When considering a single-period investment, n is one, so the PV is simply FV divided by 1+i.
Calculate the present value of a future, single-period payment
The percentage of an amount of money charged for its use per some period of time. It can also be thought of as the cost of not having money for one period, or the amount paid on an investment per year.
The length of time during which interest accrues.
The time value of money framework says that money in the future is not worth as much as money in the present. Investors would prefer to have the money today because then they are able to spend it, save it, or invest it right now instead of having to wait to be able to use it.
The difference between what the money is worth today and what it will be worth at a point in the future can be quantified. The value of the money today is called thepresent value (PV), and the value of the money in the future is called the future value(FV). There is also a name for the cost of not having the money today: the interest rateor discount rate (i or r). For example, if the interest rate is 3% per year, it means that you would be willing to pay 3% of the money to have it one year sooner. The amount of time is also represented by a variable: the number of periods (n). One period could be any length of time, such as one day, one month, or one year, but it must be clearly defined, consistent with the time units in the interest rate, and constant throughout your calculations.
FV of a single payment
The FV is related to the PV by being i% more each period.
All of these variables are related through an equation that helps you find the PV of a single amount of money. That is, it tells you what a single payment is worth today, but not what a series of payments is worth today (that will come later). relates all of the variables together. In order to find the PV, you must know the FV, i, and n.
When considering a single-period investment, n is, by definition, one. That means that the PV is simply FV divided by 1+i. There is a cost to not having the money for one year, which is what the interest rate represents. Therefore, the PV is i% less than the FV.
Multi-Period Investment
Multi-period investments are investments with more than one period, so n (or t) is greater than one.
Calculate the present value of a multi-period investment
Interest, as on a loan or a bank account, that is calculated on the total on the principal plus accumulated unpaid interest.
More than one unit of time.
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Things get marginally more complicated when dealing with a multi-period investment. That is, an investment where n is greater than 1.
Suppose the interest rate is 3% per year. That means that the value of $100 will be 3% more after one year, or $103. After the second year, the investment will be 3% more, or 3% more than $103. That means the original investment of $100 is now worth $106.09. The investment is not worth 6% more after two years. In the second year, you earn 3% interest on your original $100, but you also earn 3% interest on the $3 you earned in the first year. This is called compounding interest: interest accrues on previously earned interest.
As such, PV and FV are related exponentially, which is reflected in. Using the formula in is relatively simple. Just as with a single-period investment, you simply plug in the FV, i and n in order to find the PV. PV varies jointly with FV and inversely with i and n, which makes sense based on what we know about the time value of money.
The formula may seem simple, but there is one major tripping point: units. Sometimes, the interest rate will be something like 5% annually, and you are asked to find the PV after 24 months. The number of periods, however, is not 24--it is 2. If the interest rate is written as "percent per year" your periods must also be measured in years. If your periods are defined as "days", your interest rate must be written as "percent per day. "
Consider the $100 investment mentioned above. How much would the investment be worth in 5 years with yearly compounding interest? An expanded explanation of the calculations is as follows:
FV=PV⋅(1+i)n
FV=100⋅(1+0.03)5
FV=100⋅1.159
FV = $115.93
In order to perform the calculation for the above investment, follow these steps:
Discounting is the procedure of finding what a future sum of money is worth today.
Describe what real world costs to the investor comprise an investment's interest rate
The interest rate used to discount future cash flows of a financial instrument; the annual interest rate used to decrease the amounts of future cash flow to yield their present value.
to account for the time value of money
The process of finding the present value using the discount rate.
Another common name for finding present value (PV) is discounting. Discounting is the procedure of finding what a future sum of money is worth today. As you know from the previous sections, to find the PV of a payment you need to know the future value(FV), the number of time periods in question, and the interest rate. The interest rate, in this context, is more commonly called the discount rate.
The discount rate represents some cost (or group of costs) to the investor or creditor. The sum of these costs amounts to a percentage which becomes the interest rate (plus a small profit, sometimes). Here are some of the most significant costs from the investor/creditor's point of view:
Borrowing and lending
Banks like HSBC take such costs into account when determining the terms of a loan for borrowers.
All of these costs combine to determine the interest rate on an account, and that interest rate in turn is the rate at which the sum is discounted.
The PV and the discount rate are related through the same formula we have been using, FV[(1+i)]n.
If FV and n are held as constants, then as the discount rate (i) increases, PV decreases. PV and the discount rate, therefore, vary inversely, a fundamental relationship in finance. Suppose you expect $1,000 dollars in one year's time (FV = $1,000) . To determine the present value, you would need to discount it by some interest rate (i). If this discount rate were 5%, the $1,000 in a year's time would be the equivalent of $952.38 to you today (1000/[1.00 + 0.05]).
Number of Periods
The number of periods corresponds to the number of times the interest is accrued.
Define what a period is in terms of present value calculations
The length of time during which interest accrues.
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In , nrepresents the number of periods. A period is just a general term for a length of time. It can be anything- one month, one year, one decade- but it must be clearly defined and fixed. The length of one period must be the same at the beginning of aninvestment and at the end. It is also part of the units of the discount rate: if one period is one year, the discount rate must be defined as X% per year. If one period is one month, the discount rate must be X% per month.
FV of a single payment
The PV and FV are directly related.
The number of periods corresponds to the number of times the interest is accrued. In the case of simple interest the number of periods, t, is multiplied by their interest rate. This makes sense because if you earn $30 of interest in the first period, you also earn $30 of interest in the last period, so the total amount of interest earned is simple t x $30.
Simple interest is rarely used in comparison to compound interest . In compound interest, the interest in one period is also paid on all interest accrued in previous periods. Therefore, there is an exponential relationship between PV and FV, which is reflected in (1+i)^{n }.
Car loans, mortgages, and student loans all generally have compound interest.
For both forms of interest, the number of periods varies jointly with FV and inversely with PV. Logically, if more time passes between the present and the future, the FV must be higher or the PV lower (assuming the discount rate remains constant).
Calculating Present Value
Calculating the present value (PV) is a matter of plugging FV, the interest rate, and the number of periods into an equation.
Distinguish between the formula used for calculating present value with simple interest and the formula used for present value with compound interest
interest paid only on the principal.
Interest, as on a loan or a bank account, that is calculated on the total on the principal plus accumulated unpaid interest.
Finding the present value (PV) of an amount of money is finding the amount of money today that is worth the same as an amount of money in the future, given a certain interest rate.
Calculating the present value (PV) of a single amount is a matter of combining all of the different parts we have already discussed. But first, you must determine whether the type of interest is simple or compound interest. If the interest is simple interest, you plug the numbers into the simple interest formula.
Simple Interest Formula
Simple interest is when interest is only paid on the amount you originally invested (the principal). You don't earn interest on interest you previously earned.
If it is compound interest, you can rearrange the compound interest formula to calculate the present value.
Present Value Single Payment
Finding the PV is a matter of plugging in for the three other variables.
Once you know these three variables, you can plug them into the appropriate equation. If the problem doesn't say otherwise, it's safe to assume the interest compounds. If you happen to be using a program like Excel, the interest is compounded in the PV formula. Simple interest is pretty rare.
One area where there is often a mistake is in defining the number of periods and the interest rate. They have to have consistent units, which may require some work. For example, interest is often listed as X% per year. The problem may talk about finding the PV 24 months before the FV, but the number of periods must be in years since the interest rate is listed per year. Therefore, n = 2. As long as the units are consistent, however, finding the PV is done by plug-and-chug.
Source: Boundless
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