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Guías de estudio > Finite Math

Reading: Loans, Mortgages, and Sinking Funds

Let's start off this section by simplifying what might at first seem ambiguous, complicated, and impossible to understand:
  • A loan is money borrowed that is to be repaid along with interest.
  • A mortgage is a larger loan specifically reserved for the purchase of a home.
Purchasing a home is an important and often intimidating decision. If not executed properly, the word "foreclosure" looms directly overhead. Understanding why a mortgage functions the way it does is mainly a result of knowing how a bank defines a mortgage. The premise is basic: you take out a loan for the price of the house and any additional fees. A regular monthly payment is calculated. At the end of the month, interest is charged based on the remaining balance, and then the payment is deducted from the total balance. This process is repeated month-after-month until the principle balance reaches 0. In theory, this is straightforward, but most people either sell their home or refinance their home (take out a mortgage to pay off their current mortgage—usually due to an offer for a lower interest rate). Let's illustrate with an example: Suppose you finance a home in the amount of $100,000 at 4.25% compounded monthly for 30 years. The lender (the loaner of the funds, often a bank) calculates your payment to be $491.94. This means that at the beginning of the first month, you will receive the balance of $100,000. You will be charged a monthly rate of = .00354. This amounts to a balance of $100,000(1.00354) = $100,354. The monthly payment is applied, and the balance is reduced to $100,354 – $491.94 = $99,862.06. This means that only $137.94 of your payment went to pay down the mortgage. We show a few of the first months in the table below:
Month Initial Balance Interest Balance w/Interest Payment New Balance Principal Paid
0 100,000 .00354(100000) = 354 100,354.00 –491.94 99,862.06 137.94
1 99,862.06 .00354(99862.06) = 353.51 100,215.57 –491.94 99,723.63 138.43
2 99,723.63 .00354(99723.63) = 353.02 100,076.65 –491.94 99,584.71 139.92
3 99,584.71 .00354(99584.71) = 352.53 99,937.24 –491.94 99,445.30 139.41
4 99,445.30 .00354(99445.30) = 352.04 99,797.34 –491.94 99,305.40 139.90
5 99,305.40 .00354(99305.40) = 351.54 99,656.94 –491.94 99,165.00 140.40
6 99,165.00 .00354(99165.00) = 351.04 99,516.04 –491.94 99,024.10 140.90
7 99,024.10 .00354(99024.10) = 350.55 99,374.65 –491.94 98,882.71 141.39
When this table shows the entire lifetime of the loan, we call it an amortization table, as we are amortizing the balance, which means we are gradually reducing what is owed. Notice that the amount going towards principal begins to increase. Why is this? Since the balance is being gradually reduced the dollar amount of interest on a smaller balance is smaller. Often when people say that they are paying "all interest and no principal" they usually mean that their loan is still new and so much of their payment goes to pay interest. As we see in this example, the percentage of the first payment that went to paying principle was about 28%. We might now want to know how it is that the bank calculates the payment. Do they just make up a number that sounds good? Not in the least. The bank wants to ensure that its contract with you is finalized by a pre-arranged point in time, such as 30 years. To come up with a formula for this based on numerical patterns, let's use the numbers from our example. We will represent the balance at time n by Pn. Note that we are here using the payment to find