Chapter 5: TVM 2- Analyzing Annuity Cash Flows

PV - Several Cash Flows Ex. Assumptions: •Deposit $100 today •Deposit $125 next year •Deposit $150 at end of year 2 •Interest rates: 7%

$100 today = $100 $125 next year N=1 I=7 PV=? Pmt=0 FV=-125 PV= $116.82 $150 at end of year 2 N=2 I=7 PV=? Pmt=0 FV=-150 PV=$131.02 Total= 100 + 116.82 + 131.02 = $347.84

Time Value of Money Calculations

-Can deal with either single cash flows -Or multiple cash flows over time

Finding Payments on Amortized Loan

-Rearrange PV of annuity formula to solve for payment Payment = Present Value x Amortization

Annuity Loans

-compares payments -compares implied interest rate

PV - Multiple Annuities Ex. Assumptions (David Price's Contract): •$30 million per year from 2016 - 2018 •$31 million per year in 2019 •$32 million per year in 2020 - 2022 •Interest rates: 5% per year

30 million for 7 years N= 7 I= 5 PV= ? Pmt= -30 million FV= 0 PV= $185.16 million 1 million for 4 years N= 4 I= 5 PV= ? Pmt= -1 million FV= 0 PV= $3.55 million 1 million for 3 years N= 3 I= 5 PV= ? Pmt= -1 million FV= 0 PV= $2.72 million Total = $191.43

Compounding

Also called annuities -Value in the future -Same cash flow paid in every period

Finding FV - Several Cash Flows Example Assumptions: •Invest $100 today (compounds for 3 years) •Invest $125 at end of year 2 (compounds for 2 years) •Invest $150 at end of year 3 (compounds for 1 year) •Interest rates: 7%

Answer:

FV Multiple Annuities Example: Assumptions: •Invest $100 at end of years 1 - 3 at 8% •Invest $150 at end of years 4 - 5 at 8%

Compute $100 for 5 years and 50 for 2 years- N=5 I=8 PV=0 Pmt=-100 FV=? FV= $586.66 for $100 for 5 years N=2 I=8 PV=0 Pmt=-50 FV=? FV= $104 for $50 for 2 years Answer: 586.66 + 104 = $890.66

Future Value - Several Cash Flows

Concept: Compounding -Value in the future -Different cash flows paid in at different times

Prevent Value- Multiple Annuities

Concept: Discounting -Changing level cash flows -Example: David Price's baseball contract

Future Value of Annuity Due=

Future Value of Annuity x (1 + i)

EAR vs. APR Example Assumptions: •Borrow $100 today •12% annual interest rate •APR: Loan compounds annually - you pay 12.00% •EAR: Loan compounds monthly - you pay 12.68%

Good work

Amortized Loan

Loan in which borrower pays interest and principal over time

Present Value of Multiple Cash Flows

Multiple Cash Flows: -Car loans and home mortgage loans -Determining value of business opportunities

Effects of Compounding Frequency Assumptions: •$100 deposit today •12% annual interest rate •Bank compounds interest at six months instead of end of year •Interest is earned on interest

N= 1 x 2= 2 I= 12 / 2= 6 PV= -100 Pmt= 0 FV= ? FV= $112.36

Finding Payments on Amortized Loan Example Assumptions: •Need $10,000 to buy car •Loan term: 4 years •Interest rate: 9% APR •Use interest rate of 0.75 % (=9%/12) and 48 periods (=4× 12)

N= 4 x 12 = 48 I= 9/12 = .75% PV= -10,000 Pmt= ? FV= 0 Pmt = $248.85

Future Value of Annuity Ex. Assumptions: •Assumes cash flows at the beginning of each period •5 annuity-due cash flows of $100 each •First cash flow compounds for 5 years •Last cash flow compounds for 1 year •Interest rates: 8%

N= 5 I= 8 PV= 0 Pmt= -100 FV= ? FV= 586.66 586.66 x (1.08)= $633.59

Annuity FV Example: Assumptions: •Invest $100 at the end of each year for 5 years •Interest rates: 8%

N= 5 I=8 PV= 0 Pmt= -100 FV= ? FV= $586.66

PV of an Annuity Ex. Assumptions: •$100 payments at end of each year for 5 years •Interest rates: 8% per year

N=5 I=8 PV=? Pmt=-100 FV= 0 PV= $399.27

Ordinary Annuity vs. Annuity Due

Ordinary Annuity - Payment occurs at the end of each period Annuity Due - Payment occurs at the beginning each period (immediately) -Will say that it's realized at the beginning of period -Shifts everything 1 year

PV of a Perpetuity=

PMT/i

Present Value of Annuity Due Cont. Assumptions: •Cash flows at beginning of period •5 annuity-due cash flows of $100 •First cash flow paid today - not discounted •Last cash flow discounted 4 years •All cash flows discounted for one year less than ordinary annuity •Interest rates: 8%

Present Value of Annuity N= 5 I= 8 PV= ? Pmt= -100 FV= 0 PV= 399.27 399.27 x 1.08 = $431.21

Present Value of Annuity Due=

Present Value of Annuity x (1 + i)

Discounting

Present value of multiple cash flows -Value of future sum today -Different cash flows paid in at different times

Effective Annual Rate (EAR)

Quoted, or nominal rate -more accurate measurement of what you'll pay

Annual Percentage Rate (APR)

Rate that incorporates compounding

Perpetuity

Special Annuity -an annuity with cash flows that continues forever

Add-on Interest

a calculation of the amount of interest determined at the beginning of the loan and then added to the principal

Annuity

a stream of level and frequent cash flows paid at the end of each time period - often referred to as an ordinary annuity

Amortization schedule

a table detailing the periodic loan payment, interest payment, and debt balance over the life of the loan

Consols:

investment assets structured as perpetuities

Loan Principal

the balance yet to be paid on a loan

Annuity Due Time Line

•Cash flows at beginning, not at end of period •Five annuity-due cash flows basically same as payment today plus 4-year ordinary annuity •Payments occur one period sooner than ordinary annuity - earn extra period of interest

Future Value: Multiple Annuities

•Concept: Compounding - annuity equation to compute future value - two levels of cash flows •To solve for multiple annuities, compute FV for each separately and add them together

Perpetuity Cont.

•Concept: Discounting •Stream of level cash flows paid forever •Preferred stocks are an example •Value of investment is present value of all future annuity payments

Present Value of an Annuity Due

•Concept: Discounting •Today's value of future sum •Cash flows at beginning of each period

Future Value of Multiple Cash Flows

•Regular, evenly-spaced •Car loans and home mortgage loans •Saving for retirement •Companies paying interest on debt •Companies paying dividends

Compounding Frequency

•Used in situations that do not use yearly time periods •Semiannual bond payments •Quarterly stock dividends •Consumer loans - monthly payments