Cash Flows, Interest Rates and the Time Value of Money

1
Cash Flows, Interest Rates and the Time Value of
Money
2
Definitions

• Project an investment opportunity generating
  cash flows over time
• Cash Flow the movement of money (in or out) of a
  project
• Interest used to move money through time for
  comparisons. The rent for loaned money
• Cash Flow Diagram Describes type, magnitude and
  timing of cash flows over some horizon

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Cash Flow Diagram

• Describes type, magnitude and timing of cash
  flows over some horizon

500K
200K
200K
0
1
2
3
4
5
50K
100K
500K
4
Cash Flow Diagram

• Used to describe any investment opportunity.
• Typical investment

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Cash Flow Diagram

• Used to describe any investment opportunity.
• Typical investment

0
Make an initial investment (purchase) at time 0
P
6
Cash Flow Diagram

• Used to describe any investment opportunity.
• Typical investment

0
1
2
T
Receive revenues and pay expenses over time.
P
7
Cash Flow Diagram

• Used to describe any investment opportunity.
• Typical investment

0
1
2
T
P
Write as a NET cash flow in each period.
8
Cash Flow Diagram

• Used to describe any investment opportunity.
• Typical investment

SV
0
1
2
T
Receive salvage value at end of life of project.
P
9
Time Value of Money

• We need to compare cash flow diagrams to pick
  from a choice of projects
• To describe the same amount of money at different
  periods of time requires the use of interest.
• Generally, money grows (compounds) into larger
  future sums and is smaller (discounted ) in the
  past.

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Interest

• Cost of Money
• Rental amount charged by lender for use of money
• In any transaction, someone earns and someone
  pays
• Savings Account bank pays you
  1.5 fee to depositor
• Home/Auto Loan borrower pays bank
  7.5 fee to bank

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Interest

• Interest Rate comprised of many factors
• Example Home Mortgage 7.5
• Prime Rate Banks receive money at this rate gt
  5
• Risk Factor gt 1
• Administration Fees gt .5
• Profit gt 1

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Definitions

• Principal P capital
• Amount invested or loaned
• Interest Rate i
• Rental charge for money defined as a percentage
  of principal per time period
• Compounding Period
• Defines how often interest is calculated (may not
  be paid, however)
• Length of loan/investment n periods

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Simple Interest

• Interest earned/paid is directly proportional to
  capital involved.

I P i n Ex. 1000 loan for 2 years at
10 per year no compounding I P i n
1000 .10 2 200 Payback F P I
1000 200 1200
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Compound Interest

• Interest is paid on both the capital and accrued
  interest.

Ex. 1000 loan for 2 years at 10 per year
compounded annually Time Amt. Owe
(Begin) Interest Amt. Owe (End) 1 1000 (.10)(10
00) 1100 2 1100 (.10)(1100) 1210
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Compound Interest and Cash Flow Diagrams

• Example P1000, i10, compounded annually.
• How much accrued after one year?

F 1100
F P I P i P P (
1i )
1
0
P 1000
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Compound Interest and Cash Flow Diagrams

• Example P1000, i10, compounded annually.
• How much accrued after two years?

F 1210
F P I1 I2 P P i (P P i ) i
P ( 1i ) 2 In general F
P(1i)n
0
1
2
P 1000
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Effective Interest Rates

• Rate of interest for a given period
• i per period
• (e.g. 7 per year, 1 per month)

True cost of a loan
True return on an investment
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Compound Interest and Cash Flow Diagrams

• To perform any analysis, cash flow diagrams and
  effective interest rate must use same measure of
  a period (i.e. years, months, days, etc.)

Years on axis
Annual effective rate
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Converting Effective Interest Rates

• Match cash flows and interest rate by converting
  rate to proper period.
• Suppose the APR is i but cash flows occur
  monthly
• Convert effective annual rate to effective
  monthly rate.

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Converting Effective Interest Rates

• Cash flows occur every month. Given annual
  interest rate of 12 .

Annual Rate
F
FP(1ia)
0
1 year
P
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Converting Effective Interest Rates

• Cash flows occur every month. Given effective
  annual interest rate of 12 .

Annual Rate
F
FP(1ia)
FP(1im)12
0
1 year
12 months
1
2
3
Monthly Rate
P
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Converting Effective Interest Rates

• Cash flows occur every month. Given annual
  interest rate of 12 .

Annual Rate
F
FP(1ia)
FP(1im)12
0
1 year
12 months
1
2
3
(1ia) (1im)12
Monthly Rate
P
1.12 (1im)12
im1.121/12 -1.00949.949
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Converting Effective Interest Rates

• Cash flows occur every year. Given quarterly
  interest rate of 4 .

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Converting Effective Interest Rates

• Cash flows occur every year. Given quarterly
  interest rate of 4 .

Annual Rate
F
FP(1ia)
0
1 year
P
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Converting Effective Interest Rates

• Cash flows occur every year. Given quarterly
  interest rate of 4 .

Annual Rate
F
FP(1ia)
FP(1iq)4
0
1 year
4 quarters
1
2
3
Quarterly Rate
P
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Converting Effective Interest Rates

• Cash flows occur every year. Given quarterly
  interest rate of 4 .

Annual Rate
F
FP(1ia)
FP(1iq)4
0
1 year
4 quarters
1
2
3
(1ia) (1iq)4
Quarterly Rate
(1ia) (1.04)4
P
ia1.044 -1.169916.99
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Nominal Interest Rates

• Annual interest rate that does not incorporate
  the effects of compounding
• Examples
• r per year compounded monthly
• r compounded monthly
• Nominal interest rate r
• Misleading to consumers

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Converting Nominal Interest Rates to Effective
Interest Rates

• First, convert to compounding period
• Then convert to required period
• Nominal Rate r compounded over m periods
• Effective Rate i r/m per period

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Nominal to Effective Rates

• Example 12 compounded quarterly
• 12 / 4 3 per quarter
• ia (1.03)4 1 12.55
• Example 14 compounded monthly
• 14 / 12 1.167 per month
• ia (1.0167)12 1 14.94

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Effective to Nominal Rates

• Effective Rate i per period
• Nominal rate r i number of compounding
  periods in a year
• Example 2 per month
• r .02 12 24 compounded monthly

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Effective to Nominal Rates

• Effective Rate i per period
• Nominal rate r i number of compounding
  periods in a year
• Example 2 per month
• Nominal r .02 12 24 compounded monthly
• Effective APR (1.02)12-1 26.82
• Example 4 per quarter
• Nominal r .04 4 16 compounded monthly
• Effective APR (1.04)4-1 16.98

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Nominal vs. Effective Interest Rates

• For comparison compare effective interest rates
  over common period
• Auto Loan Which is better?
• Bank 12 compounded quarterly
• Dealer 1.5 per month

Quarterly B 12 / 4 3 per quarter D
(1.015)3 1 4.57 per quarter
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Use Formula for Compounding
i (1 r/m)nm 1 r nominal rate (e.g.
annual) m compounding periods in a year n
length of time interval in years
1 ia
0
1
2
3
4
(1 iq)4
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Discrete Cash Flows and Continuous Compounding

• So with continuous compounding, substitute
• into all of the previously derived formulas.

Recall
Taking the limit as m goes to infinity
i er - 1
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Example r5 per year compounded over m periods
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Economic Equivalence

• Two cash flows are equivalent if you are willing
  to accept either -- you do not prefer one over
  the other
• Must judge with similar diagrams over similar
  time frame
• Use formulas to convert cash flows in order to
  make sound decisions/preferences

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Typical Measures ofEconomic Equivalence

• Present Worth - Net Present Value
• Find value of cash flows at time zero
• Future Worth
• Find value of cash flows at time n
• Annual Worth
• Find n equivalent cash flows over n periods
• Internal Rate of Return
• Find interest rate such that present worth of
  cash flows is zero

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Interest Rates and Cash Flow Diagrams

• Match the effective interest rate (per period) to
  the time period of the cash flows

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Discrete Cash Flows and Discrete Compounding

• Assumptions
• Cash flows occur at the end of the period
• Effective interest rate period matches cash flow
  timing (e.g. monthly cash flows monthly
  interest rate)
• Interest rate does not change over n periods

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Single Payment Analysis
F
Want to convert a present value to a future
value, and vice versa.

• Questions to be answered
• Find F given P If I invest P now at interest
  rate i, how much will I have after n periods?
• Find P given F If I need F after n periods with
  interest rate i, how much should I invest, P, now?

n
0
1
n
0
1
P
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Compound Amount Factor

• Find F given P (assume i and n known)

F
F
1
0
0
1
2
P
P
F ?
n
0
P
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Finding F

• You put 25,000 into a secure investment earning
  10 per year. Will you have 50,000 in six
  years to purchase a new piece of equipment?

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Finding F

• You put 25,000 into a secure investment earning
  10 per year. Will you have 50,000 in six
  years to purchase a new piece of equipment?

F ?
1
2
3
6
0
25,000
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Present Worth Factor

• Find P given F (assume i and n known)

Compound Amount Factor
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Finding P

• You require 50,000 for a new piece of equipment
  to be purchased in four years. If the annual
  interest rate is 8 , how much should you invest
  at time zero?

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Finding P

• You require 50,000 for a new piece of equipment
  to be purchased in four years. If the annual
  interest rate is 8 , how much should you invest
  at time zero?

50K
1
2
3
4
0
P ?
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Multiple Cash Flows
F1
F4
F3
2
1
0
3
4
F2
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Multiple Cash Flows

• Find P or Present Value (at time zero)
• Find F or Future Value (at time four)

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Internal Rate of Return and Future worth