the payment, which seems like a logical clash. We do this only to see a patter before we establish a general formula for actually computing it. We know that: P0 = 100,000 Since this was our initial balance. The balance on month later was: P1 = 100,000(1.00354) – 491.94 = 99,862.06 One month after that: P2 = 99,862.06(1.00354) – 491.94 = 99,723.63 Note that P2 can be written in terms of P1 = 99,862.06. We use substitution to show this more clearly: [latex-display]\displaystyle{P}_{{2}}={P}_{{1}}{\left({1.00354}\right)}-{491.94}[/latex-display] [latex-display]\displaystyle{P}_{{2}}={\left[{100},{000}{\left({1.00354}\right)}-{491.94}\right]}{\left({1.00354}\right)}-{491.94}[/latex-display] [latex-display]\displaystyle{P}_{{2}}={100},{000}{\left({1.00354}\right)}^{{2}}-{491.94}{\left({1.00354}\right)}-{491.94}[/latex-display] It appears that we're beginning to see a pattern. Let's see if this pattern persists for the next term month's balance: [latex-display]\displaystyle{P}_{{3}}={99},{723.63}{\left({1.00354}\right)}-{491.94}[/latex-display] [latex-display]\displaystyle{P}_{{3}}={P}_{{2}}{\left({1.00354}\right)}-{491.94}[/latex-display] [latex-display]\displaystyle{P}_{{3}}={\left[{100},{000}{\left({1.00354}\right)}^{{2}}-{491.94}{\left({1.00354}\right)}-{491.94}\right]}{\left({1.00354}\right)}-{491.94}[/latex-display] [latex-display]\displaystyle{P}_{{3}}={100},{000}{\left({1.00354}\right)}^{{3}}-{491.94}{\left({1.00354}\right)}^{{2}}-{491.94}{\left({1.00354}\right)}-{491.94}[/latex-display] This is very interesting! Notice that we can also express the balance by "growing" the original balance by the monthly growth factor 3 times while subtracting off each of the prior and current payments, which are also increased by the growth factor 2, 1, and 0 times, respectively. To make a connection here, let's write the second part as a difference between the balance and the payments: P3 = 100,000(1.00354)3 – (491.94(1.00354)2 + 491.94(1.00354) + 491.94) The subtraction term can be reduced to the formula for an annuity with term 3 months! (Review the previous section on annuities to make this connection more clear.) So, [latex-display]\displaystyle{P}_{{3}}={100},{000}{\left({1.00354}\right)}^{{3}}-\frac{{{491.94}{\left[{\left({1.00354}\right)}^{{3}}-{1}\right]}}}{{.00354}}[/latex-display] To write this generically for month N: [latex-display]\displaystyle{P}_{{N}}={100},{000}{\left({1.00354}\right)}^{{N}}-\frac{{{491.94}{\left[{\left({1.00354}\right)}^{{N}}-{1}\right]}}}{{.00354}}[/latex-display] And to write this generically for any initial amount, interest rate, and payment: [latex-display]\displaystyle{P}_{{N}}={P}{\left({1}+{i}\right)}^{{N}}-\frac{{{P}{M}{T}{\left[{\left({1}+{i}\right)}^{{N}}-{1}\right]}}}{{i}}[/latex-display] Very cool! BUT, how do we find the payment? We know that at time N (the full term of the loan), we want the balance to be 0. That is, substitute PN = 0: [latex-display]\displaystyle{0}={P}{\left({1}+{i}\right)}^{{N}}-\frac{{{P}{M}{T}{\left[{\left({1}+{i}\right)}^{{N}}-{1}\right]}}}{{i}}[/latex-display] Finally, we isolate , the regular monthly payment: [latex-display]\displaystyle\frac{{{P}{M}{T}{\left[{\left({1}+{i}\right)}^{{N}}-{1}\right]}}}{{i}}={P}{\left({1}+{i}\right)}^{{N}}[/latex-display] [latex-display]\displaystyle{P}{M}{T}{\left[{\left({1}+{i}\right)}^{{N}}-{1}\right]}={i}\times{P}{\left({1}+{i}\right)}^{{N}}[/latex-display] [latex-display]\displaystyle{P}{M}{T}=\frac{{{i}{P}{\left({1}+{i}\right)}^{{N}}}}{{{\left[{\left({1}+{i}\right)}^{{N}}-{1}\right]}}}[/latex-display] We're done! This raises two important formulas worth mentioning:

