Chapter 4 : Annuities and Loans
Homer plans to deposit $150 in the bank in one year. He plans to make the same deposit two years from today and three years from today. How much will Homer have in the bank in four years? Homer's bank pays an interest rate of 6.4%.
"two years from today, so annuity due" i = 6.4% n = 3 PMT = $150 FVannuity due = PMT x [(1 + i)^n - 1]/ i x (1.064) FVannuity due = 150 x [(1 + 0.064)^3 - 1]/ 0.064 x (1.064) FVannuity due = $510.10
You won the lottery! The prize is $100 paid at the beginning of each of the next 4 years (starting today). What is the present value of the lottery annuity if the interest rate is 10%?
(starting today) = annuity due PMT = $100 n = 4 i = 10 PVannuity due = PMT x [(1 + i)^n - 1]/ i x ( 1 + i ) PVannuity due = $100 x [(1 + 0.10)^4 - 1]/ 0.10 x ( 1 + 0.10 ) PVannuity due = $100 x [(1 + 0.10)^4 - 1]/ 0.10 x ( 1 + 0.10 ) PVannuity due = $348.685
You borrow $200,000 over a 25 year term. The loan is structured as an amortized loan with annual payments and an interest rate of 11%. Complete the cells in the amortization schedule below.
1. Calculate Payment PMT = Principal / PVIFA Principal = $200,000 PVIFA = [ ( 1 / ( 1 + i ) ^ n ) - 1 ] / i PVIFA = [ (1/(1 + 0.11) ^ 25) - 1 ] / 0.11 PMT = - $23,748.05 2. Interest Owing = Principal Owing x i = 200,000 x 0.11 = $22,000 3. Payment = PMT =23,748.05 4. Principal Repayment = PMT - Interest Owing = $23,748.05 - $22,000 = $1748.05 5. Principal Owing at End of Year = Previous Principal Owing - PR = $200,000 - $1748.05 = $198,251.95 Until = 0
Prepare an amortization table for a $3,300 two-period loan assuming a 6% interest rate and annual payments. The annual payment will be?
1. Calculate Payment PMT = Principal / PVIFA Principal = $3,300 PVIFA = [ ( 1 / ( 1 + i ) ^ n ) - 1 ] / i PVIFA = [ (1/(1 + 0.06) ^ 2) - 1 ] / 0.06 PMT = - $1,799.94 2. Interest Owing = Principal Owing x i 3. Payment = PMT 4. Principal Repayment = PMT - Interest Owing 5. Principal Owing at End of Year = Previous Principal Owing - PR Until = 0
Two years ago you signed a 4-year lease on a car. The sticker price on the car was $22,000, you made no down payment, the lease rate was 3%, and the monthly payments (starting at the time of signing) were $278.92. The buyout was $10,560 due at the end of the lease term. Now (two years after signing the lease but just before the 25th lease payment) a new car has been released with better styling and 20% more horsepower. You want to get out of your old lease and lease the new car. The car dealer is happy to take your old car from you and cancel the lease if you have positive equity in the car. Equity is defined as the market value of the car (today) minus the principal outstanding. The market value of your old car is $17,380. What is the value of your equity in the car?
1. Calculate Principal Principal = DP + PMT x PVIFAdue + Buyout x PVIF Now solve for the Principal remaining using the equation of value for a lease, where Down Payment=$0, PMT=$278.92, PVIFAdue=23.324145, and PVIF=0.941835. 2. Equity = Market Value − Principal Owing Thus, the principal remaining on the lease payments is $16,451.35. Now find the equity by subtracting the principal remaining from the resale market value. Equity =Market Value−Principal Owing = $17,380−$16,451.35 = 928.65 dollar(s) Thus, the value of the equity in the car is $928.65.
Mister Greenjeans wants to borrow $4.1 million to purchase farm land to grow soy beans. He has approached the agriculture services division at his bank to arrange a farm mortgage. The bank has quoted him a rate of 7.5% over a 25 year amortization with quarterly payments of $90,535. Mister Greenjeans anticipates that he will have to borrow an additional $0.574 million after 12 years for land improvement costs. This extra principal will be added to the balance owing on the mortgage. If Mister Greenjeans continues making quarterly payments shown above, how long will it take him to pay off the mortgage? (Express your answer in years and round to the nearest integer value.)
1. Calculate j (periodic rate) 2. Solve for the number of years using the condition that the present value of the principal amounts is equal to the present value of the payments. PV of Principal=PV of Payments The principal has two parts—the original mortgage on the land (which is already given in terms of present value) and the land improvement costs. The latter is discounted by PVIFj, n×m, where j=0.018577 and n×m=12×4. The mortgage payments are multiplied by PVIFAj, n×m, where j=0.018577, m=4, and n needs to be solved for. Original Principal+Land Improvement Costs×PVIF = PMT×PVIFA $4,100,000+$574,000×PVIF0.018577, 12×4 = $90,535×PVIFA0.018577, n×4 The formula for the PVIFj, n×m for the land improvement principal is given below, where j is the periodic interest rate, n is the number of years to be discounted, and m is the number of payments per year. PVIFj, n×m=1(1+j)n×m Solve for PVIFj, n×m, where j=0.018577, n=12, and m=4. PVIFj, n×m = 1(1+j)n×m = 1(1+0.018577)12×4 = 0.4133280.413328 (Round to six decimal places.) The formula for the present value interest factor for an ordinary annuity, PVIFAj, n×m, is given below, where j is the periodic interest rate, n is the number of years, and m is the number of payments per year. PVIFAj, n×m=1j×1−1(1+j)n×m Combine all the known variables into one equation where n, the number of years of mortgage payments, is the only unknown variable which needs to be solved for. $4,100,000+$574,000×PVIF0.018577, 12×4 = $90,535×PVIFA0.018577, n×4 Start isolating n. $4,100,000+$574,000×0.413328 = $90,535×1j×1−1(1+j)n×m $4,100,000+$574,000×0.413328 = $90,535×10.018577×1−1(1+0.018577)n×4 1−11.018577n = 0.018577($4,100,000+$574,000×0.413328)$90,535 1−11.018577n = 0.8899660.889966 (Round to six decimal places.) Continue isolating n. 1−11.018577n = 0.889966 11.018577n = 1−0.889966 1.018577n = 11−0.889966 Take the natural logarithm of both sides and solve for n. 1.018577n = 11−0.889966 ln1.018577n = ln11−0.889966 n×ln(1.018577) = ln10.110034 n = ln10.110034ln(1.018577) n = 120120 (Round up to the nearest integer.) It will take 120 quarters for Mister Greenjeans to pay off the mortgage. Divide the number of quarters by 4 to determine the total number of years it will take to pay off the mortgage. 1204=3030 (Round to the nearest integer.) Thus, it will take 30 years for Mister Greenjeans to pay off the mortgage. Question is complete. Tap on the red indicators to see incorrect answers. VIDEO
You want to drive a new car but you really can't afford it. So, your plan is to lease it for four years and then borrow the money for the buyout. The loan for the buyout will have monthly (end-of-period) payments over a four-year term with a rate of 5.5%. How much interest will you pay over the eight-year period? (Ignore taxes.)
