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Time Value of Money Overview

Time Value of Money
The basic idea of time value of money is that a dollar today is worth more than a dollar tomorrow. This can be shown in many ways, many people find it easiest to understand if they think in terms of something they already know: food.  For example having the money today allows you to buy some food immediately. Alternatively you may be willing to forgo current consumption and wait until later to purchase your food. Thus you could lend your “food money” to another with the promise of being paid back at some future time. Since you are passing up food today you would demand a return sufficient to allow you to buy at least as much food in the future that you are giving up now.
As we do not know the future this type of deal involves risks. For example the borrower may decided to not pay you back. This is called default risk. Or the borrower may pay you back but due to rising prices you can no longer purchase the same amount of food as you had expected to be able to buy. As a result of these risks (you as a lender) would require a higher interest rate to compensate for accepting the risks. However if you ask for too high of interest rates you will not find any takers for your loan.

Present Value
The time value of money principle says that future dollars are not worth as much as dollars today.
You should be able to explain why! If you do not understand, please go back and reread the above example.  It is extremely important and influences almost everything we do from now on.
In the vernacular (I love that word) what this means is that you are unwilling to make an interest free loan.

Fortunately we can compare present and future values with a rather simple equation.
1  Present value = Future Value/(1 + required return)^ number of periods

 

(typically Present value is abbreviated PV and Future value is abbreviated FV)
This will give you the present value of a single future cash flow (CF) . In fact for ease down the road we will generally use CF instead of FV. Future Value (FV) will be reserved for when we are actually solving for a future value. (For example how much will we have in 5 years).

 

A simple Present value example follows:
What is the present value of $8,000 to be paid at the end of three years if the correct (risk adjusted interest rate) is 11%?
2  Present value = Future Value/(1 + required return)^ number of periods
= 8,000/(1.11)^3
= 8,000/1.36
=$5,849

Note that if you had so desired you could write this equation as

3 PV = CF * (1/(1+r)t )
Which would be:
PV = 8,000 * (1/1.11)3)
=8,000 * .7312
= $5,849
The second term in equation 3, (1/(1+r)t ), is known as the present value discount factor or present value interest factor. It is usually abbreviated PVIF(r%, N periods). You can find this number either mathematically or from present value tables.

(Note the higher the required interest rate, i.e. the more risk, the lower the present value.)
Continuing our example, suppose that you were willing to make a loan where you would get $8,000 back at the end of the third year, and $10,000 at the end of the fourth year.

 

What is the present value of this? Correct, you find the present value of each cash flow and then add the present values. Thus,
PV = 8,000/(1.11)3 + 10000/(1.11)4
= $12,436.84
Generically, we can thus rewrite equation #2 as:
4 Present value = ∑(CF/(1+r)^t
Whereby we calculate the present value of each cash flow and then sum the present values.

 

As you can imagine this can get quite cumbersome if we had many future cash flows. As a result many short cuts have been devised. Chief among these is when all of the cash flows are identical. This we call an annuity. When we have an annuity we do not need to add up each individual value but can use the present value (and later future value) tables.
Examples of annuities include loan payments and certain long term contracts such as pensions, leases, and certain sports contracts.
Example what is the present value of an annuity of $250 a year at the end of year for 6 years if interest rates are 12%?
To solve this we could add each individual present value up, or can use the following discount factor and then multiply by the cash flow.

5 PVIFA(r,n)=PVAF(r,n)=
Thus if interest rates are 12% and you will receive 6 payments, the discount factor is 4.114. Thus the Present Value (PV) of 6 payments of $250 if interest rates are 12% is
PV = PVAF(r,n) * CF
= 4.114 * $250
= $1,028.50
This answer will be the same whether you solve the problem mathematically, as we just did, or using the time value of money tables.  (try it!)
\An important assumption in using the annuity discount factors is that the cash flows occur at the END of each year. If the cash flows are occurring at the beginning of each year, the cash flows are called an annuity-due. Stop and think for a second what we are doing in an annuity due.

