Annuities
3.8 PV of Annuities: Formula Approach
A. Present Value of Ordinary Simple Annuity
In this section, we turn our attention to the present value of annuities, a key concept in valuing the current worth of future payments. The present value of an annuity is crucial in scenarios such as determining how much a series of future cash flows is worth today or evaluating the current worth of a long-term investment or loan. We begin by examining the present value of ordinary annuities, where payments are made at the end of the payment period. In an Ordinary Simple Annuity, the payment period and the interest compounding period are the same ([asciimath]P//Y = C//Y[/asciimath]), and the payments are made at the end of the payment period.
Consider a scenario where you have the option to receive an end-of-year payment of $1,000 for the next 5 years from a fund. This fund is expected to yield a 10% annual interest rate compounded annually. To understand the current worth of these future payments, you need to calculate the present value of this annuity. The present value in this context refers to the total value in today’s dollars of all the annuity payments you are set to receive over the 5-year term, taking into account the interest rate.
One way to calculate the present value is to calculate the present value of each payment using the formula for the present value of compound interest, [asciimath]PV=FV(1+i)^-N[/asciimath] (Formula 2.4c). The timeline for this calculation is depicted in Figure 3.8.1.
Figure 3.8.1 A Timeline for the Present Value of Ordinary Simple Annuity
In our example, since the interest is compounded annually ([asciimath]C//Y=1[/asciimath]), the periodic interest rate ([asciimath]i[/asciimath]) is 10%, and the number of compounding periods ([asciimath]N[/asciimath]) for each payment equals the number of years from the payment date until the present (i.e., the beginning of the year 0). The future value (FV) for each payment is $1,000.
[asciimath]"PV"=1000(1+0.1)^-1+1000(1+0.1)^-2+[/asciimath] [asciimath]1000(1+0.1)^-3 +1000(1+0.1)^-4+1000(1+0.1)^-5[/asciimath]
[asciimath]=909.09+826.45+751.31+683.01+620.92[/asciimath]
[asciimath]=$3790.79[/asciimath]
Therefore, the required present value in the fund to sustain payments of $1,000 at the end of each year for five years is $3790.79.
As noted earlier in the section on future value, calculating the present value for each payment individually, as demonstrated in this example, is impractical due to its time-consuming nature. Therefore, for an ordinary simple annuity, we utilize a more efficient and straightforward method to compute the overall present value.
[asciimath]PV=PMT[(1-(1+i)^-N)/i][/asciimath]Formula 3.9a
(See how the formula is derived)
where [asciimath]PMT[/asciimath] represents the size of the periodic payment, [asciimath]i[/asciimath] is the periodic interest rate (calculated as [asciimath]i=(I//Y)/(C//Y)[/asciimath], Formula 2.1a), and [asciimath]N[/asciimath] is the number of payments in the annuity term (computed as [asciimath]N=P//Y*t[/asciimath], Formula 3.1a).
B. Computing Interest Amount
In problems where the present value of an annuity is known or is calculated (usually for loan scenarios), the periodic payments of the annuity include interest. Therefore, the amount of interest is obtained by deducting the present value from the total payment amount.
[asciimath]I=(N*PMT)-PV[/asciimath] Formula 3.4
In this formula, initially introduced in Section 3.3, PV is the present value of the annuity, PMT is the periodic payment amount, and N is the total number of payments, calculated as [asciimath]N = P//Y * t[/asciimath] (Formula 3.1a).
Considering a bank offering an investment opportunity. To provide annual payments of $1200 at the end of each year for the next 7 years, at a nominal interest rate of 6% compounded annually, what is the required initial deposit that should be made by a customer today?