1. Find Payment for loan PMT = Principal / PVIFAdue PVIFA = ( i + i/m ) x .... = 42.000... PMT = $808.33 Thus, the monthly loan payment is $808.33. The total amount paid is the sum of the number of lease payments (48) times the lease payment amount ($867.99) and the number of loan payments (48) times the loan payment amount ($808.33).
Your mortgage has an amortization period of 25 years and a quoted rate of 5.6%. You elect to make 12 payments a year. What is the periodic rate on the loan which we use to calculate the payments?
1. Find j j = [( 1 + ( i / 2 ) ) ^ ( 2 / m )] - 1 j = [( 1 + ( 0.056 / 2 ) ) ^ ( 2 / 12 )] - 1 j = 0.004613136 or 0.4613%
A bank quotes you a rate of 6.5% on a home mortgage. You elect to make 52 payments per year. What periodic interest rate should you use to calculate your mortgage payments?
1. Find j j = [( 1 + ( i / 2 ) ) ^ ( 2 / m )] - 1 j = [( 1 + ( 0.065 / 2 ) ) ^ ( 2 / 52 )] - 1 j = 0.001230874
Tips for Solving Time Value of Money Problems
1. Review the information given in the problem and jot down what is known. 2. Draw a timeline. 3. Select a focal date. There are only two directions that you can move money: forward or backward. If you pick a focal date at the end of the timeline, then you are moving cash forward (a future value). If you pick a focal date at the beginning of the timeline, then you are moving cash backward (discounting). 4. Determine whether the cash flows are lump sums or annuities. If you have a single cash flow or if there are multiple cash flows of varying amounts, then you are dealing with lump sums. If there are multiple cash flows with the same value, then you have an annuity. (If the annuity goes on forever, then you have a perpetuity.) 5. Determine the compounding frequency of the problem (e.g., annually, monthly, or weekly) and decide whether the cash flows are beginning-of-period or end-of-period.
Suppose you have two 3-year annuities, one beginning one period from now (with $200 payments) and another beginning four periods from now (with $100 payments). What are the present values of each of the annuities, and what is their combined PV? Assume an interest rate of 10%.
1. apply the present value method to the annuity
Think of the annuity in the figure as if it were a three-period ordinary annuity. If you solve for the future value of the annuity using the formula for a three-period ordinary annuity, then you will get the future value as of date ______.
2 The ordinary annuity future value formula generates the future value on the date of the last payment in the annuity.
Think of the annuity in the figure as a three-period annuity due. If you solve for the future value of the annuity using the formula for a three-period annuity due, then you will get the future value as of date ______.
3 Because 2 is the last payment but 3 is when that period ends. The annuity due future value formula generates the future value on the date after the last payment in the annuity.
loan amortization schedule
A schedule of equal payments to repay a loan. It shows the allocation of each loan payment to interest and principal.
Imbedded Annuity
An annuity within a cash flow stream that includes other payments other than the series of fixed payments
Present Value of a Level Perpetuity
Annuities are streams of equal payments that go on for a fixed number of periods. If the annuity continues forever, it's called a perpetuity. The term perpetuity comes from the word perpetual, which means "continual or everlasting." PVperpetuity = PMT / i
Example 4.10 Amortized Loans, Interest, and the Reinvestment Rate Your friend, Ernest Defraud, wants to borrow $1,000 for 3 years. He has proposed an amortized loan with payments of $367.21. What is the future value of the payments if you can reinvest the first and second payments at 5%? How much interest do you earn from reinvestment and how much interest is included in the payments themselves?
At year 3, the future value of the payments can be calculated by finding the future value of a three-period ordinary annuity at a rate of 5%. The future value of the annuity is PMT = $367.21 i = 5% n = 3 FVannuity = PMT x [(1 + i)^n - 1]/ i FVannuity = $367.21 x [(1 + 0.05)^3 - 1]/ 0.05 FVannuity = $1,157.63 The amount of interest earned due to reinvestment is just the difference between the future value of the three payments and the sum of the three payments. The sum is what you would have at year 3 if you didn't earn any interest by reinvesting. For example, if you stuffed the payments under your mattress! Reinvestment interest=$1,157.63−(3×$367.21)=$56 The amount of interest that is included in the three payments is the difference between the sum of the three payments and the principal of the loan. Interest in payments=(3×$367.21)−$1,000=$101.63 This example highlights two important points about amortized loans: Interest on amortized loan payments comes in two forms: (1) each payment contains interest and (2) the lender receives the payments before the end of the term and so can earn interest by investing them. The future value of the amortized loan payments is only equal to the balloon payment if the lender can reinvest intermediate payments at the loan rate (try it with a different rate to see). Thus, lenders only earn i% on amortized loans if they can reinvest at i%. This is called the reinvestment rate assumption.
