time value of money, loan calculations and analysis chapter 3
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Time Value of Money, Loan Calculations and Analysis
Chapter 3
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
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.
Compound Interest
Assume: Passbook savings accountNo withdrawals
How do you calculate value after several periods have elapsed?
Future value of a Sum = PV * (1+i)n
FV = Ending Account ValuePV = Present Value I = periodic interest rateN = is the number of periods funds are on deposit
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
Intra Period Compounding
Intra Period CompoundingFV = PV * (1 + (i/k))n+k FV = $1000 * (1 + (.06/4))4*4 FV = $1000 * (1.015)16 FV = $1,269
This is $7 more than before, why?
Additional compounding
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 * (1+i)-n
PV = $1,000,000 * (1.10) -35 The discount rate = 10%The period = 35 yearsFV = $1,000,000
Intra Period Calculation:
PV = (future value) * (1 + (i/k) –n*k
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.
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)
FV of Annuity
FV of Annuity = (Periodic Cash Flow) * ((1+i)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 interestUse the table in back of book page 161
Future Value of Annuity Due
Future Value of Annuity Due = (Periodic Cash Flow ) * ((((1+i)n+1 – 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
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)
Present Value of Annuity Due
PV Annuity Due = (Periodic Cash Flow) * (((1- (1+i)–(n -1) )/i) +1)
4,256,782 = (500,000) * (((1-(1.10)-(20-1))/.10) +1)
Basic Loan Calculations -- use PV of annuity and algebra
Periodic Cash Flow = Loan Payment =
(Present Value of Annuity) / ((1- (1+i)-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
An alternative formulation
(Present Value of Annuity) * ( i / (1- (1+i)-n ))
Basic Loan Calculations -- use PV of annuity and algebra
Build an amortization schedule
6 Column
Year Beginning Bal. Payment Interest Principal Ending Bal.1 $4,250,000 $499,203 $425,000 $74,203 $ 4,175,7972 $4,175,797 . . . . 3 . . . . .
Loan Balance
Loan Balance = (Loan Payment) * ((1- (1+i)-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
Loan Balance
Interest Paid within a period = Total Payments – Change in Loan Balance
Need Loan Balance for two periodsEnd 5th year 3,796,978End 4th year 3,905,622
($499,203 - ($3,905,622 – $3,796,978)) = interest paid in year four.
= $390,559
Term Loan Interest
TLI = (n * Loan Payment) – Amount Borrowed
(20 * 499,203) – 4,250,000 = $5,734,060
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.
Determining APR – truth in lending
First Calculate Payment
Then use loan balance equationLoan Balance = Loan Payment * ((1-(1+i)-n)/ i )
Now subtract the points from the Loan Balance and then solve for i by trial and error.
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.
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.
Calculate the monthly payment on the loan amount after making the down payment of $50,000.
Loan Amount = $200,000 Payment = $1,199.10 IR = 6.0 N = 30 years P/Y = 12 payments per year
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.
Calculation of APR from a loan with Points
Calculate the APR based on the calculated payment and a funded loan amount of $196,000.
Loan Amount = $196,000 PMT = $1,199.10 IR = 6.19% APR N = 30 years P/Y = 12 payments per year
Calculation of APR from a loan with Points
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.
An Example from the Text
Original Loan of $200,000 at 9% for 30 years with monthly payments
Calculate Monthly Payments Loan Amount=$200,000 IR=9.0 N=30 Years, Monthly PMT= $1,609.25
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
Refinance Analysis
AMOUNT OF THE POINTS:$191,760 x 0.02 = $3,835
LOAN FEES = $1,000 TOTAL = $4,835
AMOUNT OF NEW LOAN = $191,760= $ 4,835
TOTAL OF NEW LOAN = $196,595
Refinance Analysis
Calculate the monthly payment for the new loan
Loan Amount=$196,595 IR=8.0 N=25 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.
Refinance Analysis
Calculate the Present Value of the Savings from Refinancing
Original Payment = $1,609.25 New Payment = $1,517.35
$ 91.90 PMT = $91.90 IR = 8.0 N = 25 Years, Paid monthly PV = $11,906.98
Refinance Analysis
But what if the new loan is for a term that extends the original term of the loan?
If the new loan is for 30 years at 8.0% with 2 points the new loan would extend the payoff date be 5 years.
The monthly payment would be with the
Loan Amount=$ 196,595 IR=8.0 N=30 years with payments occurring monthly
PMT = $1,442.54
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.
Refinance Analysis
TO EVALUATE THE REFINANCE IN THIS SITUATION, WE NEED TO USE DISCOUNTING.
FOR PAYMENTS 1 – 300 (25 YEARS) Monthly savings=$166.71 IR=8.0 N=25 years Paid
monthly PV= $21,599.70
THIS REPRESENTS THE PRESENT VALUE OF THE SAVINGS OVER THE 25 YEARS
Refinance Analysis
NEXT WE NEED TO CALCULATE THE PRESENT VALUE OF THE ADDITIONAL PAYMENTS.
FOR PAYMENTS 301 - 360 (5 YEARS) PMT= $1,442.54 IR=8.0 N=5 years, paid monthly PV= $71,143.81
THIS REPRESENTS THE PRESENT VALUE OF THE ADDITIONAL PAYMENTS BACK TO YEAR 25.
Refinance Analysis
NEXT WE NEED TO DISCOUNT THIS AMOUNT ($71,143.81) TO THE PRESENT. FV= 71,143.81 IR=8.0 N=25 years, paid monthly PV= $9,692.38
THE PRESENT VALUE (BACK TO YEAR 0) OF THE ADDITIONAL PAYMENTS IS $9,692.38.
Refinance Analysis
SO, WHAT IS THE NET RESULT?
LET’S 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
Refinance Analysis
Therefore, the net result is a benefit from refinancing of $11,907.32
This means that refinancing would be useful.
Refinance Analysis
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