Chapter 4: Time Value of Money Concepts and Calculations

The amount of money deposited (generally) or received when an investment or loan begins is known as what?

Present Value (PV)

Identify the Present Value (PV) amount in the following scenario:

Ten years ago, Goldie bought a gold coin for $700. She sold it this morning for $12,000.

The Present Value amount is $700.

(Note: The $12,000 represents the Future Value (FV) at the end of the 10-year holding period.)

Financial calculators will not produce accurate results unless at least _____ of the following five values are input

three

When inputting data into a financial calculator for Time Value of Money (TVM) problems, the Present Value (PV) is almost always entered as a negative number except when dealing with what type of calculation?

Amortization

(Exam Tip: For standard investments, think of PV as an immediate cash outflow or "money leaving your pocket" to buy the asset, which is why it requires a negative sign.

For a loan or amortization problem, the PV is cash received up front, making it a positive input.)

Present Value (PV) is almost always a _____ (negative or positive) input, except for amortization.

negative

Future Value (FV) is almost always a _____ (negative or positive) input, except it can also be _____.

• positive

• zero

Payment (PMT) can be a _____ or _____ input, depending on whether cash is flowing _____ or _____.

negative or positive & in or out

(Note: If your client is making recurring payments out of their pocket, PMT is negative.

If your client is receiving a stream of recurring payments, PMT is positive.)

Investments:

• Present Value (PV) = _______ ( - or + )
  
  

• Future Value (FV) / Payments to Client (PMT) = _______ ( - or + )

PV = Negative (-)

• Money leaves pocket to buy asset.

FV / PMT = Positive (+)

• Money returns to pocket at maturity.

Loans / Amortization:

• Present Value (PV) = _______ ( - or + )
  
  

• Payments (PMT) = _______ ( - or + )

PV = Positive (+)

• Lump sum received up front from lender

PMT = Negative (-)

• Monthly payments leave pocket to pay down debt.

On financial calculators, what does N or n represent?

The total number of compounding periods.

On financial calculators, what does I/YR (or i) represent?

The interest rate per compounding period.

1. A client deposits $10,000 into a short-term certificate of deposit (CD) that boasts a 7.5% annual interest rate, compounded monthly. However, the client intends to liquidate the account and withdraw the funds after exactly 3 months.

• To calculate the exact Future Value (FV) of this investment, what precise values must you input for N and I/YR?

2. Calculate the exact total dollar amount he will have after interest is accrued over that 3-month period.

• N = 3 (3 total compounding months)
  
  

• I/YR = 0.625 (7.5% annual rate ÷ 12 months)

$10,188.67

• PV = -10,000

• N = 3

• I/YR = 0.625 (7.5%/12months)

• PMT = 0

• CPT* FV* = $10,188.67

From a borrower's perspective, a $250,000 mortgage loan should be entered into a financial calculator as a _____ (negative or positive) Present Value (PV), while the recurring monthly payments are entered as a _____ (negative or positive) Payment (PMT).

positive & negative

(Exam Tip: Think of it entirely by cash direction at the moment it happens: you receive a lump sum of money upfront from the lender [+PV], and then you pay money out of your pocket every month to pay it back [-PMT].)

The process of bringing a future dollar amount or stream of cash flows back to its value today by removing the effect of compound interest is known as _____.

discounting

Exam Tip: "Compounding" moves money forward into the future to find FV, while "Discounting" moves money backward to the present to find PV. The interest rate used to do this is often called the discount rate.

To solve any TVM problem, you must learn how to write out the _____ for your specific financial calculator, using the data given in the question to fill in your variables.

keystrokes

(Exam Tip: On the CFP® exam scratch pad, always write out your 5 TVM variables vertically [N, I/YR, PV, PMT, FV] and fill in the knowns before punching buttons. This prevents input errors and keeps your logic clean.)

How do you change the BA II Plus calculator to 1 payment per year?

• Press *2ND then Press [P/Y] (located on I/Y)
  
  

• Press 1, then press ENTER
  
  

• Press 2ND then [QUIT] (located on CPT)

How do you change the BA II Plus calculator to 1 payment per year?

• Press *2ND then Press [P/Y] (located on I/Y)
  
  

• Press 1, then press ENTER
  
  

• Press 2ND then [QUIT] (located on CPT)
  
  

What are annuity payments?

Recurring amounts paid out or received at regular intervals.

Whenever a Time Value of Money (TVM) calculation involves a repeated payment (PMT), what setting must you verify first on the calculator?

The Timing Mode. You must ensure the calculator is set to either BGN (Begin) or END mode to match when the cash flows occur.

How do you determine whether to use BGN mode or END mode for an annuity?