Future Value of an Ordinary Annuity
FVannuity = PMT x [(1 + i)^n - 1]/ i where [(1 + i)^(n) - 1 ]/ i is the future value interest factor of an annuity
Future Value of an Annuity Due
FVannuity due = PMT x [(1 + i)^n - 1]/ i x ( 1 + i )
When calculating lease payments, you only need to include the buyout in the equation of value if the lessor actually intends to buy out the car at the end of the lease.
False because the equation of value for leases equates the principal of the transaction (the price of the car) to the present value of all of the payments under the lease that are necessary to pay off that obligation in full. To fully discharge this obligation, a lessor must make the monthly payments and the pay the buyout. Lessors have an option to buy out the vehicle at the end of the lease, but it is not relevant to the mathematics of solving for the payments.
The "term" of a mortgage is like the maturity of the obligation: the date at which the loan must be fully repaid.
False because the term is shorter than (or equal to) the amortization period and is the period over which the interest rate (and payments) is agreed on. A mortgage term is the length of time in which the parameters of a mortgage have legal effect, not the date at which the loan must be fully repaid.
Types of Mortgage Loans
Fixed Rate Mortgage (FRM) the interest rate, and hence the payments, remains fixed for the term of the loan. Adjustable Rate Mortgage (ARM) In an ARM, the interest rate is fixed for a period of time, after which it is periodically (quarterly or monthly) adjusted up or down to some market rate. Common rates include the prime rate (in Canada) and the London Interbank Offer Rate (LIBOR) (in the U.S.).
ordinary annuity
If the first payment occurs at the end of the period
You are considering the purchase of a BMW M5. You will borrow the money from BMW Financial Services. The terms of the deal are outlined below: BMW M5 RWD, 500hp, 0-100 in 4.7s MSRP = $90,000 Term = 24 months APR = 3.5% Down Payment = $0 Monthly Payments = $3,888.24 The amortized loan payments are a blend of interest and principal. What is the total amount of interest you would pay over the life of the loan? Assume that taxes are zero.
Interest in payments=Number of payments×Payment−Principal Number of Payments = 24 Payment = $3,888.24 Principal = $90,000 Interest in payments = 24 x $3,888.24 - $90,000 Interest in payments = $3,317.76
A car dealership offers you a four year car lease with monthly payments of $417 and a buyout of $23,050. When is the first payment due?
Now Car lease payments are structured as an annuity-due. The first payment is due on signing. The last payment is due at the beginning of the last month.
Ordinary Annuity vs. Annuity Due
Ordinary annuity - number of interest periods is equal to number of payments Ordinary = in arrears (start later) Appropriate for Receipts: Rental lease payments, life insurance premium, etc. you you pay your September credit card bill for charges incurred in August) PV: "what would I have to pay you right now to get 10% of your earnings for the rest of your life?" FV: If I have $1.00 today, what can I reasonably expect this to be worth in a year's time through investment? Annuity due - number of interest periods is one less than no of payments Annuity due = start now (-1 = ordinary) Appropriate for Payments: Housing loan, payment of mortgage, coupon bearing bonds, etc. PV: If your lease starts in January, you can budget less for your December bill, since you have 11 months to "invest" it to get to the desired level of rent. FV:
Example 4.4 PV of Ordinary Annuity You won the lottery! The prize is $100 paid at the end of each of the next 4 years. What is the present value of the lottery annuity if the interest rate is 10%?
PMT = $100 i = 10% n = 4 PVannuity = PMT x (1 - ( 1 / ( 1+i )^ n ) --------------- i PVannuity = 100 x (1 -( 1 / (1+0.10)^ 4) --------------- 0.10 PVannuity = $316.99 We've just discovered that the present value of a four-period stream of $100 cash flows is $316.99. This is the present value of the lottery prize. You would be indifferent between the four payments of $100 or a $316.99 lump sum today.
Suppose you win the lottery. The winnings consist of 20 equal annual payments of $50,000. You decide to save all of this money for your retirement, and you deposit it into an account that earns 8% per year. What is the amount of your retirement nest egg?
PMT = $50,000 i = 8% n = 20 FVannuity = PMT x [(1 + i)^n - 1]/ i FVannuity = $50,000 x [ (1 + 0.08) ^ 20 - 1 ] / 0.08 FVannuity = $2,288,098.215 FVannuity = $2,288,098.21 A good rule of thumb is to carry enough significant digits so that you can round your solution to the nearest penny.
Assume that you retire at age 65 and plan to pass away after your thirtieth withdrawal on your 95th birthday. So, enter "30" in the "Years of Retirement Income" box. Assume an interest rate of 8%. Enter "75000" as your "Periodic Income in Retirement." How much do you need at retirement?
PMT = $75,000 n = 95-65 (30) i = 8% PVannuity = PMT x (1 - ( 1 / ( 1+i )^ n ) --------------- i PVann = 75,000 x (1-(1/ (1+0.08)^ 30) --------------- 0.08 PVannuity = $844,333.75
FV of annuity vs FV of annuity due You have been offered an investment opportunity that will pay you $5,000 per year for 6 years. What is the value of this stream of cash flows at the end of year 6 if you receive payments at the end of each year and you require a return of 7%? What if you receive payments at the beginning of each year?
PMT = 5,000 i = 7% n = 6
Solving for Payments in a Future Value Annuity Problem
PMT = FVannuity / [(1 + i)^n - 1]/ i
Example 4.11 Solving for Amortized Loan Payments: The Car Loan The price of a Ferrari California is $190,000. Let's say that your bank will lend you the money to buy the car at a rate of 6% for a 6-year term with monthly payments. What are the monthly payments on the car loan?
