**The Time Value of Money**

Chapter 6

# Time lines help keep track of payments and disbursements

**-100 +100 +200
+300**

** |-------------|-------------|-------------|**

** 0 1 2 3**

**5%**

** **

# Simple
Interest Versus Compound Interest:

## u
With COMPOUND Interest, interest is earned on interest paid
previously

## u
This chapter deals with compound interest

# Future
Value of a Sum

**5% ? **

** |-------------|-------------**

**-100**

** **

# One Period Model

## FV_{1} = PV_{0} (1 + i)

## FV_{1} = 100 (1.05) = 105

##

# If deposit is left to earn for more than one period:

## FV_{2} = FV_{1} (1.05) = 105 (1.05) =
110.25

## substitute
FV_{1} = 100 (1.05) into the above

##

# FV_{2} = 100
(1.05)

(1.05) = 110.25

## So
the General Multi-Period Model is:

##

## FV_{N} = PV_{0} (1 + i)^{N}

#

# Example

## If you deposit $2000 into an

account earning 6% per year, what will be in the account after 10 years if
interest is paid annually?

#

# Solution

## FV_{10}
= $2000 (1.06)^{10}

## FV_{10} = $2000 (1.7908) = $3, 581.69

# Non-Annual Compounding

## 1. Divide the annual interest rate by the
number of payment periods per year to find the interest rate per period.

##

## 2. Multiply the number of years by the number
of periods per year to find the number of compounding periods.

##

# FV_{N} = PV
(1 + i/m)^{mn}

## m
= number of times compounding

occurs per year

## n
= number of years

##

# Example

## What is the future value of a

$2000 deposit after ten years if

it earns 6% per year and is compounded semi-annually?

#

# Solution

## FV_{10}
= $2000 (1 + .06/2)^{2 x 10}

## FV_{10}
= $2000 (1.8061) = $3,612.22

##

# As
the Number of Compounding Periods Increases, the Future Balance Increases

## FV_{10} with annual = $3,581.69

## FV_{10} with semi-annual = $3,612.22

#

# Effective Interest Rate

## Eff Rate = (1 - 1/m)^{m - 1}

## based on equation for multiple
compounding periods

#

**Compounding Effective**

Interval Equation 6.3 Rate

##

## Annual FV =
(1.12)^{1} 1 12.00%

## Semiannual FV = (1.06)^{2}
1 12.36

## Quarterly FV =
(1.03)^{4} 1 12.55

## Monthly FV =
(1.01)^{12} 1 12.68

## Weekly FV =
(1.0023)^{52} 1 12.73

## Daily FV = (1.0003288)^{365}
1 12.7475

## Continuously FV
= e^{.12} 1 12.7596

# Using Financial Tables

*n ***4% 5% 6% 7% 8% 9% 10%**

## 3 1.125 1.158 1.191 1.225 1.260 1.29 **1.331**

## 4 1.170 1.216 1.162 1.311 1.260 1.412 1.464

## 5 1.217 1.276 1.338 1.403 1.469 1.539 1.611

#

# Example: using FV to compute future balance

## How much must you invest at 6%

to have $10,000 in ten years with

semi-annual compounding?

#

# Solution

## FV_{0} = PV_{0} (1 + i/m)^{nm}

## FV_{0} = PV_{0} (1 + .06/2) ^{10 x 2}

## $10,000 = PV_{0} (1.8061)

## PV_{0} = $5,536.71

##

# Example: Inflation Adjustment

## How much will you need in 35

years when you retire to have the same purchasing power provided

by $35,000 today if inflation averages 4%?

#

# Solution

## FV = 35,000 (1.04)^{35}

## = 35,000 (3.9461) = $138,113.11

##

# Future Value of Annuity

##

## u
An annuity is a series of equal payments made at regular
intervals.

## u
Usually solve FVAs using a calculator or tables since
equation is complex.

#

## ** 10%**

** |----------|----------|----------|**

** 100 100 100**

** __________ 110**

** _________________ 121**

**
= 331**

#

#

# Table Approach

## FV_{annuity} = 100 (FVIFA_{10%,3})

##

## FV_{annuity} = 100 (3.3100) = 331

##

##

# Example

## How much will be in your retirement
account if you

deposit $2,500 per year for

35 years and it earns 10% compounded annually?

