Time Value of Money and Interest Rates

Money today is worth more than the same amount tomorrow – that one idea is the foundation of finance. In this chapter we turn that feeling into numbers, learning how interest makes money grow, how to compare rates that look different but are really the same, and how to measure what an investment truly earns.

The Big Picture#

Every financial decision – saving, borrowing, investing, pricing a bond or a derivative – comes down to moving money through time. A dollar now and a dollar a year from now are not the same, because you could invest today’s dollar and have more later. This chapter gives you the tools to move cash flows across time accurately: we build compound interest, discounting, effective rates, continuous growth, and the internal rate of return from the ground up. By the end, you will see why a small difference in interest rates can have a huge impact over time, and you will know how to spot when a single “rate of return” is misleading.

Interest and the Power of Compounding#

If you deposit 105. That is simple interest – you earn interest only on the original 5, giving 5 of interest becomes part of the balance. In the second year, you earn interest on 100. So you get 5.25 in interest, ending with 0.25 is interest on interest, and over long periods it turns into a snowball.

Compound Interest: Interest earned not only on the original principal but also on accumulated interest from previous periods.

Let’s formalise this. Suppose you invest a single amount (the present value or principal) for years at an annual interest rate . If interest is compounded once per year, the future value is:

The factor is the compound factor. Notice that growth is not linear; it is exponential. For example, $100 at 5% for 30 years becomes:

The $100 grew more than fourfold, and most of that growth came from interest earning interest. This is why starting early matters so much.

If interest is compounded more than once a year – say times per year – the formula splits the annual rate into per period and multiplies the number of periods by :

When we are back to annual compounding; when it is monthly compounding, and so on. The more frequently interest is compounded, the faster the balance grows, though there is a limit we will explore with continuous compounding.

📝 Section Recap: Compound interest turns a single sum into exponential growth by paying interest on interest. The future value of after years with an annual rate compounded times per year is .

Discounting: Bringing Future Money Back to Today#

If we can grow money forward, we can also shrink it backward. Suppose someone promises to pay you 1,000, because you could put a smaller amount in the bank today, earn interest, and have $1,000 in five years. The process of finding today’s value of a future cash flow is called discounting. The result is the present value (PV).

From we can solve for :

The term is the discount factor. It is always less than 1 for positive and , reflecting that a dollar tomorrow is worth less than a dollar today.

Discount Factor: The multiplier that converts a future cash flow into its present value; equal to when interest is compounded annually.

If the interest rate is 5% per year, the present value of $1,000 in 5 years is:

So you would be indifferent between receiving about 1,000 in five years, assuming you can earn 5% with the same risk. That logic is the core of bond pricing, capital budgeting, and virtually all asset valuation.

When compounding happens times per year, present value becomes:

The choice of – the discount rate – is critical. It should reflect the opportunity cost of capital: what return you could earn elsewhere on an investment of similar risk. We will see later that the internal rate of return is a special discount rate that makes the present value of all cash flows exactly zero.

📝 Section Recap: Discounting is the reverse of compounding. The present value of a future sum is the amount you would need to invest today at a given rate to reach that future sum. The discount factor captures the time-value decay.

Nominal and Effective Rates: The Truth About Annual Interest#

Advertised interest rates can be deceptive. A bank might offer a savings account with a “6% annual rate, compounded monthly.” That 6% is a nominal annual rate, often called the Annual Percentage Rate (APR). But because interest is added to the balance each month, the actual growth over a full year is more than 6%. The rate that truly captures the annual growth is the Effective Annual Rate (EAR).

Nominal Annual Rate (APR): The stated annual interest rate before taking compounding frequency into account. Effective Annual Rate (EAR): The actual percentage increase in money over one year after accounting for intra-year compounding.

If the nominal rate is and compounding happens times per year, then (1 + r_{\text{nom}}/m)^m$ after one year. The effective annual rate is the total proportional increase:

For 6% compounded monthly ():

So the account really earns about 6.17% per year, not 6%. The difference grows as compounding becomes more frequent.

We can also go the other way: given an effective annual rate, what nominal rate compounded times per year would produce it? Solving the EAR formula for :

These conversions are so common that spreadsheet software provides functions like EFFECT and NOMINAL, but the underlying mathematics is exactly what we have just written. The key lesson: never compare nominal rates with different compounding frequencies directly. Always convert to EAR first so you are comparing apples to apples.

