Math Skill 1 - Time Value of Money Calculations

Setting up your financial Calculator

In order to correctly solve the time value of money calculations you will be performing, it is vital that you know how to use your financial calculator. A financial calculator does not work in the same way as a traditional calculator when performing time value of money calculations.

To make sure you get the correct calculations, regardless of which calculator you use, you need to make sure your calculator is set up as follows. If you are using a calculator other than the one provided at midasclassroom.com (linked below), then you may need to consult your manual for instructions on how to make these adjustments to your calculator.

http://www.fncalculator.com/financialcalculator?type=tvmCalculator

1. The calculator needs to be in “End of Year” mode, not in “Beginning of Year” mode.

2. The calculator needs to be set to use monthly compounding.

3. If possible, your calculator should show four decimal places on calculations. Solutions have been calculated using figures that have been rounded to four decimal places, so to match the solutions precisely you will need to do the same.

In addition, we will use the following assumptions when solving the problems in this workbook:

4. All interest rates in these problems are given as the rate per year. If a problem says an account “pays 5%”, that means 5% per year.

5. Payments on loans are assumed to all be made once per month.

Every time value of money (abbreviated as “TVM”) calculation can be solved by completing all of the following steps. Please note that skipping any step is likely to result in an inaccurate calculation, and your calculator does not always warn you that you have made a mistake!

Solving TVM Calculations

1. Identify all your key data.

• What is N? PMT? FV? PV? I/Y?

2. Convert time period data to months.

• Applies to PMT, N, I/Y only

• Annual rate / 12 = Monthly Rate*

• Number of Years * 12 = Number of Months

*NOTE: If you are using the calculator app at fncalculator.com, then always enter the rate per year and leave the compounding as "Monthly". Other calculators will require you to adjust.

3. Determine the signs (+/-)

• Money that you are gaining the ability to spend gets a “positive” (+) sign.

• Money that you are losing the ability to spend today gets a “negative” (-) sign.

4. Enter data into calculator and compute.

Step 1 is to identify all of our key components.

Present value (abbreviated as “PV”). This refers to the value of a cashflow at the start of the time period in question. It is not always the value in the present. Sometimes the time period we are thinking about starts today, but sometimes it doesn’t start today. So don’t get hung up on thinking that if there is some money today, that it must be a present value. Rather, think of it as the value at the start of our time period.

Future value (abbreviated as “FV”). This refers to the value of a cashflow at the end of the time period in the question. Like present value, sometimes this cashflow will happen in the future, but sometimes it can occur today or even in the past. To identify your FV, just look for a cashflow that is going to occur at the end of the time period.

Payment (abbreviated as “PMT”). This refers to cashflows that repeat. If you have a monthly payment or a payment per year, that’s a PMT. If you see any amount of money that repeats regularly, it can only be a PMT.

N (or sometimes NPER) is the term. This refers to the amount of time between the start and the end of the period in question. It is usually easy to spot because it is expressed in units of time (months, years, weeks, etc.).

I/Y is the interest rate. Whether you are earning the interest rate or paying the interest rate (or neither), this is the rate at which interest is charged. IT is given as a percentage per year. The interest rate will always be reported per year. For example, a problem with an interest rate of 6% means that we need to use 6% per year as our interest rate.

Step 2 requires us to convert our I/Y, N, and PMT into monthly units.

Most financial calculators will assume that the amounts we put into our calculator all happen at the same frequency. If you have set your calculator to "annual" compounding (as discussed in step 2 of "setting up your financial calculator" above), then you calculator is definitely making this assumption.

In order to solve correctly, we need to input the interest rate per month, the number of months, and the payment per month. If we put in the payment per month, number of years, and interest rate per year, then we will get a bad answer. To convert the interest rate, simply divide it by 12. For example, 6% per year / 12 = 0.5% per month. To convert N, simply multiply it by 12. For example, 5 years x 12 = 60 months.

Step 3 is to determine the direction of cashflows (positive or negative)

This step is by far the most difficult in this calculation, so let’s discuss it a bit more before looking at some examples.

PV, FV, and PMT are all cashflows. They represent where cash is “flowing” from one place to another. It can be from one of your accounts to another of your accounts. It can be flowing from your account to the bank. It can flow from your friend’s account to your account.