PMT = Principal / PVIFA Principal = $190,000 PVIFA = ( 1 + 0.06/12 ) ^ 6x12 - 1 ------------------- 0.06/12 PVIFA = $60.3395 PMT = $190,000 / $60.3395 PMT = $3,148.85
What will the equal annual end-of-year payments need to be in order to fully amortize a $29,000, 12% loan over a 6-year period? How much total interest is paid by the borrower over the life of the loan?
PMT = Principal / PVIFA Principal = $29,000 PVIFA = [ ( 1 / ( 1 + i ) ^ n ) - 1 ] / i PVIFA = [ ( 1 / (1 + 0.12) ^ 6) - 1 ] / 0.12 PMT = 29000/ PMT = $7053.55 The total interest paid by the borrower over the life of the loan is: Total interest=(Payment×Number of payment)−Loan amount The total interest paid by the borrower over the life of the loan is $13321.28
Amortized Loan Payments
PMT = Principal / PVIFA Principal = PMT x PVIFA
Example 4.15 Solve for Mortgage Loan Payments What are the monthly payments on a mortgage for $300,000 with an amortization period of 25 years and a quoted rate of 5%?
PMT = Principal / PVIFA Principal = $300,000 1. Find j j = [( 1 + ( i / 2 ) ) ^ ( 2 / m )] - 1 2. Find PVIFA PVIFA = ( i / j ) x [ 1 - ( 1 /((1+j) ^nxm)] (j, 300) 3. Divide Principal by PVIFA
To finance your new house, you have obtained a 35-year, $590,000 mortgage. The bank has offered you an interest rate of 4.9%. You can't decide on a payment frequency. Complete the following table to help you make a decision.
PMT = Principal / PVIFA Principal = $590,000 1. Find j j = [( 1 + ( i / 2 ) ) ^ ( 2 / m )] - 1 j = [( 1 + ( 0.049/ 2 ) ) ^ ( 2 / each )] - 1 2. Find PVIFA PVIFA = ( i / j ) x [ 1 - ( 1 /((1+j) ^nxm)] (j, m ) 3. Divide Principal by PVIFA
For each of the following cases, calculate the present value of the annuity, assuming the annuity cash flows occur at the end of each year.
PV = 33,000 i = 13% n = 8 PVannuity = PMT x (1 - ( 1 / ( 1+i )^ n ) --------------- i PVannuity = 33,000 x (1 -( 1 / (1+0.13) ^ 8 ) --------------- 0.13 PVannuity = $158,359.42 PV = 24,000 i = 9% n = 18 PVannuity = PMT x (1 - ( 1 / ( 1+i )^ n ) --------------- i PVannuity = 24,000 x (1 -( 1 / (1+0.09) ^ 18 ) --------------- 0.09 PVannuity = $210,135.00
Present Value of an Ordinary Annuity
PVannuity = PMT x (1 - ( 1 / ( 1+i )^ n ) --------------- i
An endowed faculty chair is created when a benefactor makes a donation of sufficient size that earnings from the donation pay the salary and benefits of a professor forever. How much would have to be donated to endow a chair in your name if the salary and benefits were $78,000 and the interest rate was 7%? Assume that the first salary (with benefits) is paid in one year's time.
PVperp = ? PMT = $78,000 i = 7% PVperpetuity = PMT / i PVperpetuity = 78,000/ 0.07 PVperpetuity = $1,114,285.71
Uriah Heep celebrated his 18th birthday by opening a savings account at the Thames River Bank and depositing $3,300. He continued to deposit the same amount on every subsequent birthday until he was 31 years old. After depositing $3,300 on his 31st birthday Uriah decided to abandon his savings plan. He never saved again, but he left the accumulated savings in the bank account. The bank paid an interest rate of 11.5%. When Uriah turned 65, he withdrew the money from the bank. What was the amount of his withdrawal?
Part 1 : 18th to 31th FV of Ordinary Annuity** i = 11.5% n = 14 PMT = $3300 FVannuity due = PMT x [(1 + i)^n - 1]/ i FVannuity due = 3300 x [(1 + 0.115)^14 - 1]/ 0.115 FVannuity due = $103,028.1454 Part 2 : 31th to 65 FV of annual interest PV = $103,028.1454 n = 34 i = 11.5% $103,028.1454 x ( (1.115) ^ 34 ) FV = $4,171,587.653
Balloon Loan: Solve for the Balloon Payment Your friend, Ernest Defraud, wants to borrow $1,000 for 3 years. Let's assume that you want to earn interest at 5% compounded annually. What is the balloon payment? How much interest do you earn on the loan?
Principal = $1,000 n = 3 i = 5 % Balloon = FVn = Principal×(1+i)^n Balloon = 1000 x ( 1.05 ^ 3 ) Balloon = 1000 x ( 1.05 ^ 3 ) Balloon = $1,157.63 Since the principal of the loan is $1,000, the amount of interest earned by the lender is Interest earned=$1,157.63−$1,000=$157.63.
You are going to borrow $100,000 to fund the start of your new restaurant. The bank wants to be repaid with a balloon payment in 5 years. They will charge 11% interest. How much will your payment be?
Principal = $100,000 n = 5 i = 11% Balloon = FVn = Principal×(1+i)^n Balloon = 100,000 x ( 1.11 ^ 5 ) Balloon = $168,505.82
You are going to borrow $101,000 to fund the start of your new restaurant. The bank wants to be repaid with a balloon payment in 3 years. They will charge 10% interest. How much will your payment be?
Principal = $101,000 n = 3 i = 10% Balloon = FVn = Principal×(1+i)^n Balloon = 101,000x ( 1.10 ^ 3 ) Balloon = $134,431.00
amortized loan equation of value
Principal = PMT x PVIFA
Calculating Lease Payments
Principal0=Down payment+PMT×PVIFAdue+Buyout×PVIF Notice that the down payment is undiscounted since it occurs at time 0 when the lease contract is signed. Notice also that the buyout is included. It is included regardless of whether you choose to exercise the buyout option because the left-hand side has to equal the right-hand side of the equation. The left-hand side is the value of the car at time zero. Thus, the right-hand side has to be the present value of all of the payments required to own the car outright through the lease.
