Future value
Encyclopedia
Future value is the value
of an asset
at a specific date. It measures the nominal future sum of money that a given sum of money is "worth" at a specified time in the future assuming a certain interest rate
, or more generally, rate of return
; it is the present value
multiplied by the accumulation function
.
The value does not include corrections for inflation or other factors that affect the true value of money in the future. This is used in time value of money
calculations.
over time: $100 today is not worth $100 in five years. This is because one can invest $100 today in a bank account or any other investment, and that money will grow/shrink due to interest. Also, if $100 today allows the purchase of an item, it is possible that $100 will not be enough to purchase the same item in five years, because of inflation
(increase in purchase price).
An investor who has some money has two options: to spend it right now or to invest it. The financial compensation for saving it (and not spending it) is that the money value will accrue through the interests that he will receive from a borrower (the bank account on which he has the money deposited).
Therefore, to evaluate the real worthiness of an amount of money today after a given period of time, economic agents compound the amount of money at a given interest rate. Most actuarial calculations use the risk-free interest rate
which corresponds the minimum guaranteed rate provided the bank's saving account, for example. If one wants to compare their change in purchasing power
, then they should use the real interest rate
(nominal interest rate
minus inflation
rate).
The operation of evaluating a present value into the future value is called a capitalization
(how much will $100 today be worth in 5 years?). The reverse operation which consists in evaluating the present value of a future amount of money is called a discounting (how much $100 that will be received in 5 years- at a lottery
, for example -are worth today?).
It follows that if one has to choose between receiving $100 today and $100 in one year, the rational decision is to cash the $100 today. If the money is to be received in one year and assuming the savings account interest rate is 5%, the person has to be offered at least $105 in one year so that two options are equivalent (either receiving $100 today or receiving $105 in one year). This is because if you cash $100 today and deposit in your savings account, you will have $105 in one year.
where PV is the present value
or principal, t is the time in years (or a fraction of year), and r stands for the per annum interest
rate. Simple interest is rarely used, as compounding is considered more meaningful . Indeed, the Future Value in this case grows linearly (it's a linear function
of the initial investment): it doesn't take into account the fact that the interest earned might be compounded itself and produce further interest (which corresponds to an exponential growth
of the initial investment -see below-).
:
where PV is the present value
, t is the number of compounding periods (not necessarily an integer), and i is the interest rate for that period. Thus the future value increases exponentially
with time when i is positive. The growth rate
is given by the period, and i , the interest rate for that period. Alternatively the growth rate is expressed by the interest per unit time based on continuous compounding. For example, the following all represent the same growth rate:
Also the growth rate may be expressed in a percentage per period (nominal rate
), with another period as compounding basis; for the same growth rate we have:
To convert an interest rate from one compounding basis to another compounding basis (between different periodic interest rates), the following formula applies:
where
i1 is the periodic interest rate with compounding frequency n1 and
i2 is the periodic interest rate with compounding frequency n2.
If the compounding frequency is annual, n2 will be 1, and to get the annual interest rate (which may be referred to as the effective interest rate
, or the annual percentage rate
), the formula can be simplified to:
where r is the annual rate, i the periodic rate, and n the number of compounding periods per year.
Problems become more complex as you account for more variables. For example, when accounting for annuities
(annual payments), there is no simple PV to plug into the equation. Either the PV must be calculated first, or a more complex annuity equation must be used. Another complication is when the interest rate is applied multiple times per period. For example, suppose the 10% interest rate in the earlier example is compounded twice a year (semi-annually). Compounding means that each successive application of the interest rate applies to all of the previously accumulated amount, so instead of getting 0.05 each 6 months, one must figure out the true annual interest rate, which in this case would be 1.1025 (one would divide the 10% by two to get 5%, then apply it twice: 1.052.) This 1.1025 represents the original amount 1.00 plus 0.05 in 6 months to make a total of 1.05, and get the same rate of interest on that 1.05 for the remaining 6 months of the year. The second six month period returns more than the first six months because the interest rate applies to the accumulated interest as well as the original amount.
This formula gives the future value (FV) of an ordinary annuity
(assuming compound interest):
where r = interest rate; n = number of periods. The simplest way to understand the above formula is to cognitively split the right side of the equation into two parts, the payment amount, and the ratio of compounding over basic interest. The ratio of compounding is composed of the aforementioned effective interest rate over the basic (nominal) interest rate. This provides a ratio that increases the payment amount in terms present value.
Value
Value or values may refer to:Concepts of worth:* Value theory – overview of approaches in various disciplines* Value ** Value * Value ** Theory of value ** Value investing...
of an asset
Asset
In financial accounting, assets are economic resources. Anything tangible or intangible that is capable of being owned or controlled to produce value and that is held to have positive economic value is considered an asset...
at a specific date. It measures the nominal future sum of money that a given sum of money is "worth" at a specified time in the future assuming a certain interest rate
Interest rate
An interest rate is the rate at which interest is paid by a borrower for the use of money that they borrow from a lender. For example, a small company borrows capital from a bank to buy new assets for their business, and in return the lender receives interest at a predetermined interest rate for...