Flows of cash are much like flows of water. They need a source, and they need a destination. Just like you couldn’t tell if a bathtub is filling up or draining down without knowing whether or not water is flowing into or out of the tub, your calculator cannot tell if your account is filling up or draining down unless you tell it which direction the cash is flowing.

We tell the calculator this direction using positive and negative signs.

A negative sign is needed when we lose the ability to spend a dollar, even though we may not have actually "lost" any money. A negative cash flow occurs when the money is not being spent on goods and services, but instead being sent out of our wallets and into another account. It doesn't matter if the dollar is earning interest from an investment or paying interest on a loan. It's still a negative sign because you removed your ability to spend it.

A dollar held in a retirement account is not available to spend even though it is still our dollar and we haven't “lost” it. Therefore, any cashflow which puts money into a retirement account (or any account where a person will not be spending the money) must be entered as a negative number so the calculator knows which way the cash is flowing. Likewise, a dollar sent to the bank to pay a car loan is also negative because we lose the ability to spend that dollar on ourselves.

A positive sign is needed when we are gaining the ability to spend money in that time period. For example, money withdrawn from a retirement account is money that we gain the ability to spend (for example, on presents for the grandchildren). Therefore, in a problem where you are withdrawing money from an account so spend it, the PMT will be positive.

A dollar we gain from a loan is a dollar that we are gaining the ability to spend. For example, when you take out a student loan, you gain the ability to pay your tuition for the year. That's a positive cashflow. When you borrow money using a car loan, you gain the ability to buy a car. That's a positive cashflow.

The signs on your PV, FV, and PMT will vary depending on the situation. Students are sometimes tempted to try to memorize when cashflows should be positive and when they will be negative. While this is possible, it is also extremely difficult because there are simply too many possible scenarios to have to memorize.

The key is not to try to memorize every possible situation - the key is to recognize when you are gaining the ability to spend, and when you are losing the ability to spend. That determines the sign on the cashflow. Focus on understanding the logic of identifying the sign for your cashflows. Working through the practice problems in the textbook will help the pattern of these signs become much clearer to you. That takes time and practice. But this skill, once mastered, will enable you to perform almost limitless calculations in an unending variety of situations. It is by far the most essential and widely used of all of the financial skills.

Example Questions

Example 1

Today Leo Burke purchased an investment grade gold coin for $150,000. He expects it to increase in value at a rate of 7% for the next 5 years. How much will the coin be worth at the end of the fifth year if his calculations are correct?

Solution:

N=5*12=60

I/Y=7

PV=-150,000

PMT=0

Compute: FV= $212,639

Note: In this problem, the PV is negative. It's negative because Leo sacrificed his ability to spend this money in the beginning of the 5-year period (i.e. when the money for the PV flowed from their wallet to the gold coin) in the question. He can’t spend that money to buy food or pay rent. That makes it a negative sign.

Example 2

Lucy Brown wants to give her daughter $35,000 to start her own business in 10 years. How much should she invest today at an annual interest rate of 9% compounded annually to have $35,000 in 10 years?

Solution:

N=10*12 = 120

I/Y=9

Compute: PV= -$14,277

PMT=0

FV=35000

Note: The FV must be positive in this problem. It's positive because Lucy will withdraw the money the account so that she can give it to her daughter. Gaining the ability to spend the money means a positive sign is appropriate. Since she gains this ability at the end of the 10-year timer period in question, this is a positive FV.

Example 3

Karlie has $1,510.25 in credit card bills. If she stops making any additional charges to her credit card that charges 19.5% APR, and makes the minimum monthly payment of $50 per month, how many months will it take her to pay off her credit card?

Solution:

Compute: N= 41.87 months

I/Y= 19.5

PV= $1,510.25

PMT= -50

FV= 0

*NOTE: Since we entered the payment per month and the interest rate per month into the calculator, the answer for N will be provided as a number of MONTHS. Therefore, the answer to this question is 41.87 months.

Example 4

Your parents deposited $1,000 into a college fund for you 15 years ago. Your grandmother has also been depositing $50 into that fund every month. If the fund earns 6% annual interest, then what is the current balance of the account?