Example 4.13 Payments on a Car Lease What are the monthly payments on the BMW 650i Cabriolet? The BMW 650i Cabriolet has a manufacturer's suggested retail price (MSRP) of $105,500 (assume that this price includes all incidentals, like freight and dealer preparation, necessary to drive the car off of the lot). BMW Financial Services offers a 3-year lease with a rate of 7.9%. At the end of the 3-year term, there is a buyout of $58,025. The lease requires monthly payments with the first payment due when the contract is signed.
Principal0=Down payment+PMT×PVIFAdue+Buyout×PVIF Principal = $105,000 Down Payment = 0$ PVIFAdue = $32.1692 PMT = ? Buyout = $58,025 PVIF = 0.7896 PMT = (Principal - Down Payment - Buyout * PVIF) / PVIFAdue PMT = $1855.29
You are going to lease a new car. The sticker price for the car is $47,000. The term of the lease is 48 months, the rate on the lease is 6.3%, and the buyout is $19,000. If you decide not to make a down payment, then what are the monthly lease payments (before tax)?
Principal0=Down payment+PMT×PVIFAdue+Buyout×PVIF m = 12!!! n = 4 Principal = $47,000 Down Payment = 0$ PVIFAdue = $42.33195011 PMT = ? Buyout = $19,000 PVIF = 0.777757261 PMT = (Principal - Down Payment - Buyout * PVIF) / PVIFAdue PMT = ($47,000 - 0 - $19,000 * 0.777757261) / $42.33195011 PMT = $757.21
A car has a sticker price of $47,500. The car has a 100 hp engine and can accelerate from 0 to 60 mph in 15.8 seconds. The lease rate is 3.2%. The term of the lease is three years. The buyout is $23,750 at the end of the lease. Assume that the lease has three annual payments with the first payment due on signing. a) What are the before-tax lease payments assuming no down payment? b) With leases, sales tax is paid on the lease payments and buyout. If the sales tax rate is 4%, then what is the present value of the taxes paid on the lease when discounted at the lease rate? c) If you buy the car for its sticker price, then you will pay sales tax on the purchase price of the car. Which method of purchase generates larger retail tax payments on a present value basis?
Principal=$47,500 Down Payment=$0 Buyout=$23,750 n=3 i=0.032 m=1 i/m=0.00290901 a) What are the before-tax lease payments assuming no down payment? PMT = (Principal - Down Payment - Buyout * PVIF) / PVIFAdue PVIF = PVIFAdue = PMT = $8,903.74 If the sales tax rate is 4%, then what is the present value of the taxes paid on the lease when discounted at the lease rate? To find the present value of the taxes paid on the lease payments, apply the same formula for the present value of a lease used above, multiplying the payment and buyout amounts by the tax rate. Recall that the tax rate is 0.04, Down Payment=$0, PMT=$8,903.74, PVIFAdue=2.907938, Buyout=$23,750, and PVIF=0.909831. PVtaxes = (0.04)Down Payment+(0.04)PMT×PVIFAdue+(0.04)Buyout×PVIF = (0.04)$8,903.74×2.907938+(0.04)$23,750×0.909831 = $1,900.00 Which method of purchase generates larger retail tax payments on a present value basis? c) The sales tax on the cash purchase is 0.04×$47,500=$1,900.00 This is the same as the present value of the tax payments on the lease. Thus, a lease generates the same tax liability as a cash purchase.
You would like to lease a car. The car costs $58,000. You have $4,000 for a down payment. The dealer has offered you a 48-month lease with monthly payments of $775.23. At the end of the term, the buyout for the lease is 46% of the purchase price. What is the APR on this lease?
Principal=$58,000 Down Payment=$4,000 PMT=$775.23 Buyout=$26,680 n=4 m=12 VIDEO
You tried to sneak a cappuccino into the movie theatre. It spilled on your leg and scalded you. Your lawyer says that you can expect a payout from the coffee shop's insurance company of $8,000 per year for 12 years. The only catch is that the first payment won't happen until the court case is over in 2 years. What is the present value of your settlement as of today? Assume an interest rate of 2.7%.
Solve this in two steps. First solve for the PV of the annuity at time 1 PVan = PMT x [1 - (1 / (( 1 + i )^n)) ] / i PMT = $8,000 n = 12 i = 2.7% PVan = PMT x [1 - (1 / (( 1 + i )^n)) ] / i PVan = 8,000 x [1 - (1 / (( 1 + 0.027 )^12 )) ] / 0.027 PVan = $81,007.15 Bring that PV back to time 0. FV =$81,007.15 i = 0.027 n = 1 PV = FV / ( 1 + i ) ^ n PV = $81,007.15 / ( 1 + 0.027 ) ^ 1 PV = $78,945.62
present value interest factor (PVIF)
The acronym is useful because it is shorter than writing out the whole formula and here we use it interchangeably with the formula.
Amortization Period
The actual number of years it will take to repay a mortgage loan in full. This can be well in excess of the loan's term. For example, mortgages often have 5-year terms, but 25-year amortization periods.
Mortgage Term
The agreed-upon amount of time to pay off a mortgage. A typical residential amortization period is 25 years. Thus, it will take 25 years for the borrower to fully repay the loan. The term is less than or equal to the amortization period. At the end of the term, the interest rate is renegotiated and loan payments are recalculated.
