Additional Detail on Present and Future Values
Site:  Saylor Academy 
Course:  BUS202: Principles of Finance 
Book:  Additional Detail on Present and Future Values 
Printed by:  Guest user 
Date:  Tuesday, March 28, 2023, 11:44 PM 
Description
This section gives more detail on computing present and future values. It shows you how to compute more complex problems involving future and present values when there are multiple compounding periods and when the time duration of those problems are longer or are less than one year.
The Relationship Between Present and Future Value
Present value () and future value () measure how much the value of money has changed over time.
LEARNING OBJECTIVE

Discuss the relationship between present value and future value
KEY TAKEAWAYS
Key Points
 The future value () 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. The is calculated by multiplying the present value by the accumulation function.
 and vary jointly: when one increases, the other increases, assuming that the interest rate and number of periods remain constant.
 As the interest rate (discount rate) and number of periods increase, increases or decreases.
Key Terms
 discounting: The process of finding the present value using the discount rate.
 present value: a future amount of money that has been discounted to reflect its current value, as if it existed today
 capitalization: The process of finding the future value of a sum by evaluating the present value.
The future value () 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. The is calculated by multiplying the present value 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. The process of finding the is often called capitalization.
On the other hand, the present value () is the value on a given date of a payment or series of payments made at other times. The process of finding the from the is called discounting.
and are related , which reflects compounding interest (simple interest has multiplied by , instead of as the exponent). Since it's really rare to use simple interest, this formula is the important one.
FV of a single payment: The PV and FV are directly related.
and vary directly: when one increases, the other increases, assuming that the interest rate and number of periods remain constant.
The interest rate (or discount rate) and the number of periods are the two other variables that affect the and . The higher the interest rate, the lower the and the higher the . The same relationships apply for the number of periods. The more time that passes, or the more interest accrued per period, the higher the will be if the is constant, and vice versa.
The formula implicitly assumes that there is only a single payment. If there are multiple payments, the is the sum of the present values of each payment and the is the sum of the future values of each payment.
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Calculating Values for Different Durations of Compounding Periods
Finding the Effective Annual Rate (EAR) accounts for compounding during the year, and is easily adjusted to different period durations.
LEARNING OBJECTIVE

Calculate the present and future value of something that has different compounding periods
KEY TAKEAWAYS
Key Points
 The units of the period (e.g. one year) must be the same as the units in the interest rate (e.g. 7% per year).
 When interest compounds more than once a year, the effective interest rate (EAR) is different from the nominal interest rate.
 The equation in skips the step of solving for EAR, and is directly usable to find the present or future value of a sum.
Key Terms
 Present value: Also known as present discounted value, is the value on a given date of a payment or series of payments made at other times. If the payments are in the future, they are discounted to reflect the time value of money and other factors such as investment risk. If they are in the past, their value is correspondingly enhanced to reflect that those payments have been (or could have been) earning interest in the intervening time. Present value calculations are widely used in business and economics to provide a means to compare cash flows at different times on a meaningful "like to like" basis.
 Future Value: 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.
Sometimes, the units of the number of periods does not match the units in the interest rate. For example, the interest rate could be 12% compounded monthly, but one period is one year. Since the units have to be consistent to find the or , you could change one period to one month. But suppose you want to convert the interest rate into an annual rate. Since interest generally compounds, it is not as simple as multiplying 1% by 12 (1% compounded each month). This atom will discuss how to handle different compounding periods.
Effective Annual Rate
The effective annual rate (EAR) is a measurement of how much interest actually accrues per year if it compounds more than once per year. The EAR can be found through the formula in where i is the nominal interest rate and n is the number of times the interest compounds per year. Once the EAR is solved, that becomes the interest rate that is used in any of the capitalization or discounting formulas.
EAR with Continuous Compounding: The effective rate when interest compounds continuously.
Calculating the effective annual rate: The effective annual rate for interest that compounds more than once per year.
For example, if there is 8% interest that compounds quarterly, you plug .08 in for i and 4 in for n. That calculates an EAR of .0824 or 8.24%. You can think of it as 2% interest accruing every quarter, but since the interest compounds, the amount of interest that actually accrues is slightly more than 8%. If you wanted to find the FV of a sum of money, you would have to use 8.24% not 8%.
Solving for Present and Future Values with Different Compounding Periods
Solving for the EAR and then using that number as the effective interest rate in present and future value (/) calculations is demonstrated here. Luckily, it's possible to incorporate compounding periods into the standard timevalue of money formula. The equation in is the same as the formulas we have used before, except with different notation. In this equation, corresponds to , corresponds to Present Value, is the nominal interest rate, is the number of compounding periods per year, and is the number of years.
FV Periodic Compounding: Finding the () given the , nominal interest rate (), number of compounding periods per year (), and number of years ().
The equation follows the same logic as the standard formula. is simply the nominal interest per compounding period, and represents the total number of compounding periods.
Solving for n
The last tricky part of using these formulas is figuring out how many periods there are. If , , and the interest rate are known, solving for the number of periods can be tricky because n is in the exponent. It makes solving for manually messy. shows an easy way to solve for n. Remember that the units are important: the units on must be consistent with the units of the interest rate ().
Solving for : This formula allows you to figure out how many periods are needed to achieve a certain future value, given a present value and an interest rate.
Comparing Interest Rates
Variables, such as compounding, inflation, and the cost of capital must be considered before comparing interest rates.
LEARNING OBJECTIVE

