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2024 Curriculum CFA Program Level I Quantitative Methods

## Introduction

As individuals, we often face decisions that involve saving money for a future use, or borrowing money for current consumption. We then need to determine the amount we need to invest, if we are saving, or the cost of borrowing, if we are shopping for a loan. As investment analysts, much of our work also involves evaluating transactions with present and future cash flows. When we place a value on any security, for example, we are attempting to determine the worth of a stream of future cash flows. To carry out all the above tasks accurately, we must understand the mathematics of time value of money problems. Money has time value in that individuals value a given amount of money more highly the earlier it is received. Therefore, a smaller amount of money now may be equivalent in value to a larger amount received at a future date. The time value of money as a topic in investment mathematics deals with equivalence relationships between cash flows with different dates. Mastery of time value of money concepts and techniques is essential for investment analysts.

The reading is organized as follows: Section 2 introduces some terminology used throughout the reading and supplies some economic intuition for the variables we will discuss. Section 3 tackles the problem of determining the worth at a future point in time of an amount invested today. Section 4 addresses the future worth of a series of cash flows. These two sections provide the tools for calculating the equivalent value at a future date of a single cash flow or series of cash flows. Sections 5 and 6 discuss the equivalent value today of a single future cash flow and a series of future cash flows, respectively. In Section 7, we explore how to determine other quantities of interest in time value of money problems.

## Learning Outcomes

The member should be able to:

• interpret interest rates as required rates of return, discount rates, or opportunity costs;
• explain an interest rate as the sum of a real risk-free rate and premiums that compensate investors for bearing distinct types of risk;

• calculate and interpret the effective annual rate, given the stated annual interest rate and the frequency of compounding;

• solve time value of money problems for different frequencies of compounding;

• calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows;

• demonstrate the use of a time line in modeling and solving time value of money problems.

## Summary

In this reading, we have explored a foundation topic in investment mathematics, the time value of money. We have developed and reviewed the following concepts for use in financial applications:

• The interest rate, r, is the required rate of return; r is also called the discount rate or opportunity cost.

• An interest rate can be viewed as the sum of the real risk-free interest rate and a set of premiums that compensate lenders for risk: an inflation premium, a default risk premium, a liquidity premium, and a maturity premium.

• The future value, FV, is the present value, PV, times the future value factor, (1 + r)N.

• The interest rate, r, makes current and future currency amounts equivalent based on their time value.

• The stated annual interest rate is a quoted interest rate that does not account for compounding within the year.

• The periodic rate is the quoted interest rate per period; it equals the stated annual interest rate divided by the number of compounding periods per year.

• The effective annual rate is the amount by which a unit of currency will grow in a year with interest on interest included.

• An annuity is a finite set of level sequential cash flows.

• There are two types of annuities, the annuity due and the ordinary annuity. The annuity due has a first cash flow that occurs immediately; the ordinary annuity has a first cash flow that occurs one period from the present (indexed at t = 1).

• On a time line, we can index the present as 0 and then display equally spaced hash marks to represent a number of periods into the future. This representation allows us to index how many periods away each cash flow will be paid.

• Annuities may be handled in a similar approach as single payments if we use annuity factors rather than single-payment factors.

• The present value, PV, is the future value, FV, times the present value factor, (1 + r)N.

• The present value of a perpetuity is A/r, where A is the periodic payment to be received forever.

• It is possible to calculate an unknown variable, given the other relevant variables in time value of money problems.

• The cash flow additivity principle can be used to solve problems with uneven cash flows by combining single payments and annuities.