📝 Section Recap: A nominal rate does not tell the whole story; the effective annual rate (EAR) measures the true yearly growth after compounding. Always convert to EAR before comparing loans or investments with different compounding periods.

Continuous Compounding: When Interest Never Stops#

What if we let the compounding frequency go to infinity? That is, interest is added to the balance at every instant. This is continuous compounding, and it gives a neat, elegant formula.

Start with the factor . As , the expression approaches the mathematical constant , where . Therefore, with a continuously compounded rate , a principal grows to:

Similarly, the present value of a future sum is:

Continuous compounding is not just a theoretical curiosity. It is used a lot in option pricing and in modelling how interest rates change over time, because the math with is often neater. It also gives the highest possible future value for a given nominal rate, because interest is compounded infinitely often. For example, $100 at a 5% continuously compounded rate for one year becomes:

Compare that with annual compounding (105.12). The difference is small for low rates but becomes noticeable over long horizons.

We can relate continuous rates to effective and nominal rates. If is the continuously compounded rate, then the EAR is:

Conversely, . These relationships let us move freely between compounding conventions.

📝 Section Recap: Continuous compounding is the limit of compounding infinitely often, yielding the elegant formula . It gives the highest future value for a given rate and is a workhorse in advanced finance.

Finding the Rate: Internal Rate of Return#

So far we have taken the interest rate as given and moved cash flows through time. But often we observe a set of cash flows – an initial payment followed by future receipts – and want to know what rate of return the investment delivers. That rate is the internal rate of return (IRR).

Internal Rate of Return (IRR): The discount rate that makes the net present value (NPV) of all cash flows equal to zero.

Suppose an investment costs today (a negative cash flow) and generates positive cash flows at the end of each period. The net present value at a discount rate is:

The IRR is the specific that solves . There is usually no closed‑form solution; we find it by trial and error, or with a financial calculator or spreadsheet.

For a simple example, suppose you invest 1,100 in one year. The IRR solves:

The IRR is 10%. That matches our intuition: you earned 10% on your money.

IRR is a handy summary, but it assumes that intermediate cash flows can be reinvested at the same IRR, which may not be realistic. We will see soon that some cash flow patterns can produce multiple IRRs, making interpretation tricky.

📝 Section Recap: The internal rate of return is the discount rate that sets the net present value of a set of cash flows to zero. It answers: “What constant annual return does this project yield?”

Cash Flow Patterns and the Problem of Multiple IRRs#

Not all investments have a simple “pay now, receive later” pattern. A pure cash flow (or conventional cash flow) has exactly one sign change: an initial outflow (negative) followed by a series of inflows (positive). For such projects, the IRR is unique and meaningful.

But many real‑world projects have mixed cash flows with more than one sign change. For example, a mining project might require an initial investment, generate positive cash flows for years, and then require a large environmental cleanup cost at the end (another outflow). The sign sequence could be: –, +, +, –. When signs flip more than once, the NPV equation is a polynomial that can have multiple positive roots – that is, multiple IRRs.

Descartes’ rule of signs helps us understand how many positive IRRs might exist. It says: the number of positive real roots of a polynomial is at most the number of sign changes in its coefficients, and if not equal, it differs by an even number.

When you write the NPV equation as a polynomial in , the coefficients are the cash flows themselves. So the sign changes among the cash flows determine how many positive IRRs you could have. For a single sign change, at most one positive IRR (and usually exactly one). For two sign changes, you can have two or zero. For three sign changes, at most three, and so on.

Consider a project with these cash flows: –t=0230 at , and –t=2$. The sign sequence is –, +, – (two sign changes). The NPV equation is:

Multiply through by to get a quadratic: . Solving gives two positive IRRs: 10% and 20%. Which one is correct? Both satisfy the math, but neither tells the whole story. In such cases, a single IRR is misleading; analysts often turn to net present value at a chosen discount rate.

The lesson: always look at the cash flow pattern. If signs flip more than once, be cautious – the IRR may not be unique, and you might need a different decision tool.

📝 Section Recap: Pure cash flows (one sign change) have a unique IRR. Mixed cash flows with multiple sign changes can have multiple IRRs – Descartes’ rule of signs bounds the number of positive IRRs. When IRRs multiply, use NPV for clarity.

Summary#

We started with a simple truth: money today is worth more than money later. Compound interest makes your money grow faster and faster. Discounting brings future money back to today’s value. Effective rates and continuous compounding reveal the real yearly growth, and the internal rate of return tells you the return on an investment – but watch out when cash flows switch signs.