# AFM273 Chapter Notes - Chapter 5: Net Present Value, Annual Percentage Rate, Real Interest Rate

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16 Oct 2016

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Ch. 5 Interest Rates

Introduction

Recall the formula for the present value of a growing annuity with n payments, the first being C1:

As emphasized earlier, this formula assumes that the first payment is one period from now and

that both r and g are measured in terms of a period

We often have to adjust r (and sometimes also g) in order to correctly calculate present or future

values

This requires an understanding of how interest rates are quoted in practice

Effective Annual Rates

Effective annual rate (EAR): indicates the total amount of interest earned over a 1-year period

Adjusting the EAR to an Effective Rate over Different Periods of Time

If r is the effective rate for one period, then

1 + Equivalent n-Period Effective Rate = (1 + r)n

→ Equivalent n-Period Effective Rate = (1 + r)n − 1

o If n > 1, this is an effective rate over more than one period

o If n < 1, it is an effective rate over a fraction of a period

Annual Percentage Rates

Annual percentage rate (APR): indicates the amount of simple interest earned over a 1-year

period

Interest rates quoted by banks are APRs

The APR ignores the effect of compounding, even though compounding may occur

If compounding does happen, then the APR will not give the correct amount of interest unless it is

first converted to an EAR

o APR < actual amount of interest earned

In such cases, we cannot simply use the APR to calculate present or future values

To convert to an EAR, we have to know the compounding frequency (annual, semiannual, monthly,

etc.)

The APR with k compounding periods is actually just an indirect way of quoting the effective rate

r earned each compounding period

The APR with k compounding periods is actually just an indirect way of quoting the effective rate

earned each compounding period

APR itself cannot be used as a discount rate and it is not an EAR

In particular, the implied effective interest rate per compounding period r is given by APR /(k

periods per year)

We can then convert to an EAR using

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Example: if a bank quotes a rate of 4% per year with quarterly compounding, what is the EAR?

What if instead the quoted rate is 9% compounded monthly?

Continuous Compounding

EAR increases as the frequency of compounding increases (Table 5.1, p. 142)

In principle, we can compound as often as we like (every month, every week, every day, every

hour, every second, . . . )

An infinite number of compounding periods in a year corresponds to continuous compounding,

and in this case

If we know the EAR, we can rearrange the expression above to solve for the APR with continuous

compounding:

Continuously compounded rates are not often seen directly, but they are often used behind the

scenes in some areas of finance (e.g. fixed income, options) because some calculations are much

simpler with continuous compounding

DO NOT READ APPENDIX

Present and Future Values with Continuous Compounding

We will only consider the case with a single cash flow

Denote the continuously compounded APR as rcc

o If CT is to be received T years from today, then

o If C0 is invested today, then

Suppose that the continuously compounded interest rate rcc = 5%

o What is the present value of a bond paying $1,000 4.5 years from today?

o What is the future value 8 years and 9 months from now of an investment of $250 today?

DO NOT READ APPENDIX (ANNUITIES AND PERPETUITIES)

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