Time Value of Money and Interest Rates › Summary

This chapter covered the core concepts of time value of money and interest rates used throughout quantitative finance. With a positive applicable discount rate, earlier equal positive inflows have higher present values, while earlier equal outflows have higher present costs. Investment opportunity, expected inflation, and payment uncertainty can contribute to the valuation assumptions when relevant.

We developed the mathematics of compounding, progressing from simple interest through discrete compounding to the continuous compounding framework used in many continuous-time derivatives models. You now understand how to move between present and future values using discount factors, convert between rate quotations (nominal APR, EAR, continuous rates), and calculate year fractions under the illustrated day count conventions.

For valuing cash flow streams, you learned to calculate net present value, recognize when the annuity and perpetuity formulas apply, and understand the sensitivity of valuations to discount rate assumptions. The bond valuation example demonstrated how these tools combine in practice.

The key formulas to remember are:

• Future value (continuous): $FV = Pe^{rt}$
• Present value (continuous): $PV = FV \cdot e^{-rt}$
• Annuity PV: $PV = C \cdot \frac{1-(1+r)^{-T}}{r}$ for $r\neq0$; at $r=0$, $PV=CT$
• Perpetuity PV: $PV = C/r$ for an ordinary positive-payment perpetuity with $r>0$

These building blocks appear throughout the book as we work with more complex instruments. The next chapter develops the probability foundations needed to represent uncertain returns, cash flows, and defaults.