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Time Value of Money, Loan Calculations and Analysis Chapter 3

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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 * (1+i) 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 FV = 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

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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 * (1+i) -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) –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.

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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) * ((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 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 ) * ((((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

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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- (1+i) –(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- (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

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An alternative formulation (Present Value of Annuity) * ( i / (1- (1+i) -n )) Basic Loan Calculations -- use PV of annuity and algebra

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Build an amortization schedule 6 Column Year Beginning Bal. Payment Interest Principal Ending Bal. 1$4,250,000 $499,203 $425,000 $74,203 $ 4,175,797 2 $4,175,797.... 3.....

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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

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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-(1+i) -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. 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

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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

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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

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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 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

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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 60 th using the Loan Balance Equation the payment is $191,760.27. Which is ≈$191,760 Refinance Analysis

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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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Chapter 3 Mathematics of Finance

Chapter 3 Mathematics of Finance

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