#

# Solution

## FV = 2,500 (FVIFA_{10%, 35})

## FV = 2,500 (271.0244) = 677,561.00

#

# Ordinary Annuity versus Annuity Due

## Ordinary annuity cash flows occur
at the end of the period.

## Annuity
due cash flows occur at the beginning of each period.

## To convert an ordinary annuity to an annuity due, multiply by
(1 + i).

# Present value of lump sum

**100**

**|-------------|-------------|**

** ? Ώ**

** **

# We already know that:

## FV_{N} = PV_{O} (1 + i)^{N}

## Convert to PV equation

## PV_{O} = FV_{N} / (1 + i)^{N}

##

# Example

## You will receive $50,000 in 5 years
from a trust fund. If you can earn 8%
on investments, how much is this future payment worth today?

#

# Solution

## PV_{O} = 50,000 / (1 + .08^{5)} =
$34,029.16

#

# PV of Annuity

## u
Usually computed using tables

or calculators

#

# Example

## You won the lottery and will
receive $50,000 per year for the next 20 years. If you can invest to earn 10%, how much is the stream of lottery
payments worth today?

#

# Solution

## PV = pmt (PVIFA_{i, N})

## PV = 50,000 (PVIFA_{10, 20})

## PV = 50,000 (8.5136) = $425,680.00

##

# Perpetuity

## u
A cash flow that continues forever

## u
PV_{perp} =
Pmt / I

#

# Example

## What is the value today of a
company expected to generate

net revenues of $10,000 per

year forever, assuming a 15% interest rate?

#

# Solution

## PV = 10,000 / .15 = $66,666.67

#

# Example:

Deferred Annuities

## What is the PV of the following

cash flow stream?

##

** 100
100 100**

** |------------|------------|------------|------------|-----------
-|**

** 0 1 2
3 4 5 **

** **

# Solution

## PV_{2} = $100 (PVIFA_{3,
10%}) (PVIF_{2, 10%})

## = $100 (2.4869) (.8264)

## = $205.52

##

# LOAN PAYMENTS

## u
These are computed using the present value of annuity method.

#

#

# Example

## What is the annual payment due on a
5 year $10,000 loan if IR = 10%?

##

# Solution

## PV = pmt (PVIFA)

## 10,000 = pmt (PVIFA_{10%, 5})

## pmt = 10,000 / 3.7908 = $2,673.96

##

# GROWTH RATES

## u
To compute the
compounded

growth rate use the FV equation.

## u
The beginning value
is the PV.

## u
The ending value is
the FV.

##

# Example

## You bought stock for $15.00

per share 5 years ago. It is currently
selling for $45.00 per share. What is
the compounded average growth rate?

#

# Solution

## FV = PV (1 + g)^{N}

## 45 = 15 (1 + g)^{5}

## (45 / 15) ^{1/5 }- 1 = g

## g = .2458 = 24.58%

##

#

## u
Use future value to find the balance resulting from an
interest-earning deposit.

## u
Use future value of an annuity

to find the payment needed to achieve a
known future balance.

## u
Use either present or future value (without an annuity) to
compute growth rates.

## u
Use present value on all

loan calculations.

## u
Use present value to value assets.

## u
Use present value to evaluate investments.

#

#

#

#

# Tables,
Calculators, Equations, and Spreadsheets

## u
Tables preceded modern calculators/computers

## u
Tables pre-calculated results for equations

## u
Most calculations today done by calculators or computers

## u
Calculators and computers merging

#

# Calculators

## u
Constant memory

## u
Multiple functions of keys

## u
Algebraic or Reverse Polish Notation

## u
Clearing the calculator

## u
Changing Sign

## u
Financial Tour

#

# Calculator
Financial Tour

## u
n

## u
I/Yr

## u
PV

## u
PMT

## u
FV

## u
P/Yr

## u
xP/Yr

#

# Calculating
the Future Value of a Lump Sum

## u
Enter the proper payments per year.
You first must type in the appropriate number (eg. monthly would be 12,
quarterly would be 4, etc.), then push the cream colored key, then the P/YR key .

## u
The order of the next three steps is not important, but I
recommend that you follow across the financial tour of your calculator from
left to right. If you do this, then the next step would be to enter in the
proper number of years.

#

# Future
Value of a Lump (Continued)

## u
Next, You first type the
appropriate number of years, then push the cream colored key, then the xP/YR key.

F Now enter the appropriate
interest rate per year. This is done by entering the appropriate annual
interest rate as a whole number, not as a decimal (the calculator will convert
it to decimal automatically), then pressing the I/YR key.

.