*Chapter Recaps: my notes to chapters of the financial planning coursework material through New York University’s CFP ^{®} Program (in conjunction with Dalton Education).*

**Chapter 7: Time Value of Money**

**Introduction**- Time Value of Money (TVM) = a mathematical concept that determines the value of money at a point or over a period of time, at a given rate of interest.
- Present value = the value today of one or more future cash flows discounted to today at an appropriate interest rate.
- Future value = value at some point in the future of a present amount or amounts after earning a rate of return, for a period of time.
- TVM allows for various planning questions to be answered including the FV of retirement savings, required education savings, house or car loan monthly payments, business purchase decisions, and annual rate of return on an investment.

**Approach for Solving Time Value of Money Calculations**- The Four Step Approach:
- Start with a timeline of cash flows.
- Write down the TVM variables.
- Clear all registers in the calculator.
- Populate the TVM variables in the calculator.

- The Four Step Approach:
**Time Value of Money**__Concepts__**Present Value of $1**- Current value today of a future amount. Used to calculate how much should be deposited today to meet a financial goal in the future.

**Future Value of $1**- Value of a a present lump-sum deposit after earning interest over a period of time. Used to determine a future amount based on today’s lump-sum deposit that will earn interest.

**Periods of Compounding Other than Annual**- Need to adjust the N and I/Y variables if leave c/y in calculator at 1 (annual). Multiply N by the number of
compounded and divide I/Y by the same.**periods**

- Need to adjust the N and I/Y variables if leave c/y in calculator at 1 (annual). Multiply N by the number of
**Annuities**- Annuity = a recurring CF, of an equal amount that occurs at periodic (but regular) intervals. Annuities are reflected on a calculator as a PMT (payment).
- Ordinary annuity = occurs when the timing of the first payment is at the
of the year.**end**- Examples: debtor payments (car, student, or mortgage loans) and many savings contributions to an IRA or 401(k) if regular and recurring and made at period end.

- Annuity due = occurs when the timing of the first payment is at the
of the period.**beginning**- Examples: rents (as they are usually paid in advance), tuition payments, and retirement income.

- Ordinary annuity = occurs when the timing of the first payment is at the

- Annuity = a recurring CF, of an equal amount that occurs at periodic (but regular) intervals. Annuities are reflected on a calculator as a PMT (payment).
**PV of an Ordinary Annuity of $1 (Even CF’s)**- PV of an Ordinary Annuity = today’s value of an even CF stream received or paid over time when the payments are made at the
of the period.**end**

- PV of an Ordinary Annuity = today’s value of an even CF stream received or paid over time when the payments are made at the
**PV of an Annuity Due of $1 (Even CF’s)**- PV of an Annuity Due = today’s value of an even CF stream received or paid over time when the payments are made at the
of the period.*beginning*

- PV of an Annuity Due = today’s value of an even CF stream received or paid over time when the payments are made at the
**FV of an Ordinary Annuity of $1 (Even CF’s)**- The value of equal periodic payments or deposits made at the
**end**of a period, at some point in the future. - Example: helps calculate the value of saving contributions earning a constant compounded rate of return.

- The value of equal periodic payments or deposits made at the
**FV of an Annuity Due of $1 (Even CF’s)**- The value of equal periodic payments or deposits made at the
**beginning**of a period, at some point in the future.

- The value of equal periodic payments or deposits made at the
**FV of an Ordinary Annuity vs. Annuity Due Comparison**- The additional earnings under an annuity due are attributable to the additional compounding of the first deposit.

**Ordinary Annuity Payments from a Lump-Sum Deposit**- The payments that can be generated at the
**end**of each period based on a lump-sump amount deposited today.- Example: helps calculate the amount of payment required to repay a loan, and income payments that can be generated from a lump-sum amount.

- The payments that can be generated at the
**Annuity Due Payments from a Lump-Sum Deposit**- The payments that can be generated at the beginning of each period, based on a lump-sum amount deposited today.
- Examples: helps calculate amount of retirement income payments that can be generated from a lump-sum amount. Also, the periodic income payments that can be generated from a lump-sum amount (lottery example).

- The payments that can be generated at the beginning of each period, based on a lump-sum amount deposited today.
**Solving for Term (N)**- Provides the amount of time needed to achieve a financial goal.
- Examples: helps calculate the amount of time required to attain an account balance given a rate of return. Also, the amount of time to retire a debt.

- Provides the amount of time needed to achieve a financial goal.
**Solving for Interest Rate (i)**- Provides the rate required to attain a certain goal and also can determine the rate being charged on a debt obligation.

