Basic Time Value Concepts Study Notes

Key Concepts

Time Value of Money Principle: A dollar received today is worth more than a dollar promised at some time in the future because of the opportunity to invest and earn interest.
Importance of Time Value Concepts: Essential in financial reporting for various measurements (historical cost, net realizable value, fair value) and for personal financial decisions (buying homes, retirement planning, investment evaluation).
Present Value Techniques: Used to convert expected future cash flows into present values, providing an estimate of fair value, especially for Level 3 fair value measurements where market data is limited. Applications include valuing notes, leases, pensions, and long-term assets.
Understanding Interest: Interest is the payment for the use of money. It involves a Principal (the amount borrowed or lent), the Interest amount itself, and an Interest Rate (typically expressed as a percentage).
Simple Interest: Interest is calculated only on the original principal amount, regardless of any interest that may have accrued in the past.
Compound Interest: Interest accrues on the principal and any unpaid interest from past periods. This is known for its powerful effect over time.
Rate of Interest: Interest is usually expressed as an annual rate. However, if the compounding period is shorter than one year, the interest rate for that shorter period must be determined.
Annuity: A series of equal payments or receipts, referred to as "rents," that occur at equal intervals of time.
- Ordinary Annuity: An annuity where each payment or receipt (rent) is payable or receivable at the end of each period.
- Annuity Due: An annuity where each payment or receipt (rent) is payable or receivable at the beginning of each period.
Future Value: The value at a later date of a single sum or a series of sums (an annuity) that is invested at compound interest.
- Future Value of 1 (or value of a single sum): The future value of 1$ (or a single given sum) at the end of n $periods at an interest rate$ i $.</li><li>Future Value of an Annuity: The accumulated total that results from a series of equal deposits made at regular intervals, which are then invested at compound interest. Both the deposits themselves and the interest earned contribute to this accumulation.<ul><li>Future Value of an Ordinary Annuity: The future value calculated on the date of the last payment or receipt (rent).</li><li>Future Value of an Annuity Due: The future value calculated one period after the date of the last payment or receipt (rent).</li></ul></li></ul></li><li>Present Value: The value at an earlier date (typically now) of a given future sum or a series of sums (an annuity) discounted at compound interest.<ul><li>Present Value of 1 (or present value of a single sum): The present value (worth) of$ 1$ (or a given sum) due $n$ periods from now, discounted at an interest rate $i$ .
- Present Value of an Annuity: The present value (worth) of a series of payments or receipts (rents) discounted at compound interest. It represents the sum that, when invested at compound interest, will allow for a series of equal withdrawals at regular intervals.
 - Present Value of an Ordinary Annuity: The value now of 1$ to be received or paid at the end of each period (rents) for n $periods, discounted at an interest rate$ i $.</li><li>Present Value of an Annuity Due: The value now of$ 1$ to be received or paid at the beginning of each period (rents) for $n$ periods, discounted at an interest rate $i$ .

Key Terms

Annuity: A series of equal payments or receipts (rents) that occur at equal intervals of time.
Annuity Due: An annuity where each payment or receipt (rent) occurs at the beginning of the period.
Compound Interest: Interest that accrues on the unpaid interest of past periods as well as on the principal.
Deferred Annuity: An annuity in which the first payment or receipt is delayed until a specified future date.
Discounting: The process of determining the present value of a future cash flow by applying a discount rate.
Effective Yield: The actual annual rate of return earned on an investment, considering the effects of compounding.
Expected Cash Flow Approach: A valuation technique that uses a probability-weighted average of expected future cash flows to estimate fair value, especially in situations where a single future cash flow is uncertain (often referred to as Level 3 in the fair value hierarchy).
Face Rate / Stated Rate / Nominal Rate: The interest rate that is explicitly stated in a loan or bond agreement, before accounting for compounding frequency or fees.
Fair Value: The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date.
Future Value (FV): The value at a later date of a single sum or a series of sums (annuity) that is invested at compound interest.
Future Value of an Annuity: The accumulated total that results from a series of equal deposits made at regular intervals and invested at compound interest.
Historical Cost: The original cost of an asset recorded in accounting records.
Interest: Payment for the use of money.
Interest Rate: The percentage charged or earned on the principal amount over a period.
Net Realizable Value: The estimated selling price in the ordinary course of business less the estimated costs of completion and the estimated costs necessary to make the sale.
Ordinary Annuity: An annuity where each payment or receipt (rent) occurs at the end of the period.
Present Value (PV): The value at an earlier date (usually now) of a given future sum or series of sums (annuity) discounted at compound interest.
Principal: The initial amount borrowed or lent.
Risk-Free Rate of Return: The theoretical rate of return of an investment with zero risk, often approximated by the yield on a U.S. Treasury bond.
Simple Interest: Interest that is calculated only on the original principal amount, regardless of interest that may have accrued in the past.
Time Value of Money: The fundamental principle that a dollar received today is worth more than a dollar promised at some time in the future due to its potential earning capacity.

Formulas

Simple Interest: $\text{Interest} = p \times i \times n$
- Where: $p = \text{principal}$ , $i = \text{interest rate}$ , $n = \text{number of periods}$
Future Value Factor Formula: $FV_{Fn, i} = (1 + i)^n$
Present Value Factor (PVF) Formula: $PV_{F,n,i} = \frac{1}{(1 + i)^n}$
Future Value of an Ordinary Annuity: FVFOA=R×FVFF−n,i
- Where: $R = \text{rent/periodic payment}$ , $FVF = \text{future value factor}$
Future Value of an Annuity Due of 1 (using ordinary annuity factor): $\text{Value of annuity due of 1 for n rents} = (\text{Value of ordinary annuity for n rents}) \times (1 + \text{interest rate})$
Present Value of an Ordinary Annuity: PVOAn,i=R×PVFOAn,i
- Where: $R = \text{rent/periodic payment}$ , $PVF = \text{present value factor}$
Present Value of an Annuity Due of 1 (using ordinary annuity factor): $\text{Present value of annuity due of 1 for n rents} = (\text{Present value of an ordinary annuity of n rents}) \times (1 + \text{interest rate})$