Chapter Four -- Time Value of Money
Future Value
When you put your money in a savings account (or invest it in some fashion), you earn a certain return (sometimes called interest) in order to compensate you. Because of this, a dollar today is not worth the same amount as a dollar sometime in the future. Since you earn money on the dollar invested (or saved) today, you will have more than a dollar at some later future point. The specific amount that you will have at the future date is referred to as a Future Value.
Consider if you had $100 today and were able to earn 12% per year by putting that money in a savings account at XYZ bank. How much would you have in 1 year? 2 years? 3 years? At first, you might think that you would have $112 in 1 year, $124 in 2 years and $136 in 3 years as you would earn $12 per year in interest. However this is WRONG! It ignores the concept of compounding. After 1 year, you would indeed have $112. However, during the second year you earn 12% interest on the full $112 instead of only the $100 you started with. Therefore you will earn $13.44 in interest in the second year and have $125.44 in two years. During the third year you will earn $15.05 in interest and have $140.49 in 3 years. Therefore, the Future Value of $100 for 3 years at 12% is $140.49. $100 today is equivalent to $140.49 3 years from now assuming that you can earn 12% interest annually.
We have 3 ways to solve for the FV. The first is directly. Under this method, we use the following formula:
where
FV is the future value (in year n) for which we are trying to solve
PV is the present value (how much we have today)
k is the rate of return we are earning (also referred to as the compound rate, required return, or discount rate)
n is the number of years which we will be saving (or investing) the money.
The second method is to use the financial tables. Most textbooks include a series of time value of money tables that allow you to find present values and future values. However, these tables are cumbersome and don't allow us as much flexibility as other methods, so we are not going to cover the financial table approach in this class.
The third method (and the method we will use for this class) is to use the financial calculator or spreadsheet (we will focus on the financial calculators). Each financial calculator follows the same basic ideas, but the specifics are different for each brand of calculator. If this is your first time using your financial calculator, there are detailed instructions to set up your calculator by following this link.
The steps below are for the HP10B.
Step 1: Enter N
Step 2: Enter I/YR
Step 3: Enter PV
Step 4: Enter 0 for PMT
Step 5: Press the FV key
Note: The order of steps 1-4 is not important. The FV answer will appear as a negative number, ignore the negative sign. Also, if you are using a TI or the LeWorld calculator, you must alter step 5 to be two keys. First CPT and then FV.
EXAMPLE OF FUTURE VALUE USING THE FINANCIAL CALCULATOR
Find the Future Value of $350 invested for 25 years at 9.5% per year.
Step 1: 25 N
Step 2: 9.5 I/YR
Step 3: 350 PV Step 4: 0 PMT
Step 5: FV
You should get a solution of $3383.93
In other words, if we invest $350 today and let it compound at 9.5% per year for 25 years, we will have $3383.93 at the end of the 25th year.
Present Value
The flip side of Future Value is Present Value. Future value tells us how much a certain amount of money will be worth at some future date assuming a certain rate of return. But what if we know how much we are supposed to get at some point in the future and want to know what it is worth to us today? Now we must find the Present Value. Assume we are offered an opportunity to receive $200 at the end of two years (lets call it investment A). How much is this opportunity worth to us today assuming we could earn 8% by placing our money in a savings account? To answer this, we must ask how much we would need to place in a savings account today in order to have $200 at the end of the two years.
X(1.08)^2 = 200
X = (200)/((1.08)^2)
X = $171.47
If we had $171.47 today and placed it in a savings account earning 8%, we would have $200 in two years (the same as through investment A). Assuming that investment A had the same degree of risk as our savings account, then we would buy investment A if it was available for less than $171.47 and put our money in the savings account if investment A cost more than $171.47. We could say that the present value of investment A is $171.47.
We have 3 ways to solve for the PV. The first is directly. Under this method, we use the following formula:
where
FV is the future value (in year n) that we plan to receive
PV is the present value (how much it is worth to us today)
k is the rate of return we can earn elsewhere (also referred to as the compound rate, required return, or discount rate)
n is the number of years which we will have to wait before receiving the money.