**Uneven Cash Flows**- When an investment or project has
**uneven****dollar amounts****OR****intervals**

- When an investment or project has
**Net Present Value**- NPV is the excess or shortfall of cash flows based on the discounted present value of future cash flows minus the initial cost of the investment.
- The investor’s required rate of return is used as the discount rate.
- NPV assumes the CF’s are reinvested at the required rate of return or discount rate.

- Formula:
- NPV = PV of Future CFs – Cost (Initial Outlay)
- Similarly: PV of CF’s = NPV + Cost (Initial Outlay)

- NPV = PV of Future CFs – Cost (Initial Outlay)
- Interpretation:
- A positive NPV means the project or investment is generating CFs in
*excess*of what is required based on the required rate of return. - A negative NPV means it is not.
- A zero NPV means its generating a return rate equal to the required rate of return.

- A positive NPV means the project or investment is generating CFs in
- Used by managers in capital budgeting and investors for evaluating investment alternatives.

- NPV is the excess or shortfall of cash flows based on the discounted present value of future cash flows minus the initial cost of the investment.
**Internal Rate of Return (IRR)**- IRR = the compound rate of return that equates the cash
to the cash*inflows***outflows***.*It’s the rate that causes the sum of the present value of all discounted cash flows to equal the cost of the investment.- Important assumption: IRR assumes CFs are reinvested at the IRR.

- Allows to compare projects or investments with differing costs and CFs:
- Accept the project/investment when the IRR equals or exceeds the required rate of return. Reject if less than.

- IRR = the compound rate of return that equates the cash
**Inflation Adjusted Rate of Return**- Adjusts the nominal rate of return (the actual rate of return earned) into a real rate of return (after inflation):
- Inflation-Adjusted, aka, Real Rate of Return = {[(1+Nominal)/(1+Inflation)]-1} *100

- In-practice:
- Used when an account balance is growing at one rate of return and simultaneously an expense is growing at a different rate of return. Example: Education.
- Also, when an investment return is at one rate and inflation at another rate. Example: Retirement.

- Adjusts the nominal rate of return (the actual rate of return earned) into a real rate of return (after inflation):
**Serial Payments**- Serial payments are payments adjusted upward periodically through the payment period at a constant rate, usually to adjust for inflation’s impact. Each serial payment will increase to maintain the real dollar purchasing power of the investment.
- Calculate the payment like an ordinary annuity with an inflation-adjusted I/Y. Then add inflation to the answer for year 1, and compound for each subsequent payment (page 288 for an example).

- Serial payments are payments adjusted upward periodically through the payment period at a constant rate, usually to adjust for inflation’s impact. Each serial payment will increase to maintain the real dollar purchasing power of the investment.
**Debt Repayments**- Time value of money can be used to calculate periodic payments for retiring any type of debt obligation (student loans, credit cards, mortgages, or car loans). Can also determine interest and principal paid over a period of time with Amortization schedules or the AMORT calculator function.
- Important notes:
- Most debt repayments are ordinary annuities, so calculations are done in end mode.
- Even though most mortgage payments are made at the beginning of the month, the repayment is still an ordinary annuity (because each payment includes a portion of
repayment and**principal****interest**

**Amortization Schedule**- Shows the repayment of debt over time.
- Each payment includes both interest and principal repayment – the further along into the schedule, the bigger the amount that goes towards the remaining principal.

- Shows the repayment of debt over time.

**Other Practical**__Applications__- Intro section: Time value of money can also be used for other decisions clients face:
**Cash Rebate or Zero Percent Financing**- Figure out the total cost (calculate the payment amounts for each option and multiply by their respective total periods) for each option and pick the lower of the two.

**Payment of Points on a Mortgage**- Points on a mortgage are a percentage of the borrowed amount that is paid to the lender. Higher points amount means a lower interest rate.
- The decision depends on how long the client is going to own the property in order to recoup the points paid through the lower interest rate:
- Step 1: determine how much “paying the points” will cost:
- Total points buying * amount financed.

- Step 2: determine the payment amounts for each option using TVM on calculator.
- Step 3: determine the cost savings between the two payments and divide into the total points cost. The result is the amount of years after which it would make more sense to use the pay the points option (aka, the point when paying a higher monthly payment from not paying points starts to cost more overall).

- Step 1: determine how much “paying the points” will cost:

**Lottery Winnings – Lump Sum or Annuity**- Can use TVM to find the rate that makes the lump-sum payout option equal to the annuity payments option. If can invest the lump-sum payout and earn a return greater than that rate (without risk), it would make sense to take the lump-sum.

- Intro section: Time value of money can also be used for other decisions clients face: