Interest Rate Factor Interest Rate Factor in Financing in Financing ObjectivesObjectives

• 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 InterestCompound 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 InterestCompound Interest

• FV=PV (1 + i)n

• When using a financial calculator:• 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 CalculatorThe 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 SumFuture Value of a Lump Sum

• FV=PV(1+i)n

• This formula demonstrates the principle of compounding, or interest on interest if we know:• 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 Present Value of a Future SumSum

• 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 AnnuityFuture Value of an Annuity

• FVA=P(1+i)n-1 +P(1+i)n-2 ….. + P

• Ordinary annuity (end of period)

• Annuity due (begin of period)

Present Value of an AnnuityPresent Value of an Annuity

• PVA= R 1/(1+i)1 + R 1/(1+i)2…..

R 1/(1+i)n

Future Value of aFuture Value of a Single Lump Sum Single Lump Sum

• 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?

• Solution= $1,610.51

Future Value of an AnnuityFuture Value of an Annuity

• 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?

• Solution= $6,105.10

AnnuitiesAnnuities

• Ordinary Annuity- (e.g., mortgage payment)

• Annuity Due- (e.g., a monthly rental payment)

Sinking Fund PaymentSinking Fund Payment

• 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 aPresent Value of a Single Lump Sum Single Lump Sum

• 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?

• Solution = $1,000

Payment to Amortize Payment to Amortize Mortgage LoanMortgage Loan

• 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

• 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

Yield & IRR

Remaining Loan Remaining Loan Balance CalculationBalance 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 MortgageConventional MortgageObjectivesObjectives

• 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 Determinants of Mortgage Interest RatesInterest Rates

• 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 Determinants of Mortgage Interest RatesInterest Rates

• 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 Exhibit 4-1 to be inserted by McGraw-HillMcGraw-Hill

Development of Mortgage Development of Mortgage Payment PatternsPayment Patterns

• 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 Development of Mortgage Payment PatternsPayment Patterns

• 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 Development of Mortgage Payment Patterns Payment Patterns

• 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 Loan Closing Costs and Effective Borrowing CostsEffective Borrowing Costs

• Statutory costs

• Third party charges

• Additional finance charges i.e. loan discount fees, points

Effective Interest Cost Effective Interest Cost ExamplesExamples

• 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 Effective Interest Cost Examples ContinuedExamples Continued

• Calculator solution– n=360– PMT= -617.17– PV= 58,200– FV= 0– i=1.034324 (12.41% annualized)

Other Fixed Rate MortgagesOther Fixed Rate Mortgages

• 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 Reverse Annuity Mortgage ExampleMortgage Example

• Residential property value $500,000• Loan amount $250,000

(to be disbursed in monthly installments) • Term 10 years 120

months• Interest Rate 10%

Reverse Annuity Reverse Annuity Mortgage Example ContinuedMortgage Example Continued

• Calculator solution:– FV=-250,000– i=10%/ 12– PMT= ?– n=120– Solve for payment $1220.44