Additional Detail on Present and Future Values

Present value ($PV$) and future value ($FV$) 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 ($FV$) 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 $FV$ is calculated by multiplying the present value by the accumulation function.
• $PV$ and $FV$ 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, $FV$ increases or $PV$ 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 ($FV$) 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 $FV$ 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 $FV$ is often called capitalization.

On the other hand, the present value ($PV$) is the value on a given date of a payment or series of payments made at other times. The process of finding the $PV$ from the $FV$ is called discounting.

$PV$ and $FV$ are related , which reflects compounding interest (simple interest has $n$ multiplied by $i$, instead of as the exponent). Since it's really rare to use simple interest, this formula is the important one.

$F V=P V(1+i)^{n}$
FV of a single payment: The PV and FV are directly related.

$PV$ and $FV$ 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 $FV$ and $PV$. The higher the interest rate, the lower the $PV$ and the higher the $FV$. The same relationships apply for the number of periods. The more time that passes, or the more interest accrued per period, the higher the $FV$ will be if the $PV$ is constant, and vice versa.

The formula implicitly assumes that there is only a single payment. If there are multiple payments, the $PV$ is the sum of the present values of each payment and the $FV$ is the sum of the future values of each payment.

Source: Saylor Academy
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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 $PV$ or $FV$, 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.

$r=e^{i}-1$

$r=(1+i / n)^{n}-1$

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 ($PV$/$FV$) calculations is demonstrated here. Luckily, it's possible to incorporate compounding periods into the standard time-value of money formula. The equation in is the same as the formulas we have used before, except with different notation. In this equation, $A(t)$ corresponds to $FV$, $A0$ corresponds to Present Value, $r$ is the nominal interest rate, $n$ is the number of compounding periods per year, and $t$ is the number of years.

$A(t)=A_{0}\left(1+\frac{r}{n}\right)^{\lfloor n t\rfloor}$

FV Periodic Compounding: Finding the $FV$ ($A(t)$) given the $PV (Ao)$, nominal interest rate ($r$), number of compounding periods per year ($n$), and number of years ($t$).

The equation follows the same logic as the standard formula. $r/n$ is simply the nominal interest per compounding period, and $nt$ 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 $PV$, $FV$, 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 $n$ manually messy. shows an easy way to solve for n. Remember that the units are important: the units on $n$ must be consistent with the units of the interest rate ($i$).

$n=\frac{\log (F V)-\log (P V)}{\log (1+i)}$

Solving for $n$: 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.

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.

$i \approx r+\pi$

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.

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 $PV$ or $FV$, 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 $PV$/$FV$ 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 time-value 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 ($FV$) or present value ($PV$), 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 $FV$ or $PV$. 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 $FV$ 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.