Annuities
3.3 PV of Annuities: Calculator Approach
A. Present Value of Annuities and Interest Amount
In this section, we now 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. Consider a scenario where you have the option to receive a monthly payment of $1,000 for the next 30 years from a retirement plan. This plan is expected to yield a 10% annual interest rate compounded monthly. 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 30-year term, taking into account the interest rate.
For computing the present value in annuity problems, similar to computing future values, we use the time-value-of-money (TVM) worksheet in financial calculators. It’s also important to adhere to the cash flow sign convention (see Table 2.2.1 in Section 2.2) when inputting monetary values into the calculator to ensure consistency and accuracy in the calculations.
As we discussed in the previous section, it’s important to first verify the payment timing setting on your calculator before solving annuity problems. The default setting is usually at ‘END’ (end of the payment period), which is appropriate for ordinary annuities. However, for annuities due, where payments are made at the beginning of each period, this setting needs to be adjusted to ‘BGN’ (beginning of the payment period). Refer to Figure 3.2.1 in Section 3.2, to learn how to change the payment timing on a financial calculator.
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, and therefore, the amount of interest is obtained by
[asciimath]I=(N*PMT)-PV[/asciimath]Formula 3.4
In this formula, 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).
How to Compute the Present Value of Annuities Using a Financial Calculator
1. Clear the TVM Worksheet: Before starting, clear your TVM worksheet to remove any previous values by pressing ‘2ND’ followed by the ‘FV’ key.
2. Identify the Type of Annuity: Determine if it’s an ordinary annuity (payments at the end of each period) or an annuity due (payments at the beginning of each period). Adjust your calculator’s setting accordingly (END for ordinary annuities, BGN for annuities due).
3. Enter Key Values: Input the relevant values for the number of periods (N), the nominal interest rate in percent (I/Y), and the payment amount (PMT). Remember to adhere to the cash flow sign convention, entering payment (PMT) as a negative value (outflow) if you make payments and as a positive value (cash inflow) if you receive payments.
4. Set Future Value (FV):
- If the annuity has no remaining balance after the final payment (e.g. a loan has been fully paid off, or a retirement saving has been fully withdrawn), set FV to zero.
- If there is a lump sum to be paid or received at the end of the annuity term apart from the regular payments (e.g., a maturity value or a residual value), set FV to that amount.
5. Calculate Present Value (PV): With the other variables entered correctly, compute PV. The calculator will display the present value of the annuity, which is the current worth of all future payments, adjusted for the interest rate.
6. Compute the amount of interest (I): When required, apply Formula 3.4 to calculate the total amount of interest incurred over the entire term of the annuity.
B. Present Value of an Ordinary Annuity
We begin by examining the present value of ordinary annuities, where payments are made at the end of each period. It is important to first verify the payment timing setting on your calculator before solving annuity problems.
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]
- The payments are made (not received), so they are cash outflow: [asciimath]PMT = -$1200[/asciimath]
- No remaining balance at the end of the annuity term, so [asciimath]FV=0[/asciimath]
Enter the given values into the calculator and compute PV:
Therefore, the present value of the annuity is $6,698.86.
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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]
- For loans such as mortgages, payments are made, so they are cash outflow: [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]
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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C. Present Value of Annuity Due
In the following examples, we will determine the present value of annuities due. For these types of annuities, where payments are made at the beginning of each payment period, it is essential to adjust the payment timing setting on your calculator to ‘BGN’, which stands for the beginning of the payment period.
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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- 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]
- Shayan receives the payments, so they are cash inflow: [asciimath]PMT = $1000[/asciimath]
- Since the fund will be fully exhausted by the end of the term, [asciimath]FV=0[/asciimath]
The amount of inheritance was $1,029,401.41.
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D. Present Value Problems with Down Payment
A down payment refers to a relatively small initial payment made by a buyer at the time of purchasing a significant asset, such as real estate property. This payment is made upfront and the remaining balance of the purchase price is typically covered through a loan obtained from a financial institution. The amount of this loan corresponds to the present value (PV) of the future periodic payments that the buyer will make to pay off the loan. These periodic payments include not only the principal amount but also the interest charged on the loan. The relationship among the purchase price (also known as the cash price), the loan amount (PV), and the down payment can be expressed as follows:
[asciimath]"Purchase Price" = "Down Payment" + PV[/asciimath]Formula 3.5
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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- 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]
- For loans such as mortgages, payments are made, so they are cash outflow: [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
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 (I) and the loan principal (PV) 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 a 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]
- For loans payments are made, so they are cash outflow: [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
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 (I) and the loan principal (PV) around the ring.
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E. 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]
- The payments are made, so they are cash outflow: [asciimath]PMT = -$190[/asciimath]
- At the end of the lease, Patricia either returns the car valued at $14,600 or purchases it by paying the amount. Either way, the value is considered a cash outflow: [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]
Using a financial calculator simplifies this process, as it can compute the total present value in a single step. In the calculator, enter the residual value as the future value (FV) and the lease payment as the payment (PMT). Both FV and PMT should be input as negative values since they represent cash outflows.
Therefore, the combined present value of the lease payments and the residual amounts to $20,118.02, representing the cash value of the car.
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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]
- The payments are made, so they are cash outflow: [asciimath]PMT = -$850[/asciimath]
- Down Payment [asciimath]= $10,500[/asciimath]
- At the end of the lease, you either return the car valued at $21,000 or purchase it by paying the amount. Either way, the value is considered a cash outflow: [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]
Using a financial calculator simplifies this process, as it can compute the total present value in a single step. In the calculator, enter the residual value as the future value (FV) and the lease payment as the payment (PMT). Both FV and PMT should be input as negative values since they represent cash outflows.
Therefore, the sum of the present value of the payments and the residual is $56,538.01.
Substituting the down payment and PV yields
[asciimath]"Cash value" = "Down" + PV_("total")[/asciimath]
[asciimath]=10,500+56,538.01[/asciimath]
[asciimath]=67,038.01[/asciimath]
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Section 3.3 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