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Given information
- Interest is compounded annually so [asciimath]C//Y = 1[/asciimath]
- Payments are made at the end of every year so [asciimath]P//Y = 1[/asciimath]
[asciimath]C//Y = P//Y[/asciimath] [asciimath]=>[/asciimath] Ordinary Simple Annuity
- Investment Term: [asciimath]t = 7[/asciimath] years
- Number of payments in the term: [asciimath]N = P//Y * t = 1(7) = 7[/asciimath]
- Nominal interest rate: [asciimath]I//Y = 6%[/asciimath]
- Periodic interest rate: [asciimath]i = (I//Y)/(C//Y) = (6%)/1 =6%[/asciimath]
- Periodic payments: [asciimath]PMT = $1200[/asciimath]
- No remaining balance at the end of the annuity term, so [asciimath]FV=0[/asciimath]
- [asciimath]PV=?[/asciimath]
Substituting the values into Formula 3.9a gives
[asciimath]PV=PMT[(1-(1+i)^-N)/i][/asciimath]
[asciimath]PV=1200[(1-(1+0.06)^-7)/0.06][/asciimath]
[asciimath]=6,698.857...[/asciimath]
[asciimath]~~$6,698.86[/asciimath]
Therefore, the present value of the annuity is $6,698.86.
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C. Present Value of Ordinary General Annuity
In an Ordinary General Annuity, the payment period and the interest compounding period are NOT the same ([asciimath]P//Y != C//Y[/asciimath]) and the payments are made at the end of the payment period.
Just as we did with the future value of an ordinary general annuity in the previous section, we can modify the formula for the present value of an ordinary simple annuity by replacing the periodic interest rate per compounding period ([asciimath]i[/asciimath]) with the periodic interest rate per payment period ([asciimath]i_2[/asciimath]). The periodic interest rate per payment period can be calculated as follows:
[asciimath]i_2=(1+i)^((C//Y)/(P//Y))-1[/asciimath]
Therefore, the formula for calculating the present value of an ordinary general annuity is obtained by
[asciimath]PV=PMT[(1-(1+i_2)^-N)/i_2][/asciimath]Formula 3.10a
A 25-year mortgage on a condominium requires payments of $1000 at the end of each month. If interest is 4% compounded semi-annually, a) what was the mortgage principal? b) How much interest was charged on the mortgage?
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Given information
- Interest is compounded semi-annually so [asciimath]C//Y = 2[/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
- Mortgage Term: [asciimath]t = 25[/asciimath] years
- Number of payments in the term: [asciimath]N = P//Y * t = 12(25) = 300[/asciimath]
- Nominal interest rate: [asciimath]I//Y = 4%[/asciimath]
- Periodic interest rate: [asciimath]i = (I//Y)/(C//Y) = (4%)/2 =2%[/asciimath]
- Periodic payments: [asciimath]PMT =[/asciimath] [asciimath]$1000[/asciimath]
- The mortgage is fully paid by the end of the term (i.e., no remaining balance), so [asciimath]FV=0[/asciimath]
a) [asciimath]PV=?[/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+0.02)^(2/12)-1[/asciimath]
[asciimath]=0.003305890...[/asciimath]
Substituting the values into Formula 3.10a gives
[asciimath]PV=PMT[(1-(1+i_2)^-N)/i_2][/asciimath]
[asciimath]PV=1000[(1-(1+0.003305890 )^-300)/0.003305890 ][/asciimath]
[asciimath]=190,106.773...[/asciimath]
[asciimath]~~$190,106.77[/asciimath]
Thus, the mortgage principal was $190,106.77.
b) [asciimath]I=?[/asciimath]
The amount of interest for loans is given by Formula 3.4.
[asciimath]I=(N*PMT)-PV[/asciimath]
[asciimath]=1000(300)-190,106.77[/asciimath]
[asciimath]=300,000-190,106.77[/asciimath]
[asciimath]=$109,893.23[/asciimath]
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D. Present Value of Simple Annuity Due
In a Simple Annuity Due, the payment period and the interest compounding period are the same ([asciimath]P//Y = C//Y[/asciimath]), and the payments are made at the beginning of the payment period.
Consider a scenario we used at the start of this section for an ordinary simple annuity. This time, you receive payments of $1,000 from a fund at the beginning of every year for the next 5 years. This fund is expected to yield a 10% annual interest rate compounded annually. To understand the current worth of these future payments, you need to calculate the present value of this annuity. Similar to the previous scenario, to calculate the total present value, we calculate the present value of each payment using the formula for the present value of compound interest (Formula 2.4c). The timeline for this calculation is shown in Figure 3.8.2.