Discuss the differences between effective interest rates, real interest rates, and cost of capital
KEY TAKEAWAYS
Key Points
 A nominal interest rate that compounds has a different effective rate (EAR), because interest is accrued on interest.
 The Fisher Equation approximates the amount of interest accrued after accounting for inflation.
 A company will theoretically only invest if the expected return is higher than their cost of capital, even if the return has a high nominal value.
Key Terms
 inflation: An increase in the general level of prices or in the cost of living.
The amount of interest you would have to pay on a loan or would earn on an investment is clearly an important consideration when making any financial decisions. However, it is not enough to simply compare the nominal values of two interest rates to see which is higher.
Effective Interest Rates
The reason why the nominal interest rate is only part of the story is due to compounding. Since interest compounds, the amount of interest actually accrued may be different than the nominal amount. The last section went through one method for finding the amount of interest that actually accrues: the Effective Annual Rate (EAR).
The EAR is a calculation that account for interest that compounds more than one time per year. It provides an annual interest rate that accounts for compounded interest during the year. If two investments are otherwise identical, you would naturally pick the one with the higher EAR, even if the nominal rate is lower.
Real Interest Rates
Interest rates are charged for a number of reasons, but one is to ensure that the creditor lowers his or her exposure to inflation. Inflation causes a nominal amount of money in the present to have less purchasing power in the future. Expected inflation rates are an integral part of determining whether or not an interest rate is high enough for the creditor.
The Fisher Equation is a simple way of determining the real interest rate, or the interest rate accrued after accounting for inflation. To find the real interest rate, simply subtract the expected inflation rate from the nominal interest rate.
Fisher Equation: The nominal interest rate is approximately the sum of the real interest rate and inflation.
For example, suppose you have the option of choosing to invest in two companies. Company 1 will pay you 5% per year, but is in a country with an expected inflation rate of 4% per year. Company 2 will only pay 3% per year, but is in a country with an expected inflation of 1% per year. By the Fisher Equation, the real interest rates are 1% and 2% for Company 1 and Company 2, respectively. Thus, Company 2 is the better investment, even though Company 1 pays a higher nominal interest rate.
Cost of Capital
Another major consideration is whether or not the interest rate is higher than your cost of capital. The cost of capital is the rate of return that capital could be expected to earn in an alternative investment of equivalent risk. Many companies have a standard cost of capital that they use to determine whether or not an investment is worthwhile.
In theory, a company will never make an investment if the expected return on the investment is less than their cost of capital. Even if a 10% annual return sounds really nice, a company with a 13% cost of capital will not make that investment.
Calculating Values for Fractional Time Periods
The value of money and the balance of the account may be different when considering fractional time periods.
LEARNING OBJECTIVE

