Quantitative Methods: Basic Concepts

Reading 6. The Time Value of Money

Learning Outcome Statements

e. calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows;

CFA Curriculum, 2020, Volume 1

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Subject 5. The Future Value and Present Value of a Series of Uneven Cash Flows

A series of uneven cash flows means that the cash flow stream is uneven over many time periods. There is no single formula available to compute the present or future value of a series of uneven cash flows.

Present Value

When we have unequal cash flows, we must first find the present value of each individual cash flow and then the sum of the respective present values. (This is usually accomplished with the help of a spreadsheet.)

Future Value

Once we know the present value of the cash flows, we can easily apply time-value equivalence by using the formula to calculate the future value of a single sum of money (LOS a).


John wants to pay off his student loan in three annual installments: $2,000, $4,000 and $6,000, respectively, in the next three years. How much should John deposit into his bank account today if he wants to use the account balance to pay off the loan? Assume that the bank pays 8% interest, compounded annually.

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User Contributed Comments 9

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How are unequal future cashflows done useing a HP 12c calculator.


You can use an NPV procedure, with the CF0=0. (disclosure: HP12C calculator).


for uneven cash flow, you need to calculate pv for each period, and add them together.


You can do cash flows on the BAII also. insert under the CF function and solve for NPV.


I have a doubt with the example. which is the correct answer to the question: How much should John deposit into his bank account today if he wants to use the account balance to pay off the loan?

Considering that John should desposit at the begining of the first year (today) 1.852 as he has to pay on the second year 3.429 and the third year 4.763 in order to have at the end of the third year 12.000. So the answer is, today John should deposit only 1.852.

is that correct?


spashr...you are looking at this incorrectly. you don't have 12,000 at the end of the 3rd year as $2,000 is coming out the end of year 1, $4,000 coming out end of year 2 and $6,000 coming out end of year 3. You should technically have 0 at the end of 3 years as you have paid back the loan.


How to calculate using TI...? thanks...


(CLR WORK/TVM always)
CF, down arrow
2000 ENTER (C01 = 2000)
down, down 4000 ENTER (c02=4000)
down down 6000 ENTER (C03=6000)
NPV , 8, ENTER (I=8)
down CPT (NPV=10044.20)


Hi everyone, I am a bit confused by Example 18 in the CFA book: what is the difference between the growth rate of -2.9% and the growth rate of 0.862676 apart from the obvious calculation differences?