Loan Payment Formula

To calculate the payment, PMT, required to pay off a loan of amount P in N periods at a periodic rate of i, use [latex-display]\displaystyle{P}{M}{T}=\frac{{{i}{P}{\left({1}+{i}\right)}^{{N}}}}{{{\left({1}+{i}\right)}^{{N}}-{1}}}[/latex-display]

Balance Remaining on a Loan Formula

To calculate the balance remaining on a loan after month n, use [latex-display]\displaystyle{P}_{{n}}={P}{\left({1}+{i}\right)}^{{n}}-\frac{{{P}{M}{T}{\left[{\left({1}+{i}\right)}^{{n}}-{1}\right]}}}{{i}}[/latex-display] Commonly, financial institutions refer to the nominal annual interest rate on a loan as the Annual Percentage Rate, or APR for short.

Example 1

Alexis takes out a loan for a new car in the amount of $31,300. She receives a rate of 7.2% APR. The loan is to be repaid by regular monthly payments over the course of 5 years.
  1. Find the required monthly payment.
  2. Confirm that the balance is 0 on the ending date.
  3. Calculate the total cost of the car.

Solutions

  1. By using the formula, we have that [latex]\displaystyle{i}=\frac{{.072}}{{12}}={.006}[/latex] We can confirm this answer by using TVM Solver: Note that the present value is positive since she is receiving a car and the future value is 0, as there should be a 0 balance after 60 months.
  2. Using the Balance Remaining on a Loan formula, [latex-display]\displaystyle{P}_{{60}}={31300}{\left({1.006}\right)}^{{60}}-\frac{{{622.74}{\left[{\left({1.006}\right)}^{{60}}-{1}\right]}}}{{.006}}=-{.34}[/latex-display] Due to rounding error, it appears that she would have paid $0.34 more than she owed.
  3. Since payments include principle and interest, we calculate that she paid a total of $622.74(60) = $37,364.40, which is $4,164.40 paid toward interest alone.

Example 2

In a clever advertising scheme, an electronics store runs a television commercial offering "no payments for 24 months and full repayment not due for 72 months." What the store does not mention is that interest is still accrued over the course of the first 24 months. Assuming the rate is 11.99% APR and that a person takes "advantage" of the offer, what will he have paid, in total for a $1,500 flat-screen television?

Solution

Notice that when no payment is made, the balance formula reduces to the regular compound interest formula: [latex-display]\displaystyle{P}_{{n}}={P}{\left({1}+{i}\right)}^{{n}}-\frac{{{0}{\left[{\left({1}+{i}\right)}^{{n}}-{1}\right]}}}{{i}}[/latex-display] [latex-display]\displaystyle{P}_{{n}}={P}{\left({1}+{i}\right)}^{{n}}[/latex-display] Although we will use the TVM Solver, it is handy to understand what is happening in terms of the mathematics involved. We consider a time diagram to show what the television will cost after the first 24 months: The individual will begin making payments after 24 months. We calculate the payment required to pay off the balance in 48 months: Labeling this in the diagram gives: Since the individual made 48 payments of $50.14, he paid a total of 48($50.14) = $2,406.72 for a $1,500 television. Thus, he paid $906.72 in interest.

Example 3

On October 25, 2011, it would have been possible to finance a new $300,000 home at 4.3% APR with a 10% down payment (source: www.bankrate.com) when financed for 30 years. A down payment is simply a payment made in advance to reduce the amount financed. As a result of the risk offset created by the buyer, the bank offers a slightly lower rate. Suppose that in 14 years the interest rate falls to 3.8% and you decide to refinance.
  1. How much would you still owe on the principle at the time of reinvestment?
  2. How much did the house cost?

Solutions

Since a 10% down payment will be made initially, a total of $300,000(0.90) = $270,000 will be financed. We compute the first payment and set up a time diagram to describe the investment period: Notice that we use 360 as the number of months. We originally did not anticipate that the home would be refinanced. Thus, the payment is computed on the assumption that the home is financed for the entire home period. Otherwise, if we had used 216 for (18 years), we would've had the home paid off at the time of refinancing: Since the home is actually not paid off after 216 months, we must check to see what the outstanding balance is after 216 months: Thus, there is still a balance to be paid of $150,101.24. To be consistent with the idea of an outflow, we could interpret this as the additional outflow (lump sum payment) that would need to be made in order to "break even." We have, To get payment for the remaining 144 months, we use TVM Solver: The payment has been reduced to $1,299.65. This makes sense, since the interest rate is now lower. We complete the time diagram: We proceed to answer the questions at hand:
  1. You would still owe $150,101.24
  2. In total, was paid in out-of-pocket payments. With the down payment, the cost of the house was $505,758, which represents $205,758 in interest.
Another option for paying off a mortgage is the sinking fund method. This entails financing a home with an interest-only option. This type of option allows you to only pay interest on the home, thus never paying down the balance of the loan over its life, requiring a balloon payment, or a very large payment, at the end of the term. How, then, would you pay off the principle? Theoretically, you invest what you would pay towards principle into an investment account, so that these payments accumulate to the balance of the home at the end of the term. This sounds crazy, doesn't it? It does have benefits: your money is liquid over the course of the mortgage. That is, you have more spending money in the event you have an emergency or other financial obligation. Recall that payments made on a mortgage come out-of-pocket and, once spent, cannot be received.

Example 4

You consider a 30-year, interest-only loan to finance a $300,000 home. No down payment is required and the interest rate is 4.45% APR. Determine the interest rate that would be required on a sinking fund in order for the balloon payment to be the amount owed.

