Evaluating Lump Sums and Annuities (Calculator Approach)

Step 1: Quick Take: Evaluating Lump Sums and Annuities

Present value (PV) represents the value of a future cash flow, or series of cash flows, in todays terms. In a sense, taking the present value of a cash flow, or cash flows, is the opposite of calculating the future value of a cash flow. In calculating future values, you are compounding a cash flow to analyze how much you will have in the future given a rate of return. For present values of a future cash flow (or flows), you are calculating the amount of cash today that you would need today in order to invest and end up with the amount of that future cash flow. In other words, you are discounting future cash flows instead of compounding. That is, given a present value PV you use compounding to calculate the future value (FV). If you know the future value, you can use discounting to calculate the present value.

Since annuities involve fixed payments at regular intervals into the future, calculating the present value of these future annuity payments is a common task in finance. You can calculate the present value of an ordinary annuity using a formula, a step-by-step process, a financial calculator, or a spreadsheet. For a given fixed payment PMTPMT, interest rate II, and periods from 1 to NN, the present value of an annuity PVANPVAN is calculated as:

PVAN=PMT(1+I)1+PMT(1+I)2+...+PMT(1+I)NPVAN=PMT1+I1+PMT1+I2+...+PMT1+IN

Which represents a step-by-step process, since you would calculate the present value of each payment individual annuity payment. This simplifies to a more condenses formula of:

PVAN=PMT11(1+I)NIPVAN=PMT111+INI

Once this present value is calculated, it then becomes more intuitive to compare an annuity to a lump sum received today, since a lump sum of cash received today is already a present value amount.

However, most commonly these calculations are done in a spreadsheet or financial calculator, which you will practice in the next stage of this problem.

True or False: Given the future value of a cash flow, you can calculate the present value by discounting that future cash flow.

True

False

Step 2: Learn: Evaluating Lump Sums and Annuities

Calculating the present value of an annuity, while also comparing present values to lump sums, is an important part of finance.

Bob just won the lottery and must choose between three award options:

1.	A lump sum of $20,000,000 received today
2.	15 end-of-year payments of $2,500,000
3.	40 end-of-year payments of $1,800,000

For each option in the table, indicate which values to enter for each variable in your financial calculator.

	Option 1		Option 2		Option 3
	Lump Sum Payment		15 Payments		40 Payments
No. of Periods			PV = (options: 15,20,1,40)		PV = (options:1,20,15,40)
Annual payment			PV = (options:$2,500,000, $208,333,$1,800,000,$5,000,000)		PV = (options:$2,500,000, $208,333,$1,800,000,$5,000,000)
Future Value			FV= 0		FV = 0
Present Value	$20,000,000		?		?

Assume the interest rate is 8.00%, entered as 8 on your financial calculator.

Note: Take the absolute value of the present value when answering this question.

Using the table you just filled out, along with a financial calculator, yields a present value for option 2 of approximately (options:$23,538,566.39,$21,398,696.72, $$17,118,957.38, $19,258,827.05) and a present value for option 3 of approximately (options: $21,464,304.00,$15,025,012.80,$25,757,164.80,$27,903,595.20) (when the interest rate is 8.00%). Based on this, Bob should choose option (1,3,2) if he seeks to maximize present value.

Now assume the interest rate is 9.00%, entered as 9 on your financial calculator.

Note: Take the absolute value of the present value when answering this question.

Using the table you just filled out, along with your financial calculator, yields a present value for option 2 of approximately ($26,197,237.39,$22,166,893.18, $20,151,721.07, $24,182,065.28) and a present value for option 3 of approximately ($13,554,273.85, $17,426,923.52, $15,490,598.68, $19,363,248.35) (when the interest rate is 9.00%). Based on this, Bob should choose option (2,1,3) if he seeks to maximize present value.

Assume the interest rate is 10.00%, entered as 10 on your financial calculator.

Note: Take the absolute value of the present value when answering this question.

Using the table you just filled out, along with your financial calculator, yields a present value for option 2 of approximately ($24,719,758.40,$ 19,015,198.77, $20,916,718.65, $22,818,238.52) and a present value for option 3 of approximately ($14,081,833.03 , $12,321,603.90, $17,602,291.29,$15,842,062.16) (when the interest rate is 10.00%). Based on this, Bob should choose option (1,3,2) if he seeks to maximize present value.

As the interest rate increases, option 1 becomes (more/less) attractive.