TVM (cont)

Uneven cash flows Payment (pmt) – designates constant cash flows Cash flow (CF) – designates cash flows in general, including uneven cash flows

General Pricing equations: PV = CF1[1/(1+r)1] + CF2[1/(1+r)2]+ …+ CFn[1/(1+r)n] FV = CF1(1+r)n-1 + CF2(1+r)n-2 + …+ CFn(1+r)n

Uneven Cash Flow Example: What is the PV of this uneven cash flow stream? Assuming the opportunity cost rate is 10%? timeline: r = 10% 5 $100 $300 $300 $-50 PV = $?

Uneven Cash Flow Write an Equation for PV that you could solve using any calculator. Financial Calculator Solution: Input in Cash Flow register CF0 = 0 CF1 = 100 CF2 = 300 CF3 = 300 CF4 = -50 Then enter 10% for Interest rate, and press NPV

Uneven Cash Flow In the last example, would PV be larger or smaller if the opportunity cost rate had been bigger? Why? Check your answer using a bigger opportunity cost, say 20% PV @ 20% = ___________

Uneven Cash Flow In the last example, would PV be larger or smaller if the opportunity cost rate had been smaller? Why? Check your answer using a bigger opportunity cost, say 5% PV @ 5% = ___________

Complications: Semi-annual and other compounding periods Example: Calculate the FV of $100 invested for 2 years at 8% if interest is compounded annually, semiannually and monthly. Annually: r = 8% 1 2 $100 FV = ?

Complications: Semi-annual and other compounding periods On the financial calculator: n = i = PV = Pmt = FV =

Complications: Semi-annual and other compounding periods Semiannually: r = 8% 1/2 1 1 1/2 2 $100 FV = ?

Complications: Semi-annual and other compounding periods On the financial calculator: n = i = PV = Pmt = FV =

Complications: Semi-annual and other compounding periods Monthly: n = i = PV = Pmt = FV =

Complications: Semi-annual and other compounding periods Note: Simple (or quoted) interest rate is often what people quote as your interest rate for loans, bank accounts, credit cards and bonds. In the last example, the simple or quoted interest rate was 8% Effective annual rate (or ERA) interest rate is the annual rate actually being earned or paid, when compounding is considered.

Complications: Semi-annual and other compounding periods Effective Annual Rate (EAR) = (1+(isimple/n)n - 1

Complications: Semi-annual and other compounding periods (n is the number of times interest ins compounded each year) What was the EQAR for each compounding frequency in the last Example Annual: EAR = Semiannual: EAR = Monthly: EAR =

Fractional Time Periods Example: Calculate the PV of $100 invested for half a year in a bank account that pays an EAR of 8%

Fractional Time Periods Example: Calculate the FV of $100 invested for half a year in a bank account that pays an EAR of 8%

Fractional Time Periods Example: Calculate the PV of $100 invested for 9 months in a bank account that pays an EAR of 8%

Fractional Time Periods Example: Calculate the FV of $100 invested for 9 months in a bank account that pays an EAR of 8%

Amortized Loans These are loans that are repaid in a series of equal payments. Example: You need $1,000 today to buy a PC. IF you can get a loan at 10% per year for 3 years, how much will you annual payments be, assuming the first payment is in exactly one year. What type of annuity is this?

Amortized Loans Draw a timeline Loan Amortization Schedule Year BegBal Pmt Int Prin EndBal 1 2 3 Total

Comparison of Different types of Interest Rates Simple (or quoted) interest rate is often what people quote as your interest rate for loans, bank accounts, credit cards and bonds. Effective Annual rate (or EAR) interest rate is the annual rate actually being earned or paid, when compounding is considered. Periodic rate is the rate collected or paid each period. Annual percentage rate (APR) is the rate reported (as required by law) to borrowers - it is the periodic rate times the number of periods in the year. So the actual effects of compounding are not considered. (Same as the simple rate)

Comparison of Different types of Interest Rates Periodic Rate = isimple / n isimple = Periodic rate x n Effective Annual Rate (EAR) = (1+(isimple/n)n - 1

Comparison of Different types of Interest Rates Example: Suppose your Credit Card has an APR of 18%. What is your effective annual interest rate? (Interest is compounded once per month)

Comparison of Different types of Interest Rates Example: Suppose you have a house mortgage at 7.5%. What is your effective annual interest rate? (you make payments monthly)