## Quantitative Investment Analysis Workbook

CHAPTER 1

THE TIME VALUE OF MONEY

LEARNING OUTCOMES

After completing this chapter, you will be able to do the following:

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 OVERVIEW

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 fashion as single payments if we use annuity factors instead of 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.Learning Outcomes PROBLEMS

Practice Problems and Solutions: 1-20 taken from Quantitative Methods for Investment Analysis, Second Edition, by Richard A. DeFusco, CFA, Dennis W. McLeavey, CFA, Jerald E. Pinto, CFA, and David E. Runkle, CFA. Copyright © 2004 by CFA Institute. All other problems and solutions copyright © CFA Institute.

The table below gives current information on the interest rates for two two-year and two eight-year maturity investments. The table also gives the maturity, liqui

## Quantitative Investment Analysis Workbook

CHAPTER 1

THE TIME VALUE OF MONEY

LEARNING OUTCOMES

After completing this chapter, you will be able to do the following:

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 OVERVIEW

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 fashion as single payments if we use annuity factors instead of 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.Learning Outcomes PROBLEMS

Practice Problems and Solutions: 1-20 taken from Quantitative Methods for Investment Analysis, Second Edition, by Richard A. DeFusco, CFA, Dennis W. McLeavey, CFA, Jerald E. Pinto, CFA, and David E. Runkle, CFA. Copyright © 2004 by CFA Institute. All other problems and solutions copyright © CFA Institute.

The table below gives current information on the interest rates for two two-year and two eight-year maturity investments. The table also gives the maturity, liqui