The second method is again the financial tables which we will skip.
The third method is to use the financial calculator (or spreadsheet). Each financial calculator follows the same basic ideas, but the specifics are different for each brand of calculator. The steps below are for the HP10B.
Step 1: Enter N
Step 2: Enter I/YR
Step 3: Enter 0 for PMT
Step 4: Enter FV
Step 5: Press the PV key
Note: The order of steps 1-4 is not important. The PV answer will appear as a negative number, ignore the negative sign. Also, if you are using a TI or LeWorld, you must alter step 5 to be two keys. First CPT and then PV.
EXAMPLE OF PRESENT VALUE USING THE FINANCIAL CALCULATOR
Find the Present Value of $5000 received 15 years from today with a 9.5% discount rate.
Step 1: 15 N
Step 2: 9.5 I/YR
Step 3: 0 PMT
Step 4: 5000 FV
Step 5: PV
You should get a solution of $1281.62
In other words, if we are offered the opportunity to receive $5000 at the end of 15 years, that is equivalent to receiving $1281.62 today.
Annuities
The methods described above are for situations where we have a specific amount today and want to know what it is worth at some point in the future (FV) or when we plan to receive a certain amount at some point in the future and want to know what it is worth today (PV). These are referred to as lump sum situations because there is only one cash flow that we are discounting or compounding.
An annuity is different in that with an annuity we have the same exact amount being received (or paid) at the end of each period over a number of periods. For example, if we win the lottery, that is usually paid in equal installments over a twenty-five year period. A $1,000,000 jackpot would be paid at $40,000 per year for twenty-five years. This is said to be an annuity. (Technically the lottery is an "annuity due" because the first payment is paid today, or the beginning of the period, as opposed to the end of the period).
SOLVING FOR PRESENT VALUE OF AN ANNUITY:
An investment that pays $100 at the end of each year for 4 years is an annuity. If we wanted to know what that investment is worth to us today and we had a 10% discount rate, we would be finding the present value of that annuity.
Step 1: $100 PMT
Step 2: 10 I/YR
Step 3: 4 N
Step 4: 0 FV
Step 5: PV
You should get a solution of $316.99
Note: The order of steps 1-4 is not important. The PV answer will appear as a negative number, ignore the negative sign. Also, if you are using a TI or LeWorld, you must alter step 5 to be two keys. First CPT and then PV.
SOLVING FOR FUTURE VALUE OF AN ANNUITY:
We may alternatively be concerned with the future value of an annuity. For example, lets say we want to save $1000 per year (at the end of each year) for 10 years at 12%. How much will this be worth at the end of the 10th year?
Using a financial calculator:
Step 1: $1000 PMT
Step 2: 12 I/YR
Step 3: 10 N
Step 4: 0 PV
Step 5: FV
You should get a solution of $17,548.74
Note: The order of steps 1-4 is not important. The FV answer will appear as a negative number, ignore the negative sign. Also, if you are using a TI or LeWorld, you must alter step 5 to be two keys. First CPT and then FV.
Solving for PMT, I/YR, or N
Sometimes we will be trying to find something other than the present value or future value. For instance, we may want to know how much we have to save per year to reach a certain future value (or how much we must earn as a rate of return or how many years it will take). If we are using a financial calculator, these are relatively easy. For example, lets say we want to save $5000 per year and want to have $80,000 after 10 years. What rate of return must we earn?
Solution
Step 1: -5000 PMT
Step 2: 80000 FV
Step 3: 10 N
Step 4: 0 PV
Step 5: I/YR
Solution = 10.08%
Note: Either the PMT must be negative and the FV positive or the PMT positive and the FV negative. It doesn't matter which way you do it, but one must be negative and the other positive. Also, remember if you are using a TI or LeWorld you must hit CPT before I/YR in step 5.