Car Leases
The benefit of a lease is that it spreads payments out over time, thus keeping the periodic payments low. You buy $47,475more info worth of the car over the first three years and then you have the option to either walk away or buyout the remainder of the car at the end of the 3-year term
You are 30 years old today. You want to retire at the age of 60. You expect to live until age 85. You would like to have a monthly income of $11,000 per month in retirement. How much do you have to save per month during your working years in order to achieve your retirement goal? Assume end of period payments. Assume an annual interest rate of 3.5% in retirement and 5.5% during your working life.
The equation of value for a retirement planning problem equates the savings during the working life to the withdrawals during retirement. Any focal date will yield the same answer (for the monthly savings), but the most obvious focal date is the retirement date. STEP 1: Solve for the present value of the retirement withdrawals as of your retirement date. These withdrawals are end-of-period, so we calculate the present value of an ordinary annuity with monthly payments. What is the present value at retirement of the withdrawals during your retirement years? PV = PMT x (1 -( 1/ ( 1+ (i/m) )^n x m) --------------- i / m PMT = $11,000 i = 3.5% n = 25 PVannuity = 11k x (1-(1/( 1+0.035 / 12 )^25x12) --------------- 0.035/12 PVannuity = $2,197,259.71 STEP 2: Solve for a set of end-of-period annuity payments whose future value (at retirement) is equal to the present value from STEP 1. The generalized equation for finding the payment given the future value of an ordinary annuity is given as: FVannuity = $2,197,259.71 PMT = $11,000 m = 12 i = 5.5% / 12 i = 0.00458 n = 360 PMT = FVannuity / [(1 + i)^( n ) - 1]/ i PMT = $2,197,259.71 / [(1 + 0.00458)^ ( 360 ) - 1 ]/0.00458 PMT = $2,405.03
Your wealthy aunt passed away recently. In her will she promised you $1,400 at the end of every year, in perpetuity, starting in year 2. If the interest rate is 6%, then what is the present value of the bequest?
The first payment in the perpetuity occurs at year 2. If we find the present value of the perpetuity using the formula, then we get the present value 1 year before the first payment, year 1. We then have to discount that 1 more year to Year 0. PMT = $1,400 i = 6% PV = [ PMT / i ] x [ 1 / ( 1 + i ) ] PV = [1400/ 0.06 ] x [ 1 / ( 1 + 0.06 ) ] PV = $22,015.58
Congratulations! You won the lottery. You will receive two payments of $15,000 at the end of each of the next 2 years, followed by two more annual payments of $20,000 at the end of the following years. The interest rate is 4%. What is the present value of your prize?
The first two payments of $15,000are a simple two-period ordinary annuity. If we treat the second two payments of $20,000 as an ordinary annuity and find their present value, then we get the present value as of Year 2, 1 year before the first annuity payment. We then have to discount that 2 more years to Year 0. 1. Find Present Value of each ordinary annuity PVannuity = PMT x (1 -( 1 / ( 1+i )^ n ) --------------- i PMT = $15,000 i = 4% n = 2 PVannuity = 15k x (1-( 1/(1+ 0.04)^2) --------------- 0.04 PVannuity = $24,291.42 2. Find Present Value of each ordinary annuity PVannuity = PMT x (1 -( 1 / ( 1+i )^ n ) --------------- i PMT = $20,000 i = 4% n = 2 PVannuity = 20k x (1-( 1/(1+ 0.04)^2) --------------- 0.04 PVannuity = $37,721.89 3. Discount to year 0 = 1 / ( 1 + i ) ^ n = 1 / ( 1 + 0.04 ) ^ 2 = 0.9245 4. Add Step 1 + (Step 2 x Step 3) The present value is $63,167.4363,167.43. (Round to the nearest cent.)
Future Value of Annuity Due vs. Future Value of Ordinary Annuity
The only difference is the last term (1+i). Because of their similarity, we won't introduce a separate annuity due interest factor. Instead, we simply multiply the ordinary annuity factor by (1+i).
Homer expects to receive $250 next year. He also expects to receive $250 in two years and again in three years. What is the present value of those payments one year from today? Assume an interest rate of 5.6%.
The present value of an ordinary annuity is: PMT = $250 i = 5.6% n = 2 because it starts in one year PVannuity = PMT x (1 - ( 1 / ( 1+i )^ n ) ----------- i PVannuity = 250 x (1-(1 /(1+0.056)^2) --------------- 0.056 PVannuity = $460.93 + PMT PVannuity = $710.93
True or False? You know the future value of an annuity. It you invest that amount today, then you can make periodic withdrawals equal to the annuity payments and have nothing left at the end of the annuity's life.
The statement is false. If your goal is to generate a future stream of payments, then you need the present value of the stream, not the future value. Present values are always smaller than the future value so if you had the future value you would actually have more than you needed.
Wally, president of Wally's Burgers, is considering franchising. He has a potential franchise agreement that would allow him to receive 22 end-of-year payments starting one year from now. The first two payments would be $28,000 and $25,000 in one and two years respectively, and then $15,000 per year after that for 20 years. If Wally requires a return of 12.0%, what is the present value of this stream of cash flows?
