Interest Rate Factor
in Financing
Objectives
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Present value of a single sum
Future value of a single sum
Present value of an annuity
Future value of an annuity
Calculate the effective annual yield for
a series of cash flows
Define what is meant by the internal
rate of return
Compound Interest
• PV= present value
• i=interest rate, discount rate, rate of
return
• I=dollar amount of interest earned
• FV= future values
• Other terms:
• Compounding
• Discounting
Compound Interest
• FV=PV (1 + i)n
• When using a financial calculator:
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n= number of periods
i= interest rate
PV= present value or deposit
PMT= payment
FV= future value
n, i, and PMT must correspond to the same
period:
• Monthly, quarterly, semi annual or yearly.
The Financial Calculator
• n= number of periods
• i=interest rate
• PV= present value, deposit, or mortgage
amount
• PMT= payment
• FV= future value
• When using the financial calculator three
variables must be present in order to
compute the fourth unknown.
• PV or PMT must be entered as a negative
Future Value of a Lump Sum
• FV=PV(1+i)n
• This formula demonstrates the principle of
compounding, or interest on interest if we
know:
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1. An initial deposit
2. An interest rate
3. Time period
We can compute the values at some specified
time period.
Present Value of a Future
Sum
• PV=FV 1/(1+i)n
• The discounting process is the
opposite of compounding
• The same rules must be applied
when discounting
• n, i and PMT must correspond to
the same period
• Monthly, quarterly, semi-annually, and
annually
Future Value of an Annuity
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FVA=P(1+i)n-1 +P(1+i)n-2 ….. + P
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Ordinary annuity (end of period)
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Annuity due (begin of period)
Present Value of an Annuity
• PVA= R 1/(1+i)1 + R 1/(1+i)2…..
R 1/(1+i)n
Future Value of a
Single Lump Sum
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Example: assume Astute investor
invests $1,000 today which pays 10
percent, compounded annually. What is
the expected future value of that
deposit in five years?
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Solution= $1,610.51
Future Value of an Annuity
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Example: assume Astute investor
invests $1,000 at the end of each year in
an investment which pays 10 percent,
compounded annually. What is the
expected future value of that investment
in five years?
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Solution= $6,105.10
Annuities
• Ordinary Annuity
- (e.g., mortgage payment)
• Annuity Due
- (e.g., a monthly rental payment)
Sinking Fund Payment
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Example: assume Astute investor
wants to accumulate $6,105.10 in five
years. Assume Ms. Investor can earn
10 percent, compounded annually.
How much must be invested each year
to obtain the goal?
• Solution= $1,000.00
Present Value of a
Single Lump Sum
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Example: assume Astute investor has
an opportunity that provides $1,610.51
at the end of five years. If Ms. Investor
requires a 10 percent annual return,
how much can astute pay today for this
future sum?
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Solution = $1,000
Payment to Amortize
Mortgage Loan
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Example: assume Astute investor
would like a mortgage loan of
$100,000 at 10 percent annual interest,
paid monthly, amortized over 30 years.
What is the required monthly payment
of principal and interest?
•Solution= $877.57
Yield & IRR
• IRR (Internal Rate of Return) is the most Important
alternative to NPV. The IRR is closely related to NPV. With the
IRR, we try to find a single rate of return that summarizes the
merits of a project. Furthermore we want this rate to be an
"internal" rate in the sense that it depends only on the cash
flows of a particular investment, not on rates offered
elsewhere.
• If future value and present value are known then you can play
a guessing game.
•For example if you have a $5,639 investment that will be
worth $15,000 after 7 years. If you guess that the IRR will
be 10% you get a PV of $7,697. Is our next guess greater
than 10% or less? Why?
• Solve on calculator
Remaining Loan
Balance Calculation
• Example: determine the remaining
balance of a mortgage loan of
$100,000 at 10 percent annual interest,
paid monthly, amortized over 30 years
at the end of year four.
• The balance is the PV of the remaining
payments discounted at the contract
interest rate.
• Solution= $97,402.31
Conventional Mortgage
Objectives
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Characteristics of constant payment
(CPM), constant amortization (CAM),
and graduated payment mortgages
(GPM)
Effective cost of borrowing v.s. lenders
effective yield
Calculate discount points or loan
origination fees
Determinants of Mortgage
Interest Rates
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Real rate of interest- the required rate
at which economic units save rather
than consume
Rate of inflation
Nominal rate or constant rate i= r+f
Nominal rate= real rate plus a
premium for inflation
Determinants of Mortgage
Interest Rates
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Default risk- creditworthiness of borrowers
Interest rate risk- rate change due to market
conditions and economic conditions
Prepayment risk- falling interest rates
Liquidity risk
i=r+ f+ P…
Exhibit 4-1 to be inserted by
McGraw-Hill
Development of Mortgage
Payment Patterns
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Constant amortization mortgage
(CAM)
Constant payment
Interest computed on the monthly loan
balance
Constant amortization amount
Total payment= constant amortization
amount plus monthly interest
Development of Mortgage
Payment Patterns
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Constant payment mortgage (CPM)
Constant monthly payment on original
loan
Fixed rate of interest for a given term
Amount of amortization varies each
month
Completely repaid over the term of the
loan
Development of Mortgage
Payment Patterns
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Graduated payment mortgage (GPM)
Mortgage payments are lower in the
initial years of the loan
GPM payments are gradually increased
at predetermined rates
Loan Closing Costs and
Effective Borrowing Costs
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Statutory costs
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Third party charges
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Additional finance charges i.e. loan
discount fees, points
Effective Interest Cost
Examples
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Contractual loan amount
$60,000
Less origination fee(3%) $ 1,800
Net cash disbursed by lender $58,200
Interest rate= 12%
Term 30 years
Effective Interest Cost
Examples Continued
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Calculator solution
– n=360
– PMT= -617.17
– PV= 58,200
– FV= 0
– i=1.034324 (12.41% annualized)
Other Fixed Rate Mortgages
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Characteristics and Requirements:
Regulation Z- truth in lending (APR)
RESPA- Real Estate Settlement
Procedures Act
Prepayment penalties and other fees
Reverse annuity mortgages (RAMs)
Reverse Annuity
Mortgage Example
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Residential property value $500,000
Loan amount
$250,000
(to be disbursed in monthly installments)
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Term 10 years
Interest Rate
120 months
10%
Reverse Annuity
Mortgage Example Continued
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Calculator solution:
– FV=-250,000
– i=10%/ 12
– PMT= ?
– n=120
– Solve for payment $1220.44