Figure 3.8.2 A Timeline for the Present Value of Simple Annuity Due
[asciimath]"PV"=1000(1+0.1)^0+1000(1+0.1)^-1+[/asciimath] [asciimath]1000(1+0.1)^-2 +1000(1+0.1)^-3+1000(1+0.1)^-4[/asciimath]
If we factor out [asciimath](1+0.1)[/asciimath] from the right-hand side of the equation, we obtain
[asciimath]"PV"=(1+0.1)[1000(1+0.1)^-1+1000(1+0.1)^-2+[/asciimath] [asciimath]1000(1+0.1)^-3 +1000(1+0.1)^-4+1000(1+0.1)^-5][/asciimath]
Notice that the expression in the square brackets matches the Present Value (PV) calculated for an ordinary simple annuity in our previous example. Thus,
[asciimath]PV=(1.1)[$3790.79][/asciimath]
[asciimath]=$4169.87[/asciimath]
Similar to the case with future values, the present value of a simple annuity due equals the present value of an ordinary simple annuity multiplied by a factor of [asciimath](1 + i)[/asciimath]. Therefore, the formula for the future value of an annuity due can be expressed as [asciimath]PV_("due")=PV_("Ordinary")(1+i)[/asciimath]. This formula is defined as follows
[asciimath]PV=PMT[(1-(1+i)^-N)/i] (1+i)[/asciimath]Formula 3.11a
(See how the formula is derived)
Cai bought an appliance for a new home and opted for a payment plan. Under this plan, he will pay $490 at the beginning of each month for 3 years. The interest rate on this payment plan is 6.67%, compounded monthly. a) Determine the cash value of the appliance. b) Calculate the total amount of interest charged to Cai throughout the payment plan.
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Given information
- Interest is compounded monthly, so [asciimath]C//Y = 12[/asciimath]
- Payments are made at the beginning of every month, so [asciimath]P//Y = 12[/asciimath]
[asciimath]C//Y= P//Y[/asciimath] [asciimath]=>[/asciimath] Simple Annuity due
- Time period: [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 = 6.67%[/asciimath]
- Periodic interest rate: [asciimath]i = (I//Y)/(C//Y) = (6.67%)/12 =0.005558bar(3)[/asciimath]
- Size of the periodic payment: [asciimath]PMT = $490[/asciimath]
a) [asciimath]PV=?[/asciimath]
Substituting the values into Formula 3.11a gives
[asciimath]PV=PMT[(1-(1+i)^-N)/i](1+i)[/asciimath]
[asciimath]PV=490 [(1-(1+0.005558bar(3))^-36)/(0.005558bar(3))](1+0.005558bar(3))[/asciimath]
[asciimath]~~$16,035.82[/asciimath]
Therefore, the cash value of the appliance is $16,035.82.
b) [asciimath]I=?[/asciimath]
For loans, PMT includes interest, and the amount of interest is given by Formula 3.4.
[asciimath]I = (PMT*N)-PV[/asciimath]
[asciimath]I=(490 xx 36)-16,035.82[/asciimath]
[asciimath]=17,640.00-16,035.82[/asciimath]
[asciimath]=$1,604.18[/asciimath]
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E. Present Value of General Annuity Due
In a General Annuity Due, the payment period and the interest compounding period are different ([asciimath]P//Y != C//Y[/asciimath] ) and the payments are made at the beginning of the payment period.
Just as with the present value of ordinary annuities, the formula initially developed for the present value of a simple annuity due can be modified by including the interest rate per payment period, [asciimath]i_2[/asciimath], into the present value calculation. Thus, in the present value formula for the simple annuity due, replace [asciimath]i[/asciimath] with [asciimath]i_2[/asciimath] to make it applicable for calculating the present value of a general annuity due.