, or more generally, rate of return
Rate of return
In finance, rate of return , also known as return on investment , rate of profit or sometimes just return, is the ratio of money gained or lost on an investment relative to the amount of money invested. The amount of money gained or lost may be referred to as interest, profit/loss, gain/loss, or...
; it is the present value
Present value
Present value, also known as present discounted value, is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk...
multiplied by the accumulation function
Accumulation function
The accumulation function a is a function defined in terms of time t expressing the ratio of the value at time t and the initial investment...
.
The value does not include corrections for inflation or other factors that affect the true value of money in the future. This is used in time value of money
Time value of money
The time value of money is the value of money figuring in a given amount of interest earned over a given amount of time. The time value of money is the central concept in finance theory....
calculations.
Overview
Money value fluctuatesInflation
In economics, inflation is a rise in the general level of prices of goods and services in an economy over a period of time.When the general price level rises, each unit of currency buys fewer goods and services. Consequently, inflation also reflects an erosion in the purchasing power of money – a...
over time: $100 today is not worth $100 in five years. This is because one can invest $100 today in a bank account or any other investment, and that money will grow/shrink due to interest. Also, if $100 today allows the purchase of an item, it is possible that $100 will not be enough to purchase the same item in five years, because of inflation
Inflation
In economics, inflation is a rise in the general level of prices of goods and services in an economy over a period of time.When the general price level rises, each unit of currency buys fewer goods and services. Consequently, inflation also reflects an erosion in the purchasing power of money – a...
(increase in purchase price).
An investor who has some money has two options: to spend it right now or to invest it. The financial compensation for saving it (and not spending it) is that the money value will accrue through the interests that he will receive from a borrower (the bank account on which he has the money deposited).
Therefore, to evaluate the real worthiness of an amount of money today after a given period of time, economic agents compound the amount of money at a given interest rate. Most actuarial calculations use the risk-free interest rate
Risk-free interest rate
Risk-free interest rate is the theoretical rate of return of an investment with no risk of financial loss. The risk-free rate represents the interest that an investor would expect from an absolutely risk-free investment over a given period of time....
which corresponds the minimum guaranteed rate provided the bank's saving account, for example. If one wants to compare their change in purchasing power
Purchasing power
Purchasing power is the number of goods/services that can be purchased with a unit of currency. For example, if you had taken one dollar to a store in the 1950s, you would have been able to buy a greater number of items than you would today, indicating that you would have had a greater purchasing...
, then they should use the real interest rate
Real interest rate
The "real interest rate" is the rate of interest an investor expects to receive after allowing for inflation. It can be described more formally by the Fisher equation, which states that the real interest rate is approximately the nominal interest rate minus the inflation rate...
(nominal interest rate
Nominal interest rate
In finance and economics nominal interest rate or nominal rate of interest refers to the rate of interest before adjustment for inflation ; or, for interest rates "as stated" without adjustment for the full effect of compounding...
minus inflation
Inflation
In economics, inflation is a rise in the general level of prices of goods and services in an economy over a period of time.When the general price level rises, each unit of currency buys fewer goods and services. Consequently, inflation also reflects an erosion in the purchasing power of money – a...
rate).
The operation of evaluating a present value into the future value is called a capitalization
Capitalization
Capitalization is writing a word with its first letter as a majuscule and the remaining letters in minuscules . This of course only applies to those writing systems which have a case distinction...
(how much will $100 today be worth in 5 years?). The reverse operation which consists in evaluating the present value of a future amount of money is called a discounting (how much $100 that will be received in 5 years- at a lottery
Lottery
A lottery is a form of gambling which involves the drawing of lots for a prize.Lottery is outlawed by some governments, while others endorse it to the extent of organizing a national or state lottery. It is common to find some degree of regulation of lottery by governments...
, for example -are worth today?).
It follows that if one has to choose between receiving $100 today and $100 in one year, the rational decision is to cash the $100 today. If the money is to be received in one year and assuming the savings account interest rate is 5%, the person has to be offered at least $105 in one year so that two options are equivalent (either receiving $100 today or receiving $105 in one year). This is because if you cash $100 today and deposit in your savings account, you will have $105 in one year.
Simple interest
To determine future value (FV) using simple interest (i.e., without compounding):where PV is the present value
Present value
Present value, also known as present discounted value, is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk...
or principal, t is the time in years (or a fraction of year), and r stands for the per annum interest
Interest
Interest is a fee paid by a borrower of assets to the owner as a form of compensation for the use of the assets. It is most commonly the price paid for the use of borrowed money, or money earned by deposited funds....
rate. Simple interest is rarely used, as compounding is considered more meaningful . Indeed, the Future Value in this case grows linearly (it's a linear function
Linear function
In mathematics, the term linear function can refer to either of two different but related concepts:* a first-degree polynomial function of one variable;* a map between two vector spaces that preserves vector addition and scalar multiplication....
of the initial investment): it doesn't take into account the fact that the interest earned might be compounded itself and produce further interest (which corresponds to an exponential growth
Exponential growth
Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value...
of the initial investment -see below-).