There are two lump-sum cash flows and one 20-year annuity. To solve for the present value of mixed cash flow streams, we find the present value of the annuity, then add the present value of any other cash flows. 1. Find PV of first payment For the first payment received in year 1, substituting FV=$28,000, i = 0.120 n = 1 (one year from now) into the present value PV = (FV) / (1 + i ) ^ n PV = (28,000) / (1 + 0.120 ) ^ 1 PV = $25,000.00 Thus, the present value of $28,000 discounted at 12.0% for 1 year is $25,000.00 2. Find PV of second payment For the second payment received in year 2, substituting FV=$25,000, i = 0.120, n = 2 (two years from now) into the present value equation PV = (FV) / (1 + i ) ^ n PV = (25,000) / (1 + 0.120 ) ^ 2 PV = $19,929.85 Thus, the present value of $25,000 discounted at 12.0% for 2 years is $19,929.85. 3. Find PV of annuity The annuity portion of the cash flows has 20 payments with the first payment at year 3, so the applying the PV annuity formula will give the present value as of year 2 (one year before the first payment). Substituting PMT=$15,000, i=0.120 n=20 into the equation for the present value of an ordinary annuity gives the amount at the end of year 2 PVannuity = PMT x (1 - ( 1 / ( 1+i )^ n ) --------------- i PVann =$15,000 x (1-(1/(1+0.120)^20) --------------- 0.120 PVannuity = $112,041.65 4. Bring PV to time zero To bring the present value to time zero, the PV of the annuity will have to be discounted another two years: FV=$112,041.65 i = 0.120 n = 2 (payments started end of year 2 ) PV = (FV) / (1 + i ) ^ n PV = (112,041.65) / (1 + 0.120 ) ^ 2 PV = $89,318.92 5. Add all streams of cash flow Finally, the present value of this stream of cash flows is the sum of the present values of the payments from years 1 and 2 and the annuity in years 3 through 22: PV0 = $25,000.00 + $19,929.85 + $89,318.92 PV0 = $134,248.77
You are 62 and planning your retirement. If you retire immediately, then your Canadian Pension Plan (CPP) benefit is $1,835 per month. If you retire at age 67 then the benefit is $2,256 per month. If you expect to live to age 85, then when should you retire and start claiming retirement benefits? Since the benefits are certain as long as you live, the appropriate discount rate is the risk free rate. Since we are not incorporating inflation into the benefits we must use a real interest rate. The real risk free rate is 2% per annum. Assume that benefits are collected at the end of each month. Express your answer as the difference in the future value of the benefits. That is, subtract the future value of benefits if you retire early from the future value of benefits if you retire later.
There are two lump-sum cash flows and one 23-year annuity. To solve for the present value of mixed cash flow streams, we find the present value of the annuity, then add the present value of any other cash flows. 1. Find PV of first payment ($1,835) For the first payment received in year 1, substituting FV= $1,835 i = 0.02 n = 1 into the present value PV = (FV) / (1 + i ) ^ n PV = ($1,835) / (1 + 0.02 ) ^ 1 PV = $1,799.02 Thus, the present value of $1,835 discounted at 2.0% for 1 year is $1,799.02 2. Find PV of second payment ($2,256) For the second payment received in year 2, substituting FV=$2,256 i = 0.02, n = 2 (two years from now) into the present value equation PV = (FV) / (1 + i ) ^ n PV = (2,256) / (1 + 0.02 ) ^ 2 PV = $2,168.40 Thus, the present value of $2,256 discounted at 2% for 2 years is $2,168.40. 3. Find PV of annuity To find the present value of the ordinary annuity with payments in years 3 through 20, you first need to compute the value of the annuity in year 2 and then discount the value for 2 years to find its present value in year 0. USE EXCEL PMT= FIRST NUMBER i=0.120 n=23 PVannuity = PMT x (1 - ( 1 / ( 1+i )^ n ) --------------- i PVann =$20,00 x (1-(1/(1+0.02)^23) --------------- 0.02 PVannuity = $365,844.08 4. Bring PV to time zero To bring the present value to time zero, the PV of the annuity will have to be discounted another two years: FV=$365,844.08 i = 0.02 n = 2 (payments started end of year 2 ) PV = (FV) / (1 + i ) ^ n PV = (365,844.08) / (1 + 0.02 ) ^ 2 PV = $351,637.91 5. Add all streams of cash flow Finally, the present value of this stream of cash flows is the sum of the present values of the payments from years 1 and 2 and the annuity in years 3 through 22: PV0 = $1,799.02 + $2,168.40 + $351,637.91 PV0 = $355,605.33
The cash flows shown in the figure are ________________.
This is a three-period ordinary annuity. Because the cash flows are at the end of the period. if it started at 0 it would be annuity due.
You are going to receive $100 today, next year, and 2 years from now. This is a ________________.
This is a three-period annuity due.
Determine the equal, annual, end-of-year payment required over the life of the following loans to repay them fully during the stated term.
Use PMT = Principal/PVIFA
Future value interest factor
a factor multiplied by today's savings to determine how the savings will accumulate over time FVannuity = PMT x [ (1 + i) ^ (n) - 1 ]/ i
annuity
a series of equal regular deposits
You borrow $160,000. The loan is structured as an amortized loan to be repaid over 9 years with 24 (end-of-period) payments per year. The lender is charging you a rate of 4.7% APR. a. What are the amortized loan payments? b. After two years you want to repay the remaining principal and end the loan. How much do you owe after two years? c. After two years how much interest have you paid? d. Today is the two-year anniversary of the start of the loan and you just made your loan payment. How much interest will be included in your next loan payment?
a. What are the amortized loan payments? 1. Calculate Payment PMT = Principal / PVIFA Principal = $200,000 PVIFA = [ ( 1 / ( 1 + i ) ^ n ) - 1 ] / i PVIFA = [ (1/(1 + 0.11) ^ 25) - 1 ] / 0.11 PMT = - $23,748.05 b. After two years you want to repay the remaining principal and end the loan. How much do you owe after two years? c. After two years how much interest have you paid? d. Today is the two-year anniversary of the start of the loan and you just made your loan payment. How much interest will be included in your next loan payment? VIDEO
Some banks allow you to skip a loan payment and roll it into your principal. This is especially attractive in January when the Christmas VISA bill arrives. Consider the following simplified example. You renovated your house last year and borrowed $90,000. The term of the loan is three years, the rate is 11% (APR), and the annual (end-of-year) payments are $36,829.18. You can't afford to make the first payment. You want to roll it into your principal. Your bank has offered to increase the size of the second and third payments to allow you to do this. Answer the following two questions. a. What will your new (second and third) loan payments be? (Hold everything else in your loan constant, e.g., remaining amortization period and interest rate.) b. Compare the amount of interest in the two sets of loan payments. How much more interest do you pay over the life of the loan (if you skip the first payment) compared to what you would have paid if you hadn't skipped the first payment?
a. What will your new (second and third) loan payments be? (Hold everything else in your loan constant, e.g., remaining amortization period and interest rate.) b. Compare the amount of interest in the two sets of loan payments. How much more interest do you pay over the life of the loan (if you skip the first payment) compared to what you would have paid if you hadn't skipped the first payment?
annuity due
an annuity for which the cash flows occur at the beginning of the period
Balloon Loan
any loan in which the final payment is larger than the preceding payments Balloon = FVn = Principal×(1+i)^n
Periodic Rate for a Mortgage (j)
basically EIR for mortgage You should carry a lot of significant digits in your periodic rate because the loan payment calculation is very sensitive to errors in the interest rate. You should carry enough significant figures so that your payment calculation is accurate down to pennies.