Solution:

N= 15*12 = 180

I/Y= 6%

PV= -1,000

PMT= -50

Compute: FV= 16,995

Note: the PV and PMT are both negative in this problem. The PV is negative because your parents lost the ability to spend $1,000 at the beginning of the 15-year period. The PMT is negative because your grandmother lost the ability to spend $50 each month during the 15-year period. The FV is positive because you are gaining the ability to spend the money on tuition at the end of the 15-year period.

Example 5

You’ve been saving for retirement for the last 22 years. You started your account with a $5,000 tax return and have since contributed $230 per month. If your current balance is $165,000, what was the average annual interest rate that you earned?

Solution:

N =22*12=264

PV = -5,000

PMT = -230

FV = 165,000

Compute: I/Y = 0.5769% per month * 12 = 6.9228% per year

Note: Interest rates should always be expressed per year. Since we used N as a number of months and the PMT per month, the number produced by your calculator is the interest rate per month. We must convert it to an annual interest rate by multiplying by 12 in order to get the interest rate per year, which is a number that is meaningful to use.

Example 6

Your parents just retired this year. If they retired with $900,000 in the retirement account, and the account earns 2% per year and they withdraw $6,000 per month for living expenses, how much money will be left in their retirement account after 10 years?

Solution:

N =10*12=120

PV = -900,000

PMT = +6,000

I/Y = 2 / 12 = 0.1667

Compute: FV = $302,789

Watch it in action!

Practice Problems

1. A car costs $19,900 with an interest rate of 4.9%. If the car is financed over 5 years, what will the monthly payment be?

2. You acquired $20,000 in student loans while in school, and now you have just graduated and must begin making payments. Your interest rate is 6.8%. How much will your payment be if you make monthly payments over the standard 10-year repayment term?

3. You desire to have $50,000 to buy a car in 5 years. If your money can earn 5%, how much must you save each month to reach this goal?

4. You want to buy a $150,000 house. You decide that a 4.25% 15-year mortgage is for you. Considering a 10% down payment, how much will your monthly payment be?

5. If you have a mortgage balance of $100,000 financed at 4.9% with a $892.86 payment, how many scheduled payments are left to pay off the mortgage?

6. You have a credit card balance of $2,400. You quit using the card and make monthly payments of $50. If your interest rate is 18%, how long will it take you to pay off the balance?

7. If you save $3000 a year for retirement in an account earning 4%, how long will it take you to accumulate $250,000?

8. If you save $4000 a year for retirement in an account earning 8%, how long will it take you to accumulate $250,000?

9. You want to pay cash for a car that costs $32,000. If you save $500 a month in an account earning 4% how long will it take you to reach be able to buy the car?

10. If your grandma gave you $1,000. What rate will you need to earn, compounded annually, if you plan to pull out $2,000 in 10 years?

11. Your house is currently valued at $100,000. At what rate will it need to appreciate (annually) to be valued at $125,000 in 5 years?

12. What rate is required to double an initial investment of $1 in 9 years?

13. You plan on paying cash for a $200,000 house at retirement, in 27 years. If you can earn 6% annually, how much do you need to invest today?

14. You open an account that pays 0.5 percent interest per year. How much money must you deposit today to have $10,000 in 20 years?

15. You can earn 8 percent interest, compounded annually. How much must you deposit today to withdraw $10,000 in 6 years?

16. If you deposit $1 per month in an account that pays 12% interest, compounded monthly, what will be the balance in the account after two years if you make no withdrawals?

17. Housing values are appreciating at a rate of 3 percent a year. Approximately how much will your $100,000 house be worth in ten years if this rate of appreciation continues?

18. You deposit $2,000 in a savings account that pays 9 percent interest, compounded daily. How much will your account be worth in 15 years?

19. You just put $1,000 in a bank account which pays 6 percent annual interest, compounded quarterly. How much will you have in your account after 3 years?

20. If you put $1 in a savings account earning 10% annually, how much will it be worth in 100 years?

Solutions:

1. A car costs $19,900 with an interest rate of 4.9%. If the car is financed over 5 years, what will the monthly payment be?

N = 5*12, I/Y = 4.9, PV = 19,900, FV = 0, PMT = -374.63 per month

2. Upon graduating school you must repay $20,000 in student loans. Your interest rate is 6.8%. How much will your payment be if you make monthly payments over the standard 10-year repayment term?