Calculate the future and present value of an account when a fraction of a compounding period has passed
KEY TAKEAWAYS
Key Points
 The balance of an account only changes when interest is paid. To find the balance, round the fractional time period down to the period when interest was last accrued.
 To find the or , ignore when interest was last paid an use the fractional time period as the time period in the equation.
 The discount rate is really the cost of not having the money over time, so for / calculations, it doesn't matter if the interest hasn't been added to the account yet.
Key Terms
 time period assumption: business profit or loses are measured on timely basis
 compounding period: The length of time between the points at which interest is paid.
 time value of money: the value of an asset accounting for a given amount of interest earned or inflation accrued over a given period
Up to this point, we have implicitly assumed that the number of periods in question matches to a multiple of the compounding period. That means that the point in the future is also a point where interest accrues. But what happens if we are dealing with fractional time periods?
Compounding periods can be any length of time, and the length of the period affects the rate at which interest accrues.
Suppose the compounding period is one year, starting January 1, 2012. If the problem asks you to find the value at June 1, 2014, there is a bit of a conundrum. The last time interest was actually paid was at January 1, 2014, but the timevalue of money theory clearly suggests that it should be worth more in June than in January.
In the case of fractional time periods, the devil is in the details. The question could ask for the future value, present value, etc., or it could ask for the future balance, which have different answers.
Future/Present Value
If the problem asks for the future value () or present value (), it doesn't really matter that you are dealing with a fractional time period. You can plug in a fractional time period to the appropriate equation to find the or . The reasoning behind this is that the interest rate in the equation isn't exactly the interest rate that is earned on the money. It is the same as that number, but more broadly, is the cost of not having the money for a time period. Since there is still a cost to not having the money for that fraction of a compounding period, the still rises.
Account Balance
The question could alternatively ask for the balance of the account. In this case, you need to find the amount of money that is actually in the account, so you round the number of periods downto the nearest whole number (assuming one period is the same as a compounding period; if not, round down to the nearest compounding period). Even if interest compounds every period, and you are asked to find the balance at the 6.9999^{th} period, you need to round down to 6. The last time the account actually accrued interest was at period 6; the interest for period 7 has not yet been paid.
If the account accrues interest continuously, there is no problem: there can't be a fractional time period, so the balance of the account is always exactly the value of the money.
Loans and Loan Amortization
When paying off a debt, a portion of each payment is for interest while the remaining amount is applied towards the principal balance and amortized.
LEARNING OBJECTIVE

Discuss the process of amortizing a loan
KEY TAKEAWAYS
Key Points
 Each amortization payment should be equal in size and pays off a portion of the principal as well as a portion of the interest.
 The percentage of interest versus principal in each payment is determined in an amortization schedule.
 If the repayment model for a loan is "fully amortized," then the very last payment pays off all remaining principal and interest on the loan.
Key Terms
 amortized loan: a form of debt where the principal is paid down over the life of the debt according to some amortization schedule, typically through equal payments
 amortization: the distribution of the cost of an intangible asset, such as an intellectual property right, over the projected useful life of the asset.
 amortization schedule: a table detailing each periodic payment over the life of the loan
In order to pay off a loan, the debtor must pay off not only the principal but also the interest. Since interest accrues on both the principal and previously accrued interest, paying off a loan can seem like a dance between paying off the principal fast enough to reduce the amount of interest without having huge payments. There is an incentive to paying off the loan ahead of schedule (lower total cost due to less accrued interest), but there is also a disincentive (less use of the principal). After all, if the debtor had enough money and liquidity to pay off the loan instantly, s/he wouldn't have needed the loan.
The process of figuring out how much to pay each month is called "amortization". Amortization refers to the process of paying off a debt (often from a loan or mortgage) over time through regular payments. A portion of each payment is for interest while the remaining amount is applied towards the principal balance.
In order to figure out how much to pay off to amortize each month, many lenders offer their debtors an amortization schedule. An amortization schedule is a table detailing each periodic payment on an amortizing loan, as generated by an amortization calculator. The typical loan amortization schedule offers a summary of the number of moths left for loan, interest paid, etc. The percentage of interest versus principal in each payment is determined in an amortization schedule .These schedules makes it easier for the person who has to repay the loan, s/he can calculate and work accordingly.
Period  Interest  Principal  Balance 

1  $583.33  $191.97  $99,808.03 
2  $582.21  $193.09  $99,614.95 
3  $581.09  $194.21  $99,420.74 
4  $579.95  $195.34  $99,225.39 
5  $578.81  $196.48  $99,028.91 
6  $577.67  $197.63  $98,831.28 
7  $576.52  $198.78  $98,632.50 
8  $575.36  $199.94  $98,432.55 
9  $574.19  $20.11  $98,231.44 
10  $573.02  $202.28  $98,029.16 
11  $571.84  $203.46  $97,825.70 
12  $570.65  $204.65  $97,621.05 
13  $569.46  $205.84  $97,415.21 
14  $568.26  $207.04  $97,208.16 
15  $567.05  $208.25  $97,999.91 
16  $565.83 
$209.47  $96,790.45 
17  $564.61  $210.69  $96,579.76 
18  $563.38  $211.92  $96,367.84 
19  $562.15  $213.15  $96,154.69 
20  $560.90  $214.40  $95,940.29 
21  $559.65  $215.65  $95,724.64 
22  $558.39  $216.91  $95,507.74 
23  $557.13  $218.17  $95,289.57 
24  $555.86  $219.44  $95,070.13 
Amortization Schedule: An example of an amortization schedule of a $100,000 loan over the first two years.
If the repayment model for a loan is "fully amortized," then the very last payment (which, if the schedule was calculated correctly, should be equal to all others) pays off all remaining principal and interest on the loan.