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