N = 10*12, I/Y = 6.8, PV = 20,000, FV = 0, PMT = -230.16 per month

3. You desire to have $50,000 to buy a car in 5 years. If your money can earn 5%, how much must you save each year?

N = 5*12, I/Y = 5, PV = 0, FV = 50,000, PMT = -735.22 per month

4. You want to buy a $150,000 house. You decide that a 4.25%, 15-year mortgage is for you. Considering a 10% down payment, how much will your monthly payment be?

N = 15*12, I/Y =4.25, PV = 90%*150000, FV = 0, PMT = -1,015.60 per month

5. If you have a mortgage balance of $100,000 financed at 4.9% with a $892.86 payment, how many scheduled payments are left to pay off the mortgage?

PMT = -892.86, I/Y = 4.9, PV = 100,000, FV = 0, N = 150 MONTHLY PAYMENTS

6. You have a credit card balance of $2,400. You quit using the card and make monthly payments of $50. Assuming your interest rate is 18%, how long will it take you to pay off the balance?

PMT = -50, I/Y = 18, PV = 2,400, FV = 0, N = 85.5 MONTHS

7. If you save $3000 a year for retirement in an account earning 4%, how long will it take you to accumulate $250,000?

PV = 0, PMT = -3000 / 12, I/Y = 4, FV = 250000, N = 440.6 MONTHS / 12 = 36.7 years

8. If you save $4000 a year for retirement in an account earning 8%, how long will it take you to accumulate $250,000?

PV = 0, PMT = -4000 / 12, I/Y = 8, FV = 250000, N = 2 69.65 Months / 12 = 22.47 years

9. You want to pay cash for a car that costs $32,000. If you save $500 a month in an account earning 4% how long will it take you?

PV = 0, PMT = -500, I/Y = 4, FV = 32000, N = 58.1 MONTHS

10. If your grandma gave you $1,000. What rate will you need to earn annually if you plan to pull out $2,000 in 10 years?

N = 10*12, PV = -1,000, FV = 2,000, PMT = 0, I.Y = 0.5793% per month * 12 = 6.95% per year

11. Your house is currently valued at $100,000. At what rate will it need to appreciate (annually) to be valued at $125,000 in 5 years?

N = 5*12, PV = -100,000, FV = 125,000, PMT = 0, I.Y =0.3726 per month * 12 = 4.56% per year

12. What rate is required to double an initial investment of $1 in 9 years?

N = 9*12, PV = -1, FV = 2, PMT = 0, I.Y = 0.6439 per month * 12 = 7.73% per year

13. You plan on paying cash for a $200,000 house at retirement, in 27 years. If you can earn 6% annually, how much do you need to invest today?

N = 27*12, I/Y = 6, FV = 200,000, PMT = 0, PV = -39,739

14. You open an account that pays 0.5 percent interest per year. How much money must you deposit today to have $10,000 in 20 years?

N = 20*12, I/Y = 0.5, FV = 10,000, PMT = 0, PV =-9,048.56

15. You can earn 8 percent interest, compounded annually. How much must you deposit today to withdraw $10,000 in 6 years?

N = 6*12, I/Y = 8, FV = 10,000, PMT = 0, PV = -6,198.32

16. If you deposit $1 per month in an account that pays 12% interest, compounded monthly, what will be the balance in the account after two years if you make no withdrawals?

N = 2*12, I/Y = 12, PV = 0, PMT = -1, FV = 26.97

17. Housing values are appreciating at a rate of 3 percent a year. Approximately how much will your $100,000 house be worth in ten years if this rate of appreciation continues?

N = 10*12, I/Y = 3, PV = -100,000, PMT = 0, FV = $134,935.35

18. You deposit $2,000 in a savings account that pays 9 percent interest. How much will your account be worth in 15 years?

N = 15*12, I/Y = 9, PV = -2,000, PMT = 0, FV =7,676.09

19. You just put $1,000 in a bank account which pays 6 percent annual interest. How much will you have in your account after 3 years?

N = 3*12, I/Y = 6, PV = -1,000, PMT = 0, FV = 1,196.68

20. If you put $1 in a savings account earning 10% annually, how much will it be worth in 100 years?

N = 100*12, I/Y = 10, PV = -1, PMT = 0, FV = 21,124.03