Difficulty: Intermediate | Prerequisites: Basic algebra, comfort with exponents
Time value of money (TVM) is one of the foundational tools in financial accounting. Nearly every long-term liability or asset valuation you encounter in intermediate accounting, from pension obligations to lease payments to bond pricing, relies on discounting future cash flows to the present or compounding present values forward. If you are behind on TVM, later chapters on bonds (Ch. 14), leases and pensions will be significantly harder. You should already be comfortable with basic algebra and the idea that a pound today is worth more than a pound tomorrow.
Money received today is worth more than the same amount received later, because it can earn interest in the interim. TVM concepts let you convert between present and future values using interest rates and time periods. In accounting, you will use these tools constantly to price bonds, value lease obligations and test assets for impairment.
Present value (PV)
The current worth of a future sum of money, discounted at a given interest rate. Think of it as: "What would I need to invest today to end up with that future amount?"
Future value (FV)
The amount a present sum will grow to over a stated number of periods at a given interest rate. In simple terms, this is what your money becomes after it earns interest.
Interest rate (r)
Also called the stated rate, coupon rate or nominal rate. The percentage charged or earned per period. Always check whether it is quoted annually or per compounding period.
Number of periods (n)
The total count of compounding intervals over the life of the investment or obligation. If a bond pays semiannually for 10 years, n = 20.
Simple interest
Interest calculated only on the original principal. The base never changes. Think of it as flat, non-compounding growth.
Compound interest
Interest calculated on both the original principal and any previously accumulated interest. This is how most real-world savings accounts and financial instruments work: interest earns interest.
Annuity
A series of equal payments made at equal intervals over a defined period. In simple terms, it is a stream of identical cash flows, like rent payments or bond coupons.
Ordinary annuity
An annuity where each payment occurs at the end of the period. Most bond coupon payments follow this pattern.
Annuity due
An annuity where each payment occurs at the beginning of the period. Lease payments often fall into this category.
Interest compensates lenders for two things: the risk that the borrower may not repay, and the erosion of purchasing power through inflation.
The higher the risk or expected inflation, the higher the interest rate demanded.
Simple interest is computed on the original principal only.
Example: $500,000 at 12% for 2 years. Year 1 interest = $60,000. Year 2 interest = $60,000. Total interest = $120,000.
Compound interest is computed on principal plus any accumulated unpaid interest.
Same example at compound rates: Year 1 interest = $60,000. Year 2 base becomes $560,000, so Year 2 interest = $67,200. Total interest = $127,200.
Compound interest is the norm in financial accounting. Simple interest appears mainly in short-term or textbook illustrations.
The interest rate (r) and the number of periods (n) must always be expressed in the same time units.
If a rate is quoted as 12% per year with semiannual compounding, the per-period rate is 6% and a 10-year instrument has n = 20.
Getting this conversion wrong is one of the most common exam errors.
One period: FV = PV x (1 + r)
Multiple periods: FV = PV x (1 + r)^n
Rearranging for PV: PV = FV / (1 + r)^n
The factor (1 + r)^n is sometimes called the future value factor. Its reciprocal, 1 / (1 + r)^n, is the present value factor.
An annuity is a series of equal cash flows at equal intervals.
Ordinary annuity: payments fall at the end of each period. Bond coupons are the classic example.
Annuity due: payments fall at the beginning of each period. Many lease contracts work this way.
To value an annuity, you can either discount each payment individually and sum the results, or (far more efficiently) use the present value of annuity (PVA) factor from the tables.
The PVA factor for an annuity due is the ordinary annuity factor multiplied by (1 + r), because each payment is one period closer to the present.
Pension liabilities: the obligation to employees is a stream of future payments, discounted to the present.
Bonds payable and bonds receivable: priced as the PV of future coupon payments plus the PV of the face amount at maturity.
Lease obligations: the lessee records a liability equal to the PV of future lease payments.
Impairment testing: the recoverable amount of an asset may be estimated using the FV of expected cash flows.