[asciimath]PV=PMT[(1-(1+i_2)^-N)/i_2](1+i_2)[/asciimath] Formula 3.12a
Shayan received an inheritance which was placed in a savings account. This account pays him $1,000 at the beginning of each week for 30 years. The account accrues interest at a rate of 3% per year compounded monthly. What was the amount of the inheritance?
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Given information
- Interest is compounded monthly so [asciimath]C//Y = 12[/asciimath]
- Payments are made at the beginning of each week so [asciimath]P//Y = 52[/asciimath]
[asciimath]C//Y != P//Y[/asciimath] [asciimath]=>[/asciimath] General Annuity Due
- Annuity Term: [asciimath]t = 30[/asciimath] years
- Number of payments in the term: [asciimath]N = P//Y * t = 52(30) = 1560[/asciimath]
- Nominal interest rate: [asciimath]I//Y = 3%[/asciimath]
- Periodic interest rate: [asciimath]i = (I//Y)/(C//Y) = (3%)/12 =0.25%[/asciimath]
- Periodic payments: [asciimath]PMT = $1000[/asciimath]
- Since the fund will be fully exhausted by the end of the term, [asciimath]FV=0[/asciimath]
a) [asciimath]PV=?[/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+0.0025)^(12/52)-1[/asciimath]
[asciimath]=0.000576369...[/asciimath]
Substituting the values into Formula 3.12a gives
[asciimath]PV=PMT[(1-(1+i_2)^-N)/i_2](1+i_2)[/asciimath]
[asciimath]PV=1000[(1-(1+000576369... )^-1560)/000576369... ](1+000576369... )[/asciimath]
[asciimath]=1,029,401.416...[/asciimath]
[asciimath]~~$1,029,401.42[/asciimath]
The amount of inheritance was $1,029,401.42.
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F. Present Value Problems with Down Payment
[asciimath]"Purchase Price" = "Down Payment" + PV[/asciimath]Formula 3.5
It is worth noting that this formula was first introduced in Section 3.3.
Andy made a down payment of $40,000 on an apartment and secured a mortgage for the rest of the purchase price. He has agreed to repay this mortgage with end-of-month payments of $1,580 for 30 years at a 3.45% annual interest rate compounded monthly.
a) Calculate the original purchase price of the apartment.
b) Determine the total amount Andy will have paid by the end of the 30-year mortgage term.
c) Calculate the total amount of interest Andy will have paid on the mortgage over the 30 years.
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Given information
- Interest is compounded monthly so [asciimath]C//Y = 12[/asciimath]
- Payments are made at the end of every month so [asciimath]P//Y = 12[/asciimath]
[asciimath]C//Y = P//Y[/asciimath] [asciimath]=>[/asciimath] Ordinary Simple Annuity
- Mortgage term: [asciimath]t = 30[/asciimath] years
- Number of payments in the term: [asciimath]N = P//Y * t = 12(30) = 360[/asciimath]
- Nominal interest rate: [asciimath]I//Y = 3.45%[/asciimath]
- Periodic interest rate: [asciimath]i=(I//Y)/(C//Y)=(3.45%)/12=0.2875%[/asciimath]
- Periodic payments: [asciimath]PMT = $1580[/asciimath]
- Down payment = [asciimath]$40,000[/asciimath]
- The mortgage is fully paid off by the end of the term, so [asciimath]FV=0[/asciimath]
a) To determine the purchase price of the apartment, we first need to calculate the mortgage amount. This is done by determining the present value of the monthly payments Andy is scheduled to make.
Computing PV
Substituting the values into Formula 3.9a gives
[asciimath]PV=PMT[(1-(1+i)^-N)/i][/asciimath]
[asciimath]PV=1580[(1-(1+0.2875%)^-360)/(0.2875% )][/asciimath]
[asciimath]=354,055.043...[/asciimath]
[asciimath]~~$354,055.04[/asciimath]
Thus, the mortgage principal was $354,055.04.
Calculating the purchase price
By substituting the values of the down payment and the present value (PV) into Formula 3.5, we can calculate the total purchase price.