Compound interest
To determine future value using compound interestCompound interest
Compound interest arises when interest is added to the principal, so that from that moment on, the interest that has been added also itself earns interest. This addition of interest to the principal is called compounding...
:
where PV is the present value
Present value
Present value, also known as present discounted value, is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk...
, t is the number of compounding periods (not necessarily an integer), and i is the interest rate for that period. Thus the future value increases exponentially
Exponential growth
Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value...
with time when i is positive. The growth rate
Compound annual growth rate
Compound annual growth rate is a business and investing specific term for the smoothed annualized gain of an investment over a given time period...
is given by the period, and i , the interest rate for that period. Alternatively the growth rate is expressed by the interest per unit time based on continuous compounding. For example, the following all represent the same growth rate:
- 3 % per half year
- 6.09 % per year (effective annual rate, annual rate of returnRate of returnIn finance, rate of return , also known as return on investment , rate of profit or sometimes just return, is the ratio of money gained or lost on an investment relative to the amount of money invested. The amount of money gained or lost may be referred to as interest, profit/loss, gain/loss, or...
, the standard way of expressing the growth rate, for easy comparisons) - 2.95588022 % per half year based on continuous compounding (because ln 1.03 = 0.0295588022)
- 5.91176045 % per year based on continuous compounding (simply twice the previous percentage)
Also the growth rate may be expressed in a percentage per period (nominal rate
Nominal interest rate
In finance and economics nominal interest rate or nominal rate of interest refers to the rate of interest before adjustment for inflation ; or, for interest rates "as stated" without adjustment for the full effect of compounding...
), with another period as compounding basis; for the same growth rate we have:
- 6% per year with half a year as compounding basis
To convert an interest rate from one compounding basis to another compounding basis (between different periodic interest rates), the following formula applies:
where
i1 is the periodic interest rate with compounding frequency n1 and
i2 is the periodic interest rate with compounding frequency n2.
If the compounding frequency is annual, n2 will be 1, and to get the annual interest rate (which may be referred to as the effective interest rate
Effective interest rate
The effective interest rate, effective annual interest rate, annual equivalent rate or simply effective rate is the interest rate on a loan or financial product restated from the nominal interest rate as an interest rate with annual compound interest payable in arrears.It is used to compare the...
, or the annual percentage rate
Annual percentage rate
The term annual percentage rate , also called nominal APR, and the term effective APR, also called EAR, describe the interest rate for a whole year , rather than just a monthly fee/rate, as applied on a loan, mortgage loan, credit card, etc. It is a finance charge expressed as an annual rate...
), the formula can be simplified to:
where r is the annual rate, i the periodic rate, and n the number of compounding periods per year.
Problems become more complex as you account for more variables. For example, when accounting for annuities
Annuity (finance theory)
The term annuity is used in finance theory to refer to any terminating stream of fixed payments over a specified period of time. This usage is most commonly seen in discussions of finance, usually in connection with the valuation of the stream of payments, taking into account time value of money...
(annual payments), there is no simple PV to plug into the equation. Either the PV must be calculated first, or a more complex annuity equation must be used. Another complication is when the interest rate is applied multiple times per period. For example, suppose the 10% interest rate in the earlier example is compounded twice a year (semi-annually). Compounding means that each successive application of the interest rate applies to all of the previously accumulated amount, so instead of getting 0.05 each 6 months, one must figure out the true annual interest rate, which in this case would be 1.1025 (one would divide the 10% by two to get 5%, then apply it twice: 1.052.) This 1.1025 represents the original amount 1.00 plus 0.05 in 6 months to make a total of 1.05, and get the same rate of interest on that 1.05 for the remaining 6 months of the year. The second six month period returns more than the first six months because the interest rate applies to the accumulated interest as well as the original amount.
This formula gives the future value (FV) of an ordinary annuity
Annuity (finance theory)
The term annuity is used in finance theory to refer to any terminating stream of fixed payments over a specified period of time. This usage is most commonly seen in discussions of finance, usually in connection with the valuation of the stream of payments, taking into account time value of money...
(assuming compound interest):
where r = interest rate; n = number of periods. The simplest way to understand the above formula is to cognitively split the right side of the equation into two parts, the payment amount, and the ratio of compounding over basic interest. The ratio of compounding is composed of the aforementioned effective interest rate over the basic (nominal) interest rate. This provides a ratio that increases the payment amount in terms present value.
External links
- Disk Lectures free MBA level audio lecture with slideshow on future value and discounted cash flow.
- calculate the different FV's with one's own values