This document discusses concepts related to present value calculations including perpetuities, annuities, and uneven cash flows. It provides examples of calculating the present value of cash flows discounted at different rates as well as future value calculations. The document also discusses growing annuities and uses an example to calculate the maximum constant real annual payment from an investment of $1 million over 20 years with a 5% inflation rate and 10% expected return.

1. what $100 bond be worth if the opportunity cost rate, or discount rate, was 5%?
$100
0.05
Suppose interest Rate rose to 10 percent, what would happen to the console's value
$100
0.1
We see that the value of a perpetuity changes dramatically when interest rates changes.
$2,000
$1,000
Perpetuity
Most annuities call for payments to be made over some finite period of time-for example $100 per
years .however, some annuities go on indefinitely, or perpetually,and these are called
perpetuities.
PV (Perpetuity) = Payment
Interest Rate

2. Example Common stock typically pay an increasing stream of dividends overtime,
0 6% 1 2 3 4 5 6 7
PV= ? 100 200 200 200 200 0 1000
Description
1-Interest Rate 0.06
2-time 0 1 2 3 4 5 6 7
3-Cash flow 100 200 200 200 200 0 1000
4-Present value $1,413.19
Description
1-Interest Rate 0.06
2-time 0 1 2 3 4 5 6 7
3-Cash flow 100 200 200 200 200 0 1000
4-Future value $2,124.92
Future value of an uneven Cash flow Stream
The Future value of an uneven cash flow stream (sometime called the terminal value) is found by
compounding each payment to the end of the stream and then summing future values
the definition of an annuity inclueds the words constant payment- in other words, annunites involve
payments that are the same in every period.Although many financial decisions do not involve
constatant payments. Other important decision involve uneven ,or noncostant cash flows.
Fixed asset investment such a new equipment normally do not generate constant
cash flows
Uneven Cash Flow Streams
Present value of an uneven Cash flow Stream
The Present value of an uneven cash flow stream is found as the sum of the PVs of the individual cash
flows of the stream. For example, suppose we must find the PV of the following cash flow stream,
discounted as 6%

3. Real rete:
=
Data
N= 20
I= 5
PV= ($1,000,000)
FV= 0
PMT= ?
PMT $80,242.59
.
Thus , Portfolio worth $ 1 Million will provide 20 Annual payments with a current dollar value of $
80242.59 under the stated assumptions. The actual payment will be growing at 5% Per year to offset
inflation. The nominal value of portfolio will be growing at first and then declining and it will hit zero
at the end of 20th year
Growing Annuities
Normally, an annuity is defined as a series of constant payments to be received over a specified number of
periods. However ,the term growing annuity is used to describe a series fo payment that is growing at a
constant rate for a specified number of periods, the most common application of growing annuities is in the
area of Financial planning, where somone wants to maintain a constant real, or inflation adjusted , income
over some specified number of years.
For Example: suppose a 65 year old person is contemplating retirement, expect to live for another 20 years.
Has $ 1 million of investment funds, expect to earn 10% on the investments, expects inflation to average 5%
per year and want to withdraw a constant real amount per year, what is the maximum amount that he or she
can withdraw at the end of each year.
[(1+r)/1+inflation)]-1
5%

4. Example
year
Begning
Amount
Payment Interest
Repayment
of principal
Remaining
balance
1 1000 $374.11 $60.00 $314.11 $685.89
2 $685.89 $374.11 $41.15 $332.96 $352.93
3 $352.93 $374.11 $21.18 $352.93 ($0.00)
$1,122.33 $122.33 $1,000.00
One of the most important application of compound interest involves loans that are paid off in
installments over time,included are automobile loans,home mortgage loans , student loans and most
business loans other than very short-term loans and long term bonds.if a loan is to be repaid in equal
periodic amount (monthly ,quarterly, or annually) it is said to be amortized loan
A firm borrows $ 1000 and the loan is to be repaid in three equal payments at the end of each of the
next three years. (In this case ,there is only one payment per year, so year = periods and the stated
rate=periodic rate). The lender charges 6% interest rate on the loan balance that is outstanding at the
begining of each year.
Loan Amortization Schedule, 6% interest Rate
Amortized Loan