[asciimath]"Purchase Price" = "Down Payment" + PV[/asciimath]
[asciimath]=40,000+354,055.04[/asciimath]
[asciimath]=$394,055.04[/asciimath]
Therefore, the purchase price of the apartment was $394,055.04.
b) The total amount that Andy will pay over the 30-year term of the mortgage is equal to the number of payments (N) multiplied by the size of each payment (PMT).
[asciimath]"Total amount paid to repay loan" = N*PMT[/asciimath]
[asciimath]=360(1580)[/asciimath]
[asciimath]=$568,800[/asciimath]
c)The amount of interest for loans is given by Formula 3.4.
[asciimath]I=(N*PMT)-PV[/asciimath]
[asciimath]=568,800-354,055.04[/asciimath]
[asciimath]=$214,744.96[/asciimath]
Note: When calculating the interest charged on the mortgage, we use the present value (PV) of the loan, not the purchase price of the property.
The below figure displays a donut chart with the total payments ([asciimath]N*PMT[/asciimath]) of $741,000 at its center, breaking down into components of interest ([asciimath]I[/asciimath]) and the loan principal ([asciimath]PV[/asciimath]) around the ring.
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Bijan made a 20% down payment on the purchase of a vacation property and obtained a mortgage from a bank to cover the remaining cost. He has arranged to repay this mortgage with end-of-week payments of $570 for 25 years. The interest charged on the mortgage is 3.05% compounded monthly.
a) What was the purchase price of the property?
b) What was the total amount paid over the 25-year term to repay the mortgage?
c) How much interest was charged on the mortgage?
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Given information
- Interest is compounded monthly so [asciimath]C//Y = 12[/asciimath]
- Payments are made at the end of the week so [asciimath]P//Y = 52[/asciimath]
[asciimath]C//Y != P//Y[/asciimath] [asciimath]=>[/asciimath] Ordinary General Annuity
- Mortgage term: [asciimath]t = 25[/asciimath] years
- Number of payments in the term: [asciimath]N = P//Y * t = 52(25) = 1300[/asciimath]
- Nominal interest rate: [asciimath]I//Y = 3.05%[/asciimath]
- Periodic interest rate: [asciimath]i=(I//Y)/(C//Y)=(3.05%)/12=0.2541bar(6)%[/asciimath]
- Periodic payments: [asciimath]PMT = $570[/asciimath]
- Down payment = 20% of the purchase price
- The mortgage is fully paid off by the end of the term, so [asciimath]FV=0[/asciimath]
a)To determine the purchase price, we first need to calculate the mortgage amount, which is done by determining the present value of the weekly payments Bijan is scheduled to make.
Computing PV
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+0.002541bar(6))^(12/52)-1[/asciimath]
[asciimath]=0.000585965...[/asciimath]
Substituting the values into Formula 3.10a gives
[asciimath]PV=PMT[(1-(1+i_2)^-N)/i_2][/asciimath]
[asciimath]PV=570[(1-(1+0.000585965... )^-1300)/0.000585965... ][/asciimath]
[asciimath]=518,525.660...[/asciimath]
[asciimath]~~$518,525.66[/asciimath]
Thus, the mortgage amount was $518,525.66.
Calculating the purchase price
Since the down payment is given as a percentage of the unknown purchase price, we need to express the down payment in terms of the purchase price when applying Formula 3.5. To simplify this, let’s use ‘X’ to denote the purchase price. Thus, the down payment can be represented as 20% of X, or 0.2X.
[asciimath]"Purchase Price" = "Down Payment" + PV[/asciimath]
[asciimath]X=0.2X+518,525.66[/asciimath]
[asciimath]X-0.2X=518,525.66[/asciimath]
[asciimath]0.8X=518,525.66[/asciimath]
[asciimath]X=(518,525.66 )/0.8[/asciimath]
[asciimath]=$648,157.08[/asciimath]
Therefore, the purchase price of the vacation property was $648,157.08.
b) The total amount that Bijan will pay over the 25-year term of the mortgage is equal to the number of payments (N) multiplied by the size of each payment (PMT).
