrating Darius's retirement planning over a 50-year span. The first 30 years show quarterly RRSP contributions leading up to retirement, marked at year 30. At this point, the timeline transitions to depict the conversion of the RRSP into a RRIF. The next 20 years illustrate monthly withdrawals of $3,400 from the RRIF, ending at year 50. Key points include the start of RRSP contributions (Year 0), conversion to RRIF at retirement (Year 30), and the duration of RRIF withdrawals (20 years).

a) Term 2: RRIF – Ordinary Simple Annuity

To determine the total amount Darius needs in his RRIF at the start of retirement to afford monthly withdrawals of $3,400 for 20 years, we calculate the present value of those payments at the time of retirement.

Given Information

  • Interest is compounded monthly so [asciimath]C//Y = 12[/asciimath]
  • Withdrawals are made at the end of every month so [asciimath]P//Y_2 = 12[/asciimath]
  •  [asciimath]C//Y = P//Y_2[/asciimath]  [asciimath]=>[/asciimath]   Ordinary Simple Annuity
  • RRIF term: [asciimath]t_2 = 20[/asciimath] years
  • Number of payments in the term: [asciimath]N_2 =P//Y_2*t_2=12 (20) = 240[/asciimath]
  • Nominal interest rate: [asciimath]I//Y = 4.8%[/asciimath]
  • Periodic interest rate: [asciimath]i = (I//Y)/(C//Y) = (4.8%)/12 =0.4%[/asciimath]
  • Periodic payments: [asciimath]PMT_2 = $3400[/asciimath]
  • The fund will be depleted by the end of the second term, so [asciimath]FV_2 = 0[/asciimath]

[asciimath]PV_2=?[/asciimath]

To compute the present value of the withdrawals, we apply Formula 3.9a.

 

[asciimath]PV=PMT[(1-(1+i)^-N)/i][/asciimath]

 [asciimath]PV_2=3400[(1-(1+0.004)^-240)/0.004][/asciimath]

 [asciimath]=523,917.230...[/asciimath]

[asciimath]~~$523,917.23[/asciimath]

Therefore, the total amount Darius must have in his RRIF at the start of retirement to support his goal is $523,917.23.

b) Term 1: RRSP – Ordinary General Annuity

The value needed at retirement, determined in Part (a) becomes the target FV for the RRSP contributions. We then calculate the quarterly contribution Darius must make to his RRSP over 30 years to accumulate to that target FV at retirement.

Given Information

  • Interest is compounded monthly so [asciimath]C//Y = 12[/asciimath]
  • Withdrawals are made at the end of every quarter so [asciimath]P//Y_1 = 4[/asciimath]
  •  [asciimath]C//Y != P//Y_1[/asciimath]  [asciimath]=>[/asciimath]   Ordinary General Annuity
  • RRSP term: [asciimath]t_1 = 30[/asciimath] years
  • Number of payments in the term: [asciimath]N_1 =P//Y_1*t_1=4 (30) = 120[/asciimath]
  • Nominal interest rate: [asciimath]I//Y = 4.8%[/asciimath]
  • Periodic interest rate: [asciimath]i = (I//Y)/(C//Y) = (4.8%)/12 =0.4%[/asciimath]
  • No initial balance in RRSP, so [asciimath]PV_1 = 0[/asciimath]
  • Future value: [asciimath]FV_1 = PV_2=$523,917.230...[/asciimath]  (use unrounded value for accuracy)

[asciimath]PMT_1=?[/asciimath]

First, we need to find the interest rate per payment period [asciimath]i_2[/asciimath]:

 [asciimath]i_2=(1+i)^((C//Y)/(P//Y) )-1[/asciimath]

 [asciimath]i_2=(1+0.004)^(12/4)-1[/asciimath]

 [asciimath]=0.0120480...[/asciimath]

Given the future value is known, we use Formula 3.6b to calculate the size of the payments:

   [asciimath]PMT=(FV* i_2)/((1+i_2)^N-1)[/asciimath]

  [asciimath]PMT_1=(523,917.230... (0.0120480... ))/((1+0.0120480... )^120-1)[/asciimath]

 [asciimath]=1967.277...[/asciimath]

 [asciimath]~~$1967.28[/asciimath]

Darius will need to make quarterly deposits of $1967.28 to ensure he accumulates the required amount in his RRIF by his retirement.

 

Try an Example

 

Section 3.9 Exercises

  1. What payment is required at the end of every month for 5.5 years to repay a $11,968 loan if interest is 4.4% compounded monthly?
    Show/Hide Answer

     

    PMT = $204.49

  2. Gregory purchased a car for $53,500; he paid 5% of the cost as a down payment and financed the balance amount at 3.5% compounded monthly for 3 years. a) What is the size of payment made at the end of every month to settle the loan? b) What was the amount of interest charged?
    Show/Hide Answer

     

    a) PMT = $1,489.28

    b) I = $2,789.08

  3. To fulfill a lease contract valued at $29,000, rent payments are required at the beginning of each month for 5 years. The interest rate on this lease is assumed to be 4.79%, compounded monthly. a) What is the size of the payments? b) How much interest was charged on the lease?
    Show/Hide Answer

     

    a) PMT = $542.32

    b) I = $3,539.20

  4. Dawn entered into a lease agreement for a car with a list price of $56,460.00, which is expected to have a residual value of $31,053 after 6 years. She agreed to make a down payment of 15% of the car’s list price. Given an interest rate of 2.06% compounded semi-annually, calculate the amount of Dawn’s lease payment that is due at the beginning of each month.
    Show/Hide Answer

     

    PMT = $302.79

  5. Kristina has already saved $7,600 in her savings account as of today, and she plans to contribute equal deposits at the end of each year for the next 10 years. What deposit is required to accumulate $91,500 in total at the end of the 10 years assuming the account earns 2.42% compounded quarterly?
    Show/Hide Answer

     

    PMT = $7323.15

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