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```					Drexel University Bennett S. LeBow College of Business FIN311: Financial Management Instructor: L. Tan

DISCOUNTED CASH FLOW VALUATION
Valuing Cash Flow Bundles: Perpetuities & Annuities Perpetuity: a level stream of cash flows of \$C, beginning one period from today and going on forever.

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The present value of a perpetuity is
PV  C 
t 1 t 

1 C  t r (1  r )

Quick check: How much would you pay for an investment that will generate a cash flow of \$500 per year forever, if your required return is 10% per year?

Annuity: a level stream of cash flows of \$C, beginning one period from today and ending at date T. An annuity is equivalent to buying a perpetuity today, and agreeing to sell it at date T.

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The present value of an annuity is

PV  C 

t T

1 C C 1 C 1      1   t T r r (1  r ) r  (1  r )T  t 1 (1  r )

Quick check: You won the state lottery, which will pay you \$50,000 per year for the next 20 years. Quickcash Services offers to give you a lump sum today instead. How much of a lump sum would you need, if your required return is 10% per year?

The formulas described above serve as the basic building blocks for discounting cash flows. These formulas are for cash flow streams where the first cash flow occurs one period from today. Our approach will always be to compute the present value of the cash flows as of today (time zero). We can then move to any time in the future by multiplying by (1+r)t.

Effective annual rates and annual percentage rates The effective annual rate (EAR) converts a quoted rate to an annual rate after considering compounding. As an example, the EAR of loan quoted at 12%, compounded monthly, is

 0.12  EAR  1    1  0.1268, or 12.68% 12  
The annual percentage rate (APR) is the interest rate per period multiplied by the number of periods per year. The APR of a loan at 1% per month as noted above is simply APR = 0.01*12 = 0.12, or 12%

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Present Value: A Retirement Savings Example Susan Jones is currently 40 years old and is planning for her retirement. She plans to retire at age 65, and receive an annual benefit payment (pension) of \$20,000 for the following 15 years, i.e., until she is 80. What must her yearly contribution (occurring one year from today) be (assume occur yearly before retirement)? Assume a constant interest rate of 10%. Solution: Step 1 - draw the timeline of cash flows:

Step 2 - compute the present value of her pension (level annuity). Note that she will receive the first annuity payment at age 66, and therefore the PV calculated with the annuity formula is as of age 65, one period before the first payment.

Step 3 - compute the present value of her pension as of age 40 by discounting the above sum at 10% for 25 years EMBED "Equation" "Word Object1" \* mergeformat PV 40 ( pension) 
\$152 ,122  \$14 ,040 (1.10 ) 25

Step 4 - the present value of her pension must equal the present value of her contributions, so equate the two and solve for C. 14,040 = C(1.1)25 => C = \$1,295.64

Loan amortization Three common types of loans:


Pure discount



Interest only



Fully amortized

Example of loan amortization: Suppose you borrow \$10,000 at an interest rate of 14% and plan to repay the loan by making equal annual payments for 5 years. The value of each payment is \$2,912.84, computed as
\$10 ,000  C   1   1   0.14   1.14  
5

   

The amortization schedule is given in the following table: Year 1 2 3 4 5 Total Beginning balance 10,000.00 8,487.16 6,762.53 4,796.45 2,555.12 Total payment 2,912.84 2,912.84 2,912.84 2,912.84 2,912.84 14,565.17 Interest paid 1,400.00 1,188.20 946.75 671.50 357.72 4,564.17 Principal paid 1,512.84 1,724.63 1,966.08 2,241.33 2,555.12 10,000.00 Ending balance 8,487.16 6,762.53 4,796.45 2,555.12 0.00

Example of loan amortization: Construct an amortization table for a three-year loan of \$20,000 at an interest rate of 10%, assuming annual payments. Round all amounts to the nearest dollar. The value of each payment is:

The amortization schedule is given in the following table:

Year 1 2 3 Total

Beginning balance 20,000

Total payment

Interest paid

Principal paid

Ending balance

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