Interest Rates, Present Value, and Future Value

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Published

June 25, 2025

Introduction

What is the value of an asset? It basically reflects the worth of a stream of future cash flows. To determine the value of an asset, we need to understand the “time value of money”. The “time value of money” deals with equivalence relationships between cash flows with different dates.

Interest Rates

Let’s start with a quick example: Do you agree to pay $1000 today, and, in return, receive $950 today? Of course you don’t! You are basically losing $50 here! But… what if you need to pay it one year from now? This might be okay. A $1000 in one year from now should be worth less than $1000 today. So, it sounds to be fair to discount the (future) $1000 to calculate the amount that is received today.

The interest rate, \(r\), is a rate of return that reflects the relationship between differently dated cash flows. In our example above, $500 is the compensation required for paying $10000 one year from now.

\(r = \frac{500}{9500} = 0.0526\)

So, in this example, the rate of return is 5.26%.

We can look at the interest rate from three different perspectives. 1. It can be considered as “required” rate of return, which is the minimum interest rate that an investor should receive to accept an investment. 2. It can be considered as “discount rate”. In the example above, 5.26% is the discount rate that discounts the future cash flow of $1000 to determine the present value of $950. 3. It can be considered as “opportunity cost”. For instace, if we decide to not invest $950, and spend it instead, we are foregoing the opportunity to earn 5.26% on that amount.

Components of Interest Rates

Interest rates are composed of several components, which can be summarized as follows:

\({ r = r_{real} + r_{inflation} + r_{risk} + r_{liquidity} + r_{maturity} }\)

  • \(r_{real}\): is “real risk-free interest rate”. This is a single-period interest rate that is expected when there is no other risks or inflation.

  • \(r_{inflation}\): is the inflation premium. It compensates investors for the expected inflation rate.

  • \(r_{risk}\): is the default risk permium. It compensates investors for the risk of default. This is the risk that the borrower will not be able to repay the loan.

  • \(r_{liquidity}\): is the liquidity premium. Itcompensates investors for the risk of not being able to sell the asset quickly at a fair price.

  • \(r_{maturity}\): is the maturity premium. It compensates investors for the increased sensitivity of the market value of debt to a change in market interest rates as maturity is extended.

Note that \(r_{real} + r_{inflation}\) is known as the nominal risk-free interest rate.

Present Value and Future Value

In this section, we will try to understand the relationship between Present Value (PV) and Future Value (FV)

FV of a single cash flow

For the given inital investment, \(PV\), and a given interest rate, \(r\), the future value of the investment after one period is:
\(FV = PV \times (1 + r)\)
and after \(n\) periods is:
\(FV_{n} = PV \times (1+r)^{n}\)

Non-annual compounding

Some banks may offer an interest that is compounded more frequently than annually. For instance, they might offer a monthly compounding interest rate. Financial institutions usually quote the annual interest rate, \(r_{s}\), which is known as the stated interest rate (a.k.a. quoted interest rate). For instance, a bank may state that a particular CD pays 7 percent compounded monthly. This means that the stated interest rate is 7 percent. The monthly interest rate is then calculated by dividing the stated interest rate by the number of compounding periods per year, \(m\). So, in this case, the monthly interest rate is: \[r_{m} = \frac{r_{s}}{m} = \frac{0.07}{12} = 0.0058333\].

The formula for future value is as follows:
\(FV_{n} = PV \times (1 + \frac{r_{s}}{m})^{nm}\)

where \(m\) is the number of compounding periods per year, and \(n\) is the number of years.

Let’s consider an example: Suppose you invest $1000 in a CD that pays 7 percent interest compounded monthly for 5 years. The future value of the investment at the end of 5 years is:

\(FV_{5} = 1000 \times (1 + \frac{0.07}{12})^{5 \times 12}\)
\(FV_{5} = 1000 \times (1 + 0.0058333)^{60}\)
\(FV_{5} = 1000 \times (1.0058333)^{60}\)
\(FV_{5} = 1000 \times 1.48985\)
\(FV_{5} = 1489.85\)

NOTE: In a special case where the interest rate is compounded continuously, the future value can be calculated using the following formula:
\(FV_{n} = PV \times e^{r_{s} \cdot n}\)

Effective Annual Rate (EAR)

The Effective Annual Rate (EAR) is the annual interest rate that is equivalent to a given stated interest rate, \(r_{s}\), compounded more frequently than annually. So, basically we are trying to find \(EAR\) that satisfies the following equation:
\(PV \times (1 + r_{s} / m)^{m} = PV \times (1 + EAR)\)

This can be simplified to:
\(EAR = (1 + r_{s} / m)^{m} - 1\)

And if the stated interest rate is compounded continuously, the EAR can be calculated as:
\(EAR = e^{r_{s}} - 1\)

Future Value of a Series of Cash Flows

When we have a series of cash flows, we can calculate the future value of the series by summing the future values of each individual cash flow. Let’s start with defining some terminolgies:

  • Annuity: A series of equal cash flows received at regular intervals. For instance, if you receive $1000 every year for 5 years, this is an annuity.
    • Ordinary Annuity: An annuity where the cash flows are received at the end of each period. For instance, if you receive $1000 at the end of each year for 5 years, this is an ordinary annuity.
    • Annuity Due: An annuity where the cash flows are received at the beginning of each period. For instance, if you receive $1000 at the beginning of each year for 5 years, this is an annuity due.
  • Perpetuity: A series of cash flows that continues indefinitely. For instance, if you receive $1000 every year forever, this is a perpetuity.

