Time Value of Money, Loan Calculations and Analysis

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Time Value of Money, Loan Calculations and
Analysis

• Chapter 3

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Time Value of Money

• Time Value of Money
• Interest is paid over time for the use of money
• 1000 in 1976 is equal to what in 2006? How do
  you go about calculating that?
• Future value of a sum

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Compound Interest

• Compound Interest is interest added to
  principal which from that point on earns interest
  too.
• Most interest earning checking and savings
  accounts earn compound interest.

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Compound Interest
Assume Passbook savings account No
withdrawals How do you calculate value after
several periods have
elapsed? Future value of a Sum PV (1i)n FV
Ending Account Value PV Present Value I
periodic interest rate N is the number of
periods funds are on deposit
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Compound Interest
Example 1000 invested for four years earning 6
interest with annual compounding FV 1000
(1.06)4 1000 X 1.26247 1,262
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Intra Period Compounding
Intra Period Compounding FV PV (1
(i/k))nk FV 1000 (1 (.06/4))44 FV
1000 (1.015)16 FV 1,269 This is 7
more than before, why? Additional compounding
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The Process of Discounting
Discounting is the compounding of interest in
reverse for a future value to determine its
present value. Present Value Future Value
(1i)-n PV 1,000,000 (1.10) -35 The
discount rate 10 The period 35 years FV
1,000,000
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Intra Period Calculation
PV (future value) (1 (i/k) nk Do two
problems Lottery 8 discount rate 20,000,000
(1.08)-10 9,263,870 20,000,000
(1.04)-20 9,127,739 If you have more
periods of compounding then the present value is
lower for the same 20 million.
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Annuities
Ordinary Annuity has cash flows at the end of
the period. (Loan Repayment) Annuities Due
have cash flows at the beginning of the period.
(Insurance, retirement, investment)
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FV of Annuity
FV of Annuity (Periodic Cash Flow) ((1i)n
1)/i) 1,000 annually, 8 IR, 40 years FV of
Annuity (1,000) ((1.08)40 1)/.08)
259,057 How much of this is interest
earned? 40,000 deposited so 219,057 is
interest Use the table in back of book page 161
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Future Value of Annuity Due
Future Value of Annuity Due (Periodic Cash
Flow ) ((((1i)n1 1)/i) -1) Put 1,000 in
for 2 years at 10 Annuity 1,000 .10 1,000
2,100 Annuity due 1,000 .10 2,100 .10
2,310
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Present Value of an Annuity
PV Annuity (Periodic Cash Flow) ((1- (1 i)-n
)/ i) 4,256,782 (500,000) ((1-(1.10)-20)/.10)
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Present Value of Annuity Due
PV Annuity Due (Periodic Cash Flow) (((1-
(1i)(n -1) )/i) 1) 4,256,782 (500,000)
(((1-(1.10)-(20-1))/.10) 1)
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Basic Loan Calculations -- use PV of annuity and
algebra
Periodic Cash Flow Loan Payment (Present
Value of Annuity) / ((1- (1i)-n ) / i) Loan
Payment 4,250,000/ ((1-(1.10)-20 )/ .10) The
principle balance will be 0 at the end of its
Term, 20 years
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Basic Loan Calculations -- use PV of annuity and
algebra
An alternative formulation (Present Value of
Annuity) ( i / (1- (1i)-n ))
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Build an amortization schedule
6 Column
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Loan Balance
Loan Balance (Loan Payment) ((1- (1i)-n)/ i
) Where n is years left on term Calculate the
loan balance for year 5, n would equal 15 on a 20
year loan
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Loan Balance
Interest Paid within a period Total Payments
Change in Loan Balance Need Loan Balance for two
periods End 5th year 3,796,978 End 4th year
3,905,622 (499,203 - (3,905,622 3,796,978))
interest paid in year four.
390,559
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Term Loan Interest
TLI (n Loan Payment) Amount
Borrowed (20 499,203) 4,250,000
5,734,060
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APR - Annual Percentage Rate

• APR is the true or effective interest rate for a
  loan. It is the actual yield to the lender.
• The APR is calculated using the stated interest
  rate, any prepaid interest (points) or other
  lender fees.

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Determining APR truth in lending
First Calculate Payment Then use loan balance
equation Loan Balance Loan Payment
((1-(1i)-n)/ i ) Now subtract the points from
the Loan Balance and then solve for i by trial
and error.
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Points

• Points are loan fees that are viewed as prepaid
  interest and raise the APR of the loan. One
  point is 1 of the loan amount.

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Calculation of APR from a loan with Points

• Your are purchasing a residence that has a
  purchase price of 250,000. You plan on making a
  down payment of 20. Your mortgage lender has
  agreed to finance the loan at 6 for 30 years,
  monthly payments, and wants 2 points.