Payments are either positive (+) or negative (-).

What determines whether a payment (PMT) is entered as a positive (+) or negative (-) number in financial calculator calculations?

The direction of the cash flow.

• Cash coming in (received) is positive (+).
  
  

• Cash going out (paid) is negative (-).

How do you determine the positive (+) or negative (-) sign for PMT in annuity questions?

Based on who receives or pays the cash flow.

• Positive (+): Annuity payouts/distributions received by the client (money entering their checking account).
  
  

• Negative (-): Premium payments or systematic contributions made by the client into the annuity (money leaving their checking account).

If money is deposited into a client’s checking account, the PMT is _____ (positive or negative).

Positive.

If money moves out of the client’s checking account, the PMT is _____ (positive or negative).

Negative

If a client receives interest from a CD or bond, is the PMT positive or negative?

Positive

If a client receives dividends, is the PMT positive or negative?

Positive

If a client invests money into mutual funds, stocks, bonds, or an IRA from a checking account, is the PMT positive or negative?

Negative

If a clients employer deposits money into their 401(k) is the PMT positive or negative?

Negative

For scenarios like college tuition being paid, retirement benefits being received, and family needs, do you use Begin Mode or End Mode?

Begin Mode

For scenarios like 401(k) deferrals, pension and profit-sharing contributions, bond interest paid, and mortgage payments, do you use Begin Mode or End Mode?

End Mode

For Pension and Profit-Sharing contributions, do you use Begin mode or End mode, and what is the underlying logic?

END mode.

• Explanation: Just like 401(k) deferrals, these contributions are tied to company profits and employee service.
  
  A business cannot calculate or deposit profit-sharing or pension funding for a period until that period has closed and the work has been performed.
  
  Therefore, the funding happens at the end of the cycle.

When calculating standard mortgage payments, which timing mode must you use on your financial calculator? Begin mode or End mode?

END mode.

• Explanation: Because when you buy a house and move in on October 1st, you do not pay the bank for October right away.
  
  You live in the house for the entire month first. Your November 1st payment pays for the month of October that just passed.
  
  Because you pay after occupying the space (in arrears), it is an END mode calculation.

When calculating bond interest paid to a client, which timing mode must you use on your financial calculator?

END mode.

• Explanation: Bonds pay interest in arrears.
  
  The issuer holds and uses your principal investment for the period first, then pays you the interest at the very end of that timeframe.

If a questions asks you to solve for how much the client paid for an annuity, what are you solving for on the calculator?

Present Value

A client expects to receive an annuity payment of $1,000 at the end of each of the next 3 years. If the annuity pays 8% per year, what are your inputs for N, I/Y, and FV?

N = 3

I/Y = 8

FV = 0

• The account hits zero after the last payment is made.

Calculate the upfront price a client pays for an 8% annuity delivering $1,000 at the end of each year for 3 years.

PV = - $2,577.10

• 3 → N

• 8 → I/Y

• ? → PV

• 1000 → PMT

• 0 → FV

• Press CPT → PV

Note: The result shows as a negative number because it represents the cash the client must pay out of pocket today to buy the contract.

In a standard annuity payout question with a fixed term, if no leftover lump sum or residual value is mentioned, what is FV?

Zero!

Susan Nelson deposited $500 from her checking account into her bond account at the end of each month for 4 years, at which point she had accumulated $27,000. If the interest was compounded monthly, what was the annual compound rate of return on the account over the 4-year period?

N =

IY =

PV =

PMT =

FV =

5.91%

N = 48

IY = *CPT *N = 0.4926% x 12 months = 5.91% Annual Return

PV = $0

PMT = -$500

FV = $27,000

Note 1: Make sure to change the mode to End Mode

Note 2: "Compounded monthly" means multiplying years by 12 to get total monthly periods. Because your periodic inputs match months, the solved I/Y is a monthly rate that must be annualized (x 12).

Doug Spencer wants to accumulate $150,000 for retirement in 8 years. He expects to earn an annual return of 13% (compounded semiannually) on investments. How much should Doug invest at the beginning of each 6-month period to attain his goal?

N =
I/Y =
PV =
PMT =
FV =

$5,264.45

N = 16 (8 years x 2 semiannual periods)
I/Y = 6.5% (13% annual / 2 semiannual periods)
PV = $0
PMT = *CPT *PMT = - $5,264.45
FV = $150,000

Note 1: Make sure to change the mode to BGN Mode (payments occur at the "beginning of each 6-month period").

Note 2: "Compounded semiannually" means modifying both N (years \(\times \) 2) and I/Y (annual interest / 2) to match the 6-month period lengths before solving for the payment.