Solving for N and PMT is done along similar lines. However this is a good point to discuss the role of the negative (+/- key) on your calculator when dealing with TVM. When you are using your financial calculator, it is important to recognize how the calculator "thinks." The calculator uses a framework of cash INFLOWS and cash OUTFLOWS. Cash INFLOWS (money coming to us) are positive and cash OUTFLOWS (money going away from us) are negative. In order to receive a cash INFLOW today, I have to balance that with a cash OUTFLOW later. Or to receive a cash INFLOW later, I need to balance that with a cash OUTFLOW today. Whenever you solve for I/Y or N, you must have both a cash INFLOW (positive) and cash OUTFLOW (negative). If you are using all three of the cash flow keys (PV, FV, and PMT) then you must be careful about which are INFLOWS (positive) and which are OUTFLOWS (negative). One factor that can make this easier (or more confusing) is that whatever is an INFLOW (positive) for us is an OUTFLOW (negative) for others and vice-versa. That is why in the above example I said it doesn't matter whether PMT is positive or negative as long as the FV is the opposite sign.
Videos of Using the 5-Key Approach to Solve TVM Problems on the Financial Calculators
Perpetuities
A Perpetuity is an annuity that lasts forever. While it is difficult to imagine a situation where we could buy a cash flow stream that will pay us a fixed amount per year through infinity, perpetuities can be useful tools when dealing with long, constant cash flow streams.
How much would a perpetuity of $100 be worth assuming a discount rate of 10%? Remember this is $100 per year forever. It would seem that it would be worth an infinite amount. However, consider what would happen if you had $1000 today and could put it in the bank to earn 10% interest. You would receive $100 per year and never touch your principal. You would essentially be buying a $100 perpetuity (assuming the bank didn't change the interest rate). Therefore, a perpetuity has a finite value. The formula for finding the present value of a perpetuity is as follows:
PV = (PMT/k) ==> (Hint -- in this formula, make sure you always plug in k as a decimal so that 10% is 0.10)
Uneven Cash Flow Streams
Sometimes we will encounter a situation where we have more than one payment, but it is not the same each year. Remember that an annuity requires the payment to be the same each year. If we have multiple cash flows, but they are not the same, we have an uneven cash flow stream. In order to solve this, we must treat it as a series of single cash flows (or possibly a series of smaller annuities).
Consider the following example: We have an investment project that will pay the following cash flows
Year 1 $1000
Year 2 $500
Year 3 $2000
Year 4 $2000
The discount rate is 15%. Find the Present Value.
If we are using a financial calculator (HP10B -- see below for TI and LeWorld), we can proceed as follows:
Step 1: Clear All
Step 2: 0 CFj
Step 3: 1000 CFj
Step 4: 500 CFj
Step 5: 2000 CFj
Step 6: 2 Nj
Step 7: 15 I/YR
Step 8: NPV
Solution $3706.18
Note: The Nj key is used to tell the calculator the number of times we have that same cash flow consecutively. If the cash flow only occurs once (in a row) then we do not need to use the Nj key. However, when we have the same cash flow multiple times in a row (such as the 2000 for two years), we use the Nj key to tell this to the calculator.
If we are using the TI-BAII+ or LeWorld
Step 1: CF
Step 2: CLR Work (this is just CLEAR for the LeWorld)
Step 3: 0 Enter Down Arrow
Step 4: 1000 Enter Down Arrow
Step 5: Down Arrow
Step 6: 500 Enter Down Arrow
Step 7: Down Arrow
Step 8: 2000 Enter Down Arrow
Step 9: 2 Enter
Step 10: NPV
Step 11: 15 Enter Down Arrow
Step 12: CPT
Solution $3706.18
Note: The F screen that appears after you enter a cash flow and down arrow is used to tell the calculator the number of times we have that same cash flow consecutively. If the cash flow only occurs once (in a row) then we do not F screen and just down arrow past it. However, when we have the same cash flow multiple times in a row (such as the 2000 for two years), we use the F screen to tell this to the calculator. The calculator does not have a F screen after the initial cash flow, so we do not need the double down arrow after entering the initial CF.
The above calculator methods are referred to as your Cash Flow Register or Cash Flow Worksheet. It is essential that you always clear all/clear work before entering any cash flows. If you do not do this you will be adding cash flows to a previous problem instead of starting a new problem.