The total amount paid to repay the loan [asciimath]= N*PMT[/asciimath]
[asciimath]=570(1300 )[/asciimath]
[asciimath]=$741,000[/asciimath]
c) The amount of interest for loans is given by Formula 3.4.
[asciimath]I=N.PMT-PV[/asciimath]
[asciimath]=741,000-518,525.66[/asciimath]
[asciimath]=$222,474.34[/asciimath]
The below figure displays a donut chart with the total payments ([asciimath]N*PMT[/asciimath]) of $741,000 at its center, breaking down into components of interest charged ([asciimath]I[/asciimath]) and the loan principal ([asciimath]PV[/asciimath]) around the ring.
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G. Lease and Residual Value
A lease is a contractual agreement where one party, the lessee, pays the lessor (the owner) for the use of an asset over a specified period. Common in real estate, vehicles, and equipment, leases often include terms that specify payments, duration, and conditions of use. A key component of many lease agreements, particularly in auto leasing, is the residual value. This term refers to the projected value of the leased asset at the end of the lease term. It’s an estimate of the asset’s worth after depreciation over the lease period.
When calculating the present value (PV) of lease payments, the residual value plays a significant role. The present value of a lease is essentially the sum of the discounted values of all lease payments and the discounted residual value. Therefore, the calculation of PV in leasing scenarios must account for both the periodic payments and the residual value. This approach helps in comparing different lease options. For lease calculations, the residual value is often treated as a future value (FV). Since the residue value represents the value of the asset (such as a car) that needs to be returned to the owner at the end of the term, it is considered a cash outflow and should be entered as a negative value in the financial calculator.
Patricia’s car lease agreement entails monthly payments of $190.00, due at the start of each month for 3 years. At the end of the lease term, she can choose to either return the car or buy it for a residual value of $14,600. The lease is subject to a 2.55% interest rate compounded monthly. Determine the original cash value of the car based on this lease agreement.
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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
- Lease 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 = 2.55%[/asciimath]
- Periodic interest rate = [asciimath]i=(I//Y)/(C//Y)=(2.55%)/12=0.2125%[/asciimath]
- Periodic payments: [asciimath]PMT = $190[/asciimath]
- The residual value: [asciimath]FV = $14,600[/asciimath]
To determine the original cash value of the car, we need to calculate the present value (PV) of both the lease payments and the residual value. The car’s cash value is equal to the combined present value of these amounts:
[asciimath]"Cash value" = PV_("PMT") +PV_("Residual")[/asciimath]
Finding [asciimath]PV_"PMT"[/asciimath]:
Substituting the values into Formula 3.11a gives
[asciimath]PV=PMT[(1-(1+i)^-N)/i](1+i)[/asciimath]
[asciimath]PV=190 [(1-(1+0.002125)^-36)/0.002125 ](1+0.002125 )[/asciimath]
[asciimath]~~6,592.171[/asciimath]
Finding [asciimath]PV_"Residual"[/asciimath] (Compound Interest):
In compound interest problems, [asciimath]N[/asciimath] represents the number of compounding periods within the term, calculated as [asciimath]N = C//Y * t = 12(3) = 36[/asciimath]. In this particular problem, [asciimath]N[/asciimath] coincidentally matches the [asciimath]N[/asciimath] used for the Present Value (PV) of an annuity. This is because the annuity is a simple annuity where the number of compounding periods per year (C/Y) equals the number of payment periods per year (P/Y). Also for compound problems, [asciimath]P//Y[/asciimath] should be set to the value of [asciimath]C//Y[/asciimath].
Substituting the values into Formula 2.4c yields
[asciimath]PV=FV(1+i)^-N[/asciimath]
[asciimath]PV=14,600(1+0.002125 )^-36[/asciimath]
[asciimath]~~13,525.850[/asciimath]
Therefore, the cash value of the car will be the sum of those present values:
[asciimath]"Cash value" = PV_("PMT") +PV_("Residual")[/asciimath]
[asciimath]=6,592.171 +13,525.850[/asciimath]
[asciimath]=$20,118.02[/asciimath]
Try an Example
Slavica has a car lease agreement that requires an initial down payment of $10,500 and lease payments of $850 at the beginning of every month for 4 years. At the end of 4 years, she has the option to either return the car or purchase it for a residual value of $21,000. If the interest charged on the lease is 3.4% compounded monthly, what was the cash value of the car?