Sabrina deposits $700 in an account at the beginning of each year for 6 years. If the account pays 2% interest annually, how much money will be in Sabrina's account at the end of 6 years?
beginning of each year = annuity due i = 2% n = 6 PMT = $700 FVannuity due = PMT x [(1 + i)^n - 1]/ i FVannuity due = 700 x [(1 + 0.02)^6 - 1]/ 0.02 x (1.02) FVannuity due = $4503.998
What is the future value of a perpetuity with annual payments of $1,000 compounding at 10%?
future value of a perpetuity is infinity if it was present value: PMT = $1,000 i = 10% PVperpetuity = PMT / i PVperpetuity = 1000 / .10 PVperpetuity = $ 10,000
Example 4.14 Solve the Periodic Rate for a Mortgage A bank quotes a mortgage rate of 5%. You choose monthly payments. What monthly periodic rate should you use to calculate the mortgage loan payments?
i = 0.05 m = 12 j = [ ( 1 + ( i / 2 ) ) ^ ( 2 / m ) ] - 1 j = [ ( 1 + ( 0.05 / 2 ) ) ^ ( 2 / 12 ) ] - 1 j = 0.004123915 Keep in mind that j = 0.004123915 is a monthly rate in decimal form
Suppose you want to know the future balance in your interest-bearing bank account after 3 years if you make three annual $100 deposits. We will assume that your bank pays interest at the rate of 10% each year.
i = 10% n = 3 PMT = 100 FVannuity = PMT x [(1 + i)^n - 1]/ i FVannuity = 100 x [ ( 1 + 0.10 ) ^ (3) - 1 ] / 0.10 FVannuity = $331
A young couple wishes to accumulate $16,000 at the end of 5 years so that they can make a down payment on a house. What should their equal end-of-year deposits be to accumulate the $16,000, assuming a 3% rate of interest?
i = 3% n = 5 FVannuity = 16,000 PMT = FVannuity / FVIFAi,n PMT = FVannuity / [(1 + i)^n - 1]/ i PMT = 16,000 --------- [ ( 1 + 0.03 ) ^ (5) - 1 ] / 0.03 PMT = $3,013.67 You will have to save $3,013.67 per year to have $16,000 for a down payment in 5 years.
Calculate the future value of the following annuity
i = 4% n = 4 PMT = $18,000 FVannuity = PMT x [(1 + i)^n - 1]/ i FVannuity = 18,000 x [(1 + 0.04)^4 - 1]/ 0.04 FVannuity = $76,436.35
You hope to retire in 35 years and you need $1,000,000 in the bank to live the life you want after retirement. You can earn 7% per year on your investments. How much must you save every year to reach your goal?
i = 7% n = 35 FVannuity = 1,000,000 PMT = FVannuity / FVIFAi,n PMT = FVannuity / [(1 + i)^n - 1]/ i PMT = 1,000,000 / [ ( 1 + 0.07 ) ^ (35) - 1 ] / 0.07 PMT = $7,233.96 You will have to save $7,233.96 per year to have $1 million in your retirement fund after 35 years.
Mortgages
long-term debt obligations created to finance the purchase of real estate
You have been making $2,400 per-year contributions to your company retirement plan for 25 years. It now has a balance of $248,000. What average compounded return have you earned?
n = 25 FV = $248,000. PMT = -($2,400) PV = 0 cpt i = 10.33%
Jed wants to borrow $1,000 from you. He is proposing to repay you with three annual payments of $325.52 starting one year from now. In addition, he will make a final lump-sum payment of $120 three years from today. What rate of return are you earning on the loan?
n = 3 payments PV = -($1,000) PMT = $325.52 FV =$120 cpt i i = 4.50%
Let's look at the retirement question from another viewpoint. Suppose you have saved $562,889 and want to withdraw $50,000 per year for 30 years. What rate of return would you need to earn?
n = 30 PV = $562,889 PMT = -($50,000) cpt i i = 8% If you start with $562,889 and invest the remaining balance each year at 8%, then you will be able to withdraw $50,000 per year for 30 years.
Finding the Interest Rate in a PV Annuity Problem
n = ? PV = ? PMT = -(?) cpt i
You've graduated from college and landed a good job. You want to replace your car, but don't want to take out a car loan. Instead, you decide to invest $100 per month in the stock market and hope to earn 8%. If the market performs as you're hoping, how many years will it take to accumulate $40,000? Ignore taxes.
n = ln ( 1 + ((FV x i/m ) / PMT)) --------------------------- x (1/m) ln ( 1 + ( i / m ) ) Financial Calculator i = i/m pv = 0 fv = -fv pmt = pmt cpt n / 12 i = 8/12 *not decimal for calc* pv = 0 fv = -40,000 pmt = $100 cpt n then / 12 =16.30 = 17 years
Present Value of an Annuity Due
the present value of a series of equal amounts to be withdrawn or received at equal intervals, periodic rents occur at the beginning of the period PVannuity due = PMT x [(1 + i)^n - 1]/ i x ( 1 + i )