We can also find the discount rate (I/Y) if we have uneven cash flows. Consider the following example: We have an investment project that will pay the following cash flows
Year 1 $1000
Year 2 $500
Year 3 $2000
If the present value of this investment is $3000, what is the discount rate?
With a HP-10B
Step 1: 2nd Clear All
Step 2: -3000 CFj
Step 3: 1000 CFj
Step 4: 500 CFj
Step 5: 2000 CFj
Step 6: IRR/YR
Solution 7.06%
Note: The IRR/YR is not the same key as you used for the I/YR, but it serves a similar role -- finding the discount rate (or rate of return) for a cash flow stream. The difference is that they I/YR key only works with single cash flows or annuities while the IRR/YR key works with uneven cash flows.
With a TI-BAII+ or LeWorld
Step 1: CF
Step 2: 2nd CLR Work (or CLEAR for LeWorld)
Step 3: -3000 Enter down arrow
Step 4: 1000 Enter down arrow down arrow
Step 5: 500 Enter down arrow down arrow
Step 6: 2000 Enter
Step 7: IRR CPT
Note: The IRR is not the same key as you used for the I/Y, but it serves a similar role -- finding the discount rate (or rate of return) for a cash flow stream. The difference is that they I/Y key only works with single cash flows or annuities while the IRR key works with uneven cash flows.
Non Annual Compounding
The more frequently interest is compounded, the greater the effective yield on our savings. Many banks use non annual compounding periods (monthly, daily, etc). In order to make comparisons, we must find the effective annual yield. This tells us how much we are earning on an annual basis. The formula for effective annual yield is as follows:
where
keff is the effective annual yield
knom is the nominal or stated yield
m is the number of compounding periods per year
For example, what is the effective interest rate of 8% compounded daily?
keff = [1 + (.08/365)]^365 - 1 = 8.33%
Note: Be careful not to round when you take .08/365 or you will end up with significant error after compounding it 365 times.
As an alternative, you could use your financial calculator to find the effective interest rate. Again, using 8% compounded daily
With an HP-10B
Step 1: 365 2nd P/YR
Step 2: 8 2nd NOM%
Step 3: 2nd EFF%
Solution 8.33%.
Note: You have changed your payments per year when doing this calculation. If you go back to another TVM problem, be sure to reset your payments per year to one.
With a TI-BAII+ or LeWorld
Step 1: 2nd I Conv
Step 2: 8 Enter down arrow down arrow
Step 3: 365 Enter up arrow
Step 4: CPT
Solution 8.33.
Note, the I Conv is the shift of the 2 key.
We could also look at non-annual compounding with loans or investments. For example, consider a mortgage loan. You are borrowing $80,000 at an 8% rate with monthly payments for 30 years (note that non-annual annuities and lump sums work best with calculators), what is your monthly payment?
Step 1: Convert your calculator to monthly payments by entering 12 P/YR
Step 2: -80000 PV
Step 3: 8 I/YR
Step 4: 360 N (30 years at 12 months per year)
Step 5: 0 FV
Step 6: PMT
Solution = $587.01 per month
Be VERY careful if you change your payments per year to change it back to 1 P/YR when you are done.
Videos of Uneven Cash Flow Stream and Effective Annual Rate Problems on the Financial Calculators
Complex Time Value of Money Problems
Everything above this point completes your "Time Value of Money Toolbox." All the examples to this point have been straight-forward situations. However, sometimes we have what I refer to as complex time value of money problems where there are multiple issues that need addressed within one problem. One of the most common examples of this would be a retirement problem where you have X dollars available today, want to be able to withdraw a certain cash flow stream at retirement throughout your retirement years and want to find out how much you need to save each month until retirement between now and the day you retire to achieve your goal. In order to solve a problem like this, you need to visualize (a time line is very helpful) what information you have and what you are missing (that you need to solve for). You will often need to break this down into multiple steps. We will go through a couple of complex time value of money problems in class. |