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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
- Lease Term: [asciimath]t = 4[/asciimath] years
- Number of payments in the term: [asciimath]N = P//Y * t = 12(4) = 48[/asciimath]
- Nominal interest rate: [asciimath]I//Y = 3.4%[/asciimath]
- Periodic interest rate = [asciimath]i=(I//Y)/(C//Y)=(3.4%)/12=0.28bar(3)%[/asciimath]
- Periodic payments: [asciimath]PMT = $850[/asciimath]
- Down Payment [asciimath]= $10,500[/asciimath]
- Residual value: [asciimath]FV = $21,000[/asciimath]
To determine the original cash value of the car, we need to calculate the present value (PV) of both the lease payments and the residual value. The car’s cash value is equal to the combined present value of these amounts plus the down payment:
[asciimath]"Cash value" =[/asciimath] [asciimath]"Down"+PV_("PMT") +PV_("Residual")[/asciimath]
Finding [asciimath]PV_"PMT"[/asciimath]:
Substituting the values into Formula 3.11a gives
[asciimath]PV=PMT[(1-(1+i)^-N)/i](1+i)[/asciimath]
[asciimath]PV=850 [(1-(1+0.0028bar(3) )^-48)/(0.0028bar(3)) ](1+0.0028bar(3) )[/asciimath]
[asciimath]~~38,204.791[/asciimath]
Finding [asciimath]PV_"Residual"[/asciimath] (Compound Interest):
- [asciimath]N = C//Y * t = 12(4) = 48[/asciimath]
- [asciimath]P//Y=[/asciimath] [asciimath]C//Y=12[/asciimath]
Substituting the values into Formula 2.4c yields
[asciimath]PV=FV(1+i)^-N[/asciimath]
[asciimath]PV=21,000(1+0.0028bar(3) )^-48[/asciimath]
[asciimath]~~18,333.223[/asciimath]
Substituting the down payment and PVs yields
[asciimath]"Cash value" =[/asciimath] [asciimath]"Down"+PV_("PMT") +PV_("Residual")[/asciimath]
[asciimath]=10,500+38,204.791 +18,333.223[/asciimath]
[asciimath]=67,038.01[/asciimath]
Try an Example
Section 3.8 Exercises
- A mortgage with a 30-year term has monthly payments of $5,896.97, due at the end of each month. The interest rate on this mortgage is 3.02%, compounded semi-annually. a) Calculate the original principal amount of the mortgage. b) Determine the total amount of interest that will be charged over the life of the mortgage.
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a) PV = $1,398,488.92
b) I = $724,420.28
- Trinity paid $30,400 as a down payment toward a vacation property purchase and received a mortgage from a bank for the remaining amount. She agreed to pay end-of-month payments of $3,005.14 for 15 years to repay the mortgage. The interest charged on the mortgage was 3.72% compounded annually. a) Calculate the amount of the mortgage. b) What was the purchase price of the property? c) What was the total amount paid over the 15-year term to repay the mortgage? d) How much interest was charged on the mortgage?
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a) PV = $415,839.19
b) Purchase price = $446,239.19
c) Total amount paid = $540,925.20
d) I = $125,086.01
- Carmen inherited a sum of money that was invested in a savings account providing her with $613.29 at the beginning of each quarter for 12 years. The account earned interest of 5.46% compounded semi-annually. a) What was the amount of the inheritance? b) How much interest was earned in the account during the term?
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a) PV = $21,827.09
b) I = $7,610.83
- Cheyenne has a car lease agreement that requires an initial down payment of $8,900 and lease payments of $260.00 at the beginning of every month for 5 years. At the end of 5 years, she has the option to either return the car or purchase it for a residual value of $14,800. If the interest charged on the lease is 2.65% compounded semi-annually, what was the cash value of the car?
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Cash value of the car = $36,507.58