The first cash flow occurs today. Thus the present value of the first cash flow is equal to the cash flow. One year from now you will receive another cash flow. This second cash flow occurs at the same time (or technically 1 day later) than the first cash flow of a regular annuity. To the present value of an annuity due is
6 PV = CF + PVAF(r,n-1) * CF
Using the above example but assuming the first payment is made today (rather than in one year). we can value the cash flows using the annuity-due equation.
PV = 250 + PVAF(12%, 5)* CF
= 250 + 3.6048 * 250
= $1,151.20

Note that the present value is greater than before. Why? Because the payments were all shifted up one year, thus allowing you (the lender) to reinvest sooner and make more money. Alternatively if you were the borrower you are paying earlier so you lose interest that you could have earned by keeping your money invested.

Another shortcut that you will be responsible for is a perpetuity. A perpetuity is a stream of cash flows that goes on forever. Or at least we assume it does. This may or may not be a good assumption (why?-forever is a very long time!) but is very easy to use.
7 PV of a perpetuity = CF/r
Note the cash flow does not have a subscript. Why? Because the definition of a perpetuity says that all cash flows are identical.
Example: To pay for a new highway the local government sells a perpetuity that promises to pay $1000 a year from now until the end of time. If interest rates are 10%, what is the most that you would be willing to pay to get these future cash flows?
PV = CF / r
= $1000 / (.1)
=$10,000

An interesting (this is interesting, right?!?!?!) extension of this is that you can also value a growing perpetuity. This is a series of cash flows that grows at a constant rate. For example suppose that the above perpetuity were promised to grow by 4% per year. Thus this year you get $1000, the next year you get 1000(1.04)=$1,040 etc.

PV=CF1 / (r-g) where CF1 is the cash flow you will receive in one year.
Example: suppose you JUST received $1000 and now want to sell your growing perpetuity. What is the least you should accept for the claim on these future cash flows if the growth rate is 4% and the correct risk adjusted interest rate is 10%.
PV =
= $1,040/(.1-.04)
= $17,333

Future Value
Future Value is largely the same as present value but in reverse. The basic idea is the same except here instead of determining what something is worth today, we want to find out how much something is worth in the future. For example how much will I have if I invest today.
The basic formula is
8 FV = PV* (1+r)^T
Example: you invest $1000 today at 10% in one year you will have 1000*(1.1)1=$1,100
In two years you will have 1,000*(1.1)2= 1,210.

In three years you will have $1,331,

This is based on the implicit assumption of compound interest. Which means you earn interest on your interest. This is a powerful concept and can lead to very large amounts when you have enough time periods over which to accumulate more interest.
Like in the present value discussion we also can use tables to determine a future value factor . These are abbreviated as FVIF(r,n). Thus
FV= PV * (FVIF(r,n)) If r=10%, n=3
= $1,000 * 1.331 = $1,331 which is the same we calculated above.

We also have annuities when calculating future values. These are often used in retirement planning. For example if you invest $1000 a year for three years how much will you have at the end of three years? Use table A4. If r=10%, n=3 (as before)
FV= CF * (FVAF(r,n))
= $1000 * 3.3100
= $3,310.00

How is the future value of an annuity calculated? (that is where are the numbers coming from?) Remember the future values for single payments at 10% for 1 and 2 years these plus the last payment of $1000 sum to $3310. Still uncertain as to the logic? Draw a timeline. Your first payment occurs at the end of year one and earns interest for two years ($1210), the second cash flow occurs at the end of the second year and earns interest for 1 year ($1100), while the third cash flow occurs at the end of the third year and therefore earns no interest ($1000).

When planning your retirement you must account for inflation. We generally use the nominal rate of interest. Thus although you may have a million dollars in the future, that money will be worth less than a million dollars today.


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