NOTE: Another way to understand the difference between Ordinary Annuity and Annuity Due is that Ordinary Annuity is a series of cash flows that are received starting at t=1, and Annuity Due is a series of cash flows that are received starting at t=0.

Now that we know the definition, we can compute their value.

Ordinary Annuity with payment $A for n periods at interest rate r

timeline: t=0, t=1 ($A), t=2 ($A), …, t=n ($A)

  • Calculate the future value of this asset at the end:
    \(FV_{n} = A(1 + r)^{n-1} + A(1 + r)^{n-2} + ... + A(1 + r) + A\)
    \(FV_{n} = A \left( (1 + r)^{n-1} + (1 + r)^{n-2} + ... + (1 + r) + 1 \right)\)
    \(FV_{n} = A \times \frac{(1 + r)^{n} - 1}{r}\)

  • Calculate the present value of this asset (at t=0):
    \(PV = \frac{A}{1 + r} + \frac{A}{(1+r)^2} + ... + \frac{A}{(1+r)^{n}}\)
    \(PV = A \left( \frac{1}{1 + r} + \frac{1}{(1+r)^2} + ... + \frac{1}{(1+r)^{n}} \right)\)

    with \(d = \frac{1}{1 + r}\), we will have:
    \(PV = A \left( d + d^2 + ... + d^{n} \right)\)
    \(PV = A \cdot d \left(1 + d + ... + d^{n-1}\right)\)
    \(PV = A \cdot d \cdot \frac{d^{n} - 1}{d - 1}\)

    Alternaively, we could have just used the future value \(FV_{n} = A \times \frac{(1 + r)^{n} - 1}{r}\) to compute the present value as follows:
    \(PV = \frac{FV_{n}}{(1 + r)^{n}}\)

Annuity Due with payment $A for n periods at interest rate r

timeline: t=0 ($A), t=1 ($A), t=2 ($A), …, t=n-1 ($A)

Let’s Calculate the present value of this asset at t=0:

\(PV = A + A(1+r)^{-1} + A(1+r)^{-2} + ... + A(1+r)^{n-1}\)
\(PV = A \left(1 + (1+r)^{-1} + ... + (1+r)^{n-1}\right)\)
\(PV = A \left(1 + d + ... + d^{n-1}\right)\), where \(d = (1+r)^{-1}\)
\(PV = A \frac{(1 + d)^{n} - 1}{d - 1}\), where \(d = (1+r)^{-1}\)

Perpetuity with payment $A at interest rate r

timeline: t=0, t=1 ($A), t=2 ($A), …

\(PV = A(1+r)^{-1} + ...\)
\(PV = Ad(1 + d + ...)\), where \(d = \frac{1}{1+r} < 1\)
\(PV = \frac{Ad}{1 - d}\), where \(d = \frac{1}{1+r}\)
\(PV = \frac{A}{r}\)

NOTE: We can use the simple formula of Perpetuity to compute PV for Annuity. Let’s see how that works. Suppose we have an ordinary annuity with payment $A for n periods at interest rate r. So, the timeline looks like this: timeline: t=0, t=1 ($A), t=2 ($A), …, t=n-1 ($A)

This can be considered as the substraction of two perpetuities: 1. A perpetuity with timeline t=0, t=1 (A), t=2 (A), … 2. A perpetuity with timeline t=0, t=1, t=2, …, t=n (A), t=n+1 (A), …

\(PV = PV1 - PV2\), where the PVs are at time t=0
\(PV = \frac{A}{r} - \frac{A}{r} \frac{1}{(1+r)^{n-1}}\)

Regarding PV2, note that \(\frac{A}{r}\) gives the present value of the second perpetuity at time t=n-1. So, it needs to be discounted to time t=0 by dividing it by \((1+r)^{n-1}\).

Some examples

Let’s go through some examples…

Example 1
Pension fund manager says that $1 Million per year mush be paid to retirees, starting in 10 years, and continuing for 30 years. If the annual discount rate for plan liabilities is 5% compounded annually, what is the present value of the pension fund?

Let’s start with drawing the timeline:
t=0, t=1, t=2, …, t=9, t=10 ($1M), t=11 ($1M), …, t=39 ($1M).

Let’s denote the annual payment as \(A = 1,000,000\), the interest rate as \(r = 0.05\), and the number of periods as \(n = 30\).

\(PV(@t=9) = \frac{A}{r} - \frac{A}{r} \frac{1}{(1+r)^{n-1}}\)
\(PV(@t=0) = PV(@t=9) \cdot \frac{1}{(1+r)^{9}}\)

and we will get the following value:
\(PV(@t=0) \approx 9,909,219\)