Future Value with Multiple Cash Flows
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Because of the principle of Value Additivity, the future value of a set of cash flows is equal to the sum of the future values of the individual flows.
Present Value with Multiple Cash Flows
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Similarly, the present value of a set of cash flows is equal to the sum of the present values of the individual flows.
A Note on Cash Flow Timing In solving time value problems, it is important to specify when each cash flow occurs: at the beginning each period, or at the end. Unless told otherwise, we generally assume cash flows occur at the end of each period.
Valuing Level Cash Flows: Annuities and Perpetuities
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Present Value for Annuity Cash Flows The fundamental present value equation for an ordinary annuity is: Annuity Present Value = C ´ (1 - 1/(1 + r)t)/r, where C is the cash flow, the second is called "PVIFA(r,t)", and the Annuity Present Value equation is: Annuity Present Value = C ´ PVIFA(r,t). Suppose you have decided to purchase a new supercharged Jaguar XJR. You borrow $70,000, and promises to pay the loan back in equal installments at the end of each of the next 6 years. The loan rate is 10%. The size of each payment is $ _____. CONCEPT CHECK
Future Value for Annuities The fundamental future value equation for an ordinary annuity is: Annuity Future Value = C ´ [(1 + r)t - 1]/r. Again, C is the cash flow, and the second term is called "FVIFA(r,t)." The Annuity Future Value equation is: Annuity Future Value = C ´ FVIFA(r,t). Example: Assume you are saving for a child's college education. You have $50,000 now, can add $100 monthly for the next four years, and will earn 5% on your money. How much will you accumulate? Check your calculations by going to this financial calculator website and plugging in the same numbers.
A Note on Annuities Due
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When payments occur at the beginning of each time period, we have an annuity due. The present value equation for an annuity due is C ´ PVIFA(r,t) ´ (1 + r). The future value equation for an annuity due is C ´ FVIFA(r,t) ´ (1 + r).
Perpetuities A perpetuity is an annuity where t is equal to infinity. The present value of a perpetuity is C/r. What is the value today of $1000 perpetuity if the discount rate is 10%? CONCEPT CHECK
Comparing Rates: The Effect of Compounding Periods
Effective Annual Rates and Compounding
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When there is more than one compounding period per year, the stated rate will not be equal to the effective annual rate.
Calculating and Comparing Effective Annual Rates
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The effective annual rate (EAR) is equal to (1 + Quoted rate/m)m - 1.
EARs and APRs
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The annual percentage rate (APR) is equal to the interest rate charged per period times the number of periods per year. MortgageBanker.com will loan you money to buy a home at a stated rate of 8%. Assume you make monthly payments. The APR for this loan is _____; the EAR for this loan, however, is _____ . CONCEPT CHECK
D. EARs, APRs, Financial Calculators, and Spreadsheets
Loan Types and Loan Amortization
Pure Discount Loans
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With a pure discount loan, a borrower receives money today, and repays a single lump sum (representing principal and interest) at a future date.
Interest-Only Loans
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By contrast, one who borrows via an interest-only loan pays interest each period and the principal at a future date.
Amortized Loans
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Finally, an amortized loan calls for one to make a series of payments which include both interest and reductions of principal.
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