The time Value of money
Before addressing Annuities and Perpetuities it is important to understand the time value of money concept. They say “time is money” but how much do these really have to do with one another? Let’s assume you put some money in a savings account at the bank for 1 year. By the end of that year the bank will have given you a percentage of that money back in addition to the amount you originally deposited. The money you put in the bank is called the principle and the money they give you is referred to as the interest. This simple example illustrates the concept of the Time Value of Money. Another way to think of this is in terms of inflation. $4.00 today might be able to buy you a gallon of milk but 4 years from now that same gallon of milk could cost $4.50.
This difference in money from how much it is today (and how much it can buy) to what it is in the future is where the “time is money” saying comes from. The value of the milk 4 years from now, or money deposited into the bank (with interest) is referred to as the future value of money while the amount presently (without interest) is referred to as the present value of money.
For example, you put 100 dollars into a bank with high savings rates (formally called APY) and they give you a 1% return every year. At the end of the year you will have 101 dollars, detailed below:
$100 (initial deposit)
$1 (interest gained over 1 year period)
$101 (total savings 1 year in the future)
In this case the 100 dollars is the “Present Value of Money” the 101 dollars is the “Future Value of Money” and the 1 dollar is the “Time Value of Money”
You can calculate the future value of money using the following formula
Future Value = Present Value + (Present Value x Interest Rate)
In Our Example:
101 = 100 + (100 x 1%) = 100 + 1
Now what about if you waited even longer? In a process called “Compounding Interest” you would get a 1% return on both the initial deposit (also called “Principle”) AND on the interest you earned (in our example $1)
Using algebra with the above equation you can find the value of money with a fixed interest rate for any number of years. To see how money compounds over time we use the following equation
Future Value = Present Value x (1 + Interest Rate)^(Number of Years)
With the effects of compounding interest the Future Value of money will grow following an exponential curve with time while the Future Value without compounding interest will have a linear growth. The equation without compounding interest is show below and an example with graph is included as well.
Future Value = Present Value + (Present Value x Interest Rate) x (Number of Years)
Example Compounding Interest
A neat trick that these equations can do for you is helping you calculate how much you need to put away today to have a set amount of money in the future (assuming constant interest rates). In this case, the required Present Value is simply the Future Value desired, divided by one plus the Interest Rate raised to the number of years you have to save.
Present Value = Future Value/((1+Interest Rate)^Years)
So if you want to know how much money to put away today to have $1 million in 30 years with a static interest rate of 10% just plug the values in to find…
1,000,000/1.130=57,308.55
So if you invest $57,308.55 today, in 30 years, without touching it again (adding to or subtracting from) you would have $1million saved (assuming you could receive a risk free return of 10%)
Annuities and Perpetuities
An Annuity is a fixed set of payments over a given timeframe, for instance $50,000 per year for 20 years, for a total of 1,000,000 in payments.
A Perpetuity is a cashflow that will be received forever, for instance $50,000 per year forever.
The present value of and Annuity (and a Perpetuity) is actually less than the sum of payments due to inflation and expected return on investments.
For instance, if you can expect to receive a risk free constant return of 5% per year then then the value of $1 million payed out over 20 years is actually closer to a present value of $623,111. This is because the payment next year would have a present value of 50K/1.05 and the following year would be 50K/(1.05^2) all the way to year 20 at 50K/(1.05^20) which summed would equal $623,111 in today’s dollars.
To calculate the present value of an annuity use the following
Present Value = (Cashflow Payments Per Year/Interest Rate) x (1-(1/((1+Interest Rate)^Number of Years that Payments are Received)))
The Present Value of a Perpetuity is whatever the yearly cashflow is, divided by the Interest Rate you could otherwise receive. For instance, If you can get a perpetuity of $50,000 per year, but can also get a 5% risk free return then the present value of the Perpetuity is 50k/0.05 which is $1 million
PV(Normal Perpetuity) = Cashflow/Interest Rate
A Growing Perpetuity is one that grows with time (for instance to match inflation) and the present value can be calculated as follows
PV(Growing Perpetuity) = Cashflow/(Interest Rate - Growth Rate)
For the Future payment on a growing Perpetuity use the equation below to calculate
FV = Cashflow x (1+Growth Rate)^Number of Years
Lastly, a growing Annuity has a set number of payments but grows with each (similar to the growing perpetuity) this means that a growing annuity can be adjusted for expected inflation rates over the time period of the returns. The equation to calculate the Present Value of a growing annuity is shown here.
Before addressing Annuities and Perpetuities it is important to understand the time value of money concept. They say “time is money” but how much do these really have to do with one another? Let’s assume you put some money in a savings account at the bank for 1 year. By the end of that year the bank will have given you a percentage of that money back in addition to the amount you originally deposited. The money you put in the bank is called the principle and the money they give you is referred to as the interest. This simple example illustrates the concept of the Time Value of Money. Another way to think of this is in terms of inflation. $4.00 today might be able to buy you a gallon of milk but 4 years from now that same gallon of milk could cost $4.50.
This difference in money from how much it is today (and how much it can buy) to what it is in the future is where the “time is money” saying comes from. The value of the milk 4 years from now, or money deposited into the bank (with interest) is referred to as the future value of money while the amount presently (without interest) is referred to as the present value of money.
For example, you put 100 dollars into a bank with high savings rates (formally called APY) and they give you a 1% return every year. At the end of the year you will have 101 dollars, detailed below:
$100 (initial deposit)
$1 (interest gained over 1 year period)
$101 (total savings 1 year in the future)
In this case the 100 dollars is the “Present Value of Money” the 101 dollars is the “Future Value of Money” and the 1 dollar is the “Time Value of Money”
You can calculate the future value of money using the following formula
Future Value = Present Value + (Present Value x Interest Rate)
In Our Example:
101 = 100 + (100 x 1%) = 100 + 1
Now what about if you waited even longer? In a process called “Compounding Interest” you would get a 1% return on both the initial deposit (also called “Principle”) AND on the interest you earned (in our example $1)
Using algebra with the above equation you can find the value of money with a fixed interest rate for any number of years. To see how money compounds over time we use the following equation
Future Value = Present Value x (1 + Interest Rate)^(Number of Years)
With the effects of compounding interest the Future Value of money will grow following an exponential curve with time while the Future Value without compounding interest will have a linear growth. The equation without compounding interest is show below and an example with graph is included as well.
Future Value = Present Value + (Present Value x Interest Rate) x (Number of Years)
Example Compounding Interest
A neat trick that these equations can do for you is helping you calculate how much you need to put away today to have a set amount of money in the future (assuming constant interest rates). In this case, the required Present Value is simply the Future Value desired, divided by one plus the Interest Rate raised to the number of years you have to save.
Present Value = Future Value/((1+Interest Rate)^Years)
So if you want to know how much money to put away today to have $1 million in 30 years with a static interest rate of 10% just plug the values in to find…
1,000,000/1.130=57,308.55
So if you invest $57,308.55 today, in 30 years, without touching it again (adding to or subtracting from) you would have $1million saved (assuming you could receive a risk free return of 10%)
Annuities and Perpetuities
An Annuity is a fixed set of payments over a given timeframe, for instance $50,000 per year for 20 years, for a total of 1,000,000 in payments.
A Perpetuity is a cashflow that will be received forever, for instance $50,000 per year forever.
The present value of and Annuity (and a Perpetuity) is actually less than the sum of payments due to inflation and expected return on investments.
For instance, if you can expect to receive a risk free constant return of 5% per year then then the value of $1 million payed out over 20 years is actually closer to a present value of $623,111. This is because the payment next year would have a present value of 50K/1.05 and the following year would be 50K/(1.05^2) all the way to year 20 at 50K/(1.05^20) which summed would equal $623,111 in today’s dollars.
To calculate the present value of an annuity use the following
Present Value = (Cashflow Payments Per Year/Interest Rate) x (1-(1/((1+Interest Rate)^Number of Years that Payments are Received)))
The Present Value of a Perpetuity is whatever the yearly cashflow is, divided by the Interest Rate you could otherwise receive. For instance, If you can get a perpetuity of $50,000 per year, but can also get a 5% risk free return then the present value of the Perpetuity is 50k/0.05 which is $1 million
PV(Normal Perpetuity) = Cashflow/Interest Rate
A Growing Perpetuity is one that grows with time (for instance to match inflation) and the present value can be calculated as follows
PV(Growing Perpetuity) = Cashflow/(Interest Rate - Growth Rate)
For the Future payment on a growing Perpetuity use the equation below to calculate
FV = Cashflow x (1+Growth Rate)^Number of Years
Lastly, a growing Annuity has a set number of payments but grows with each (similar to the growing perpetuity) this means that a growing annuity can be adjusted for expected inflation rates over the time period of the returns. The equation to calculate the Present Value of a growing annuity is shown here.