Solution

We first determine the payment required: Next, we calculate the amount of interest that will be charged each month. Remember that the exact interest amount will be paid off each month, so that the interest amount remains the same. We first find the monthly rate of [latex]\displaystyle\frac{{.0445}}{{12}}={.003708}[/latex]. Multiplied by the balance, we have .003708(300,000) = $1,112.50 of interest each month. Thus, you will save $1,511.16 – $1,112.50 = 398.66 each month, which will go to a liquid investment account. We will put this payment in 360 times. We calculate what interest rate is required for payments of this size to reach $300,000: Interestingly, this rate is about 4.45% (we have some rounding error). This is not surprising, since it should require at least as high of an interest rate as you are being charged in order to make up the difference. What is the point? Clearly, if you can find a rate that is higher than 4.45%, you can actually profit from the sinking fund. Additionally, you save nearly $400 per month, which can be liquid, should you require any of those payments. As a side note, sinking funds are often used by companies to establish a savings for some future repayment.

Practice Problems

    1. You open up a credit card through Best Buy at 14.99% APR compounded monthly in order to buy a new car stereo. The stereo costs you $492.78. You would like to repay it within 24 months, in order to avoid additional rate hikes.
      1. What will be your monthly payment?
      2. How much will the car stereo cost you, in total?
    1. A car is purchased for $34,000. It is financed at 7.99% APR for 72 months.
      1. Calculate your monthly payment in two ways:
        1. Using the Loan Repayment Formula
        2. Using TVM Solver
      2. What will be the total cost of the car?
      3. How much interest did the person pay over the loan period?
        1. Suppose the car is refinanced after 30 months due to the discovery of a lower interest rate, 5.5% APR for the remaining 42 months. How much lower will the payment be?
    1. A credit card is much like a loan, except that there are no strict terms of repayment other than the requirement that an individual should make a minimum monthly payment based on his remaining balance. Suppose a person has a credit card that charges 14.99% APR. Additionally, any fees that take place over the course of the repayment period can be considered a form of interest. Suppose a person starts with a $250 balance and accepts an offer from the bank to make a balance transfer of $700 from another credit card that has a higher interest rate. The fee for a balance transfer is 3% of the balance to be transferred. How long would it take to pay the credit card off with the card's required minimum payment of $30?
    1. Dizzy purchases a house for $200,000. He can either choose to make a required 5% down payment at 4.25% APR or a 10% down payment for a reduced rate of 4.10% APR. The home is to be financed with a 30-year fixed-rate mortgage.
        1. Calculate the payment on each of the two options.
        2. Create an amortization table for each of the two options by using Excel. For instructions on how to do this, watch the following video:
      1. Using your amortization table, determine the amount of time it will take you to see the extra 5% down payment returned to you. This can be thought of as the point at which you have seen a savings of interest dollars equivalent to this amount.
      2. Calculate your total interest paid for each option in two ways:
          1. Using TVM Solver
          2. Using your amortization table and the sum() function in Excel. To see more about the sum function, watch the following video:
  1. Go to www.realtor.com and select a home in the location of your choice that you would like to finance. Find the price (in green). Go to www.bankrate.com to determine the current national average mortgage rates. You will see a box on the right side of the screen that look like this: (No Video Solution—Answers Vary) Write down the 30-year and 15-year rates from the rate column.
    1. Calculate the payment for your home under each of the two options.
    2. What is the payment difference?
    3. How much will you save in the total cost by going with the 15-year mortgage over the 30-year mortgage?
    4. Discuss the pros and cons of each of the two options. Which one do you think you would choose?
    5. Suppose you go with the 30-year mortgage. However, after 20 years, you are able to refinance. What should your APR be for the last 10 years so that you pay just as much, in total, as the 15-year mortgage alone?
  2. Do a web search on the following terms: 5/1 ARM, 5/2/5 Cap Structure, mortgage points, and rate buy-down.
    1. Provide the definition of each of the four terms above in your own words. Give an example, if needed.
    2. Using your new knowledge, determine the balance after 7 years on a $90,000 financed house through a 5/1 ARM with a 5/2/5 Cap Structure on which you paid 1 point to get a .125% rate reduction. Suppose that the pre-point APR is 4.1% and that the rate maxes out (and stays maxed out) when the first renewal period takes place. Assume a 30-year pay-off period.
Milos Podmanik, By the Numbers, "The Mathematics of Finances," licensed under a CC BY-NC-SA 3.0 license. MathIsGreatFun, "4 3 P1 MAT217," licensed under a Standard YouTube license. MathIsGreatFun, "4 3 P2 MAT217," licensed under a Standard YouTube license. MathIsGreatFun, "4 3 P3 MAT217," licensed under a Standard YouTube license. MathIsGreatFun, "Amortization Table.avi," licensed under a Standard YouTube license. MathIsGreatFun, "Using Formulas in Excel.avi," licensed under a Standard YouTube license. MathIsGreatFun, "4 3 P4 MAT217," licensed under a Standard YouTube license.