Formula | Expression | Use |
|---|---|---|
Future value of a single sum | FV = PV x (1 + r)^n | Growing a lump sum forward |
Present value of a single sum | PV = FV / (1 + r)^n | Discounting a lump sum back |
Simple interest (annual) | Interest = Principal x Rate | Flat interest, no compounding |
PV of an ordinary annuity | PV = PMT x PVA factor | Valuing end-of-period payment streams |
PV of an annuity due | PV = PMT x PVA factor x (1 + r) | Valuing beginning-of-period payment streams |
Remember: r and n must be in the same compounding units. If a rate is annual and payments are semiannual, divide r by 2 and multiply n by 2.
Banks use TVM to price every loan and mortgage they issue: the monthly payment on a 30-year mortgage is simply the annuity payment that makes the PV of the payment stream equal to the loan amount. Corporate treasurers use the same maths when deciding whether to invest in a new factory: if the PV of expected cash inflows exceeds the cost, the project adds value.
Students often forget to convert r and n into the same compounding frequency. If the annual rate is 12% and compounding is quarterly, the per-period rate is 3%, not 12%.
Students sometimes confuse an ordinary annuity with an annuity due. The difference is one period of discounting. Mixing them up shifts every valuation answer.
Some students treat simple interest and compound interest as interchangeable. They are not. Compound interest grows faster because interest earns interest.
Students occasionally apply the PV of a single sum formula when they should be using the PV of an annuity formula, or vice versa. A single lump sum at maturity is not the same as a stream of payments.
⚠️ Converting annual rates to per-period rates is tested constantly. Expect a question where the rate is annual but payments are semiannual or quarterly.
⚠️ Know the difference between ordinary annuity and annuity due. The exam may not label which one applies; you need to determine it from the payment timing described.
⚠️ Be comfortable moving between PV and FV in both directions. A question may give you the future value and ask for the present value, or the reverse.
⚠️ TVM is the foundation for bond pricing in Chapter 14. If you cannot discount cash flows confidently, bond questions become very difficult.
True or False: Simple interest and compound interest will always produce the same total interest over multiple periods.
False. Compound interest produces more because interest earns interest.
Fill in the blank: The formula for future value over n periods is FV = PV x ______.
(1 + r)^n
True or False: An ordinary annuity has payments at the beginning of each period.
False. That describes an annuity due. Ordinary annuity payments fall at the end.
Fill in the blank: If a bond pays semiannually for 20 years, the number of periods (n) is ______.
40
True or False: When the annual interest rate is 10% with quarterly compounding, the per-period rate is 2.5%.
True. 10% / 4 = 2.5%.
Q: A company invests $100,000 at 8% compounded annually for 3 years. What is the future value?
A: FV = $100,000 x (1.08)^3 = $100,000 x 1.2597 = $125,971.20.
Q: You need $50,000 in 5 years. The discount rate is 6% compounded annually. How much must you invest today?
A: PV = $50,000 / (1.06)^5 = $50,000 / 1.3382 = $37,363.
Q: A bond pays $4,000 semiannually for 10 years. The market rate is 10% per year. What is the PV of the coupon payments?
A: The per-period rate is 5% (10% / 2), n = 20. PV = $4,000 x PVA factor (5%, 20 periods). Look up the PVA factor in the table and multiply.
Q: What is the key difference between an ordinary annuity and an annuity due, and how does it affect present value?
A: An ordinary annuity has payments at the end of each period; an annuity due has payments at the beginning. Because annuity-due payments are received one period earlier, their present value is higher by a factor of (1 + r).
Q: Why must r and n be stated in the same terms?
A: Because the formula FV = PV x (1 + r)^n assumes r is the rate per compounding period and n is the total number of those periods. Mixing an annual rate with semiannual periods (or vice versa) produces an incorrect answer.
This connects directly to Chapter 14 (Bonds and Long-Term Notes) because bond pricing is simply the PV of future coupon payments plus the PV of the face value at maturity. The effective interest method of amortisation is a direct application of compound interest principles covered here.
It also connects to lease accounting: the lessee's right-of-use asset and lease liability are measured at the PV of future lease payments, using the same annuity mechanics.
Pension accounting relies on TVM as well. The projected benefit obligation is a discounted stream of expected future payments to retirees.
Time value of money, TVM, present value, future value, PV, FV, discount rate, compound