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Calculate the monthly payment on the loan amount
after making the down payment of 50,000.
Calculation of APR from a loan with Points

• Loan Amount 200,000
• Payment 1,199.10
• IR 6.0
• N 30 years
• P/Y 12 payments per year

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Calculation of APR from a loan with Points

• The amount of the points that is being required
  is 200,000 x 0.02 4,000.
• Therefore the amount of the funded loan is
  200,000 less the 4,000 196,000.

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Calculate the APR based on the calculated payment
and a funded loan amount of 196,000.
Calculation of APR from a loan with Points

• Loan Amount 196,000
• PMT 1,199.10
• IR 6.19 APR
• N 30 years
• P/Y 12 payments per year

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Refinance Analysis

• The proper perspective for refinancing is to
  weigh the discounted cash flow savings of the
  new, lower payment against the cost of the
  transaction.

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An Example from the Text
Refinance Analysis

• Original Loan of 200,000 at 9 for 30 years with
  monthly payments
• Calculate Monthly Payments
• Loan Amount200,000 IR9.0 N30 Years, Monthly
• PMT 1,609.25

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Refinance Analysis

• Refinance the balance after 5 years at 8 with 2
  Points and 1,000 In other loan fees for 25 years
  with monthly payments. The lender will finance
  the cost of the points and fees.
• What is the payoff amount of the original loan?
• Calculating the principal balance following the
  60th using the Loan Balance Equation the payment
  is 191,760.27. Which is 191,760

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Refinance Analysis

• AMOUNT OF THE POINTS191,760 x 0.02
  3,835
• LOAN FEES 1,000
• TOTAL 4,835
• AMOUNT OF NEW LOAN 191,760
  4,835TOTAL OF NEW LOAN 196,595

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Refinance Analysis

• Calculate the monthly payment for the new loan
• Loan Amount196,595 IR8.0 N25 years
• Paid monthly
• PMT 1,517.35
• Since the new loan is paid off at the same time
  as the original loan, the fact that the new
  monthly payment is less means the refinance would
  be profitable.

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Calculate the Present Value of the Savings from
Refinancing
Refinance Analysis

• Original Payment 1,609.25
• New Payment 1,517.35 91.90
• PMT 91.90 IR 8.0 N 25 Years, Paid monthly
• PV 11,906.98

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But what if the new loan is for a term that
extends the original term of the loan?
Refinance Analysis

• If the new loan is for 30 years at 8.0 with 2
  points the new loan would extend the payoff date
  be 5 years.
• The monthly payment would be with the
• Loan Amount 196,595 IR8.0 N30 years with
  payments occurring monthly
• PMT 1,442.54

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Refinance Analysis

• The new loan would reduce the payment by 166.71
  per month from the original loan over 25 Years or
  300 Payments.
• However, there would be an additional 5 years or
  60 payments in the amount of 1,442.54 each.

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TO EVALUATE THE REFINANCE IN THIS SITUATION, WE
NEED TO USE DISCOUNTING.
Refinance Analysis

• FOR PAYMENTS 1 300 (25 YEARS)
• Monthly savings166.71 IR8.0 N25 years Paid
  monthly
• PV 21,599.70
• THIS REPRESENTS THE PRESENT VALUE OF THE SAVINGS
  OVER THE 25 YEARS

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Refinance Analysis

• NEXT WE NEED TO CALCULATE THE PRESENT VALUE OF
  THE ADDITIONAL PAYMENTS.
• FOR PAYMENTS 301 - 360 (5 YEARS)
• PMT 1,442.54 IR8.0 N5 years, paid monthly
• PV 71,143.81
• THIS REPRESENTS THE PRESENT VALUE OF THE
  ADDITIONAL PAYMENTS BACK TO YEAR 25.

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Refinance Analysis

• NEXT WE NEED TO DISCOUNT THIS AMOUNT (71,143.81)
  TO THE PRESENT.
• FV 71,143.81 IR8.0 N25 years, paid monthly
• PV 9,692.38
• THE PRESENT VALUE (BACK TO YEAR 0) OF THE
  ADDITIONAL PAYMENTS IS 9,692.38.

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SO, WHAT IS THE NET RESULT?
Refinance Analysis

• LETS EXPRESS THE PV IN TERMS WHERE SAVINGS IS
  POSITIVE AND AN ADDITIONAL COST IS NEGATIVE.
• PV OF SAVINGS FOR 25 YEARS 21,599.70
• PV OF ADDITIONAL PAYMENTS FOR 5 YEARS -9,692.38

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Refinance Analysis

• Therefore, the net result is a benefit from
  refinancing of 11,907.32
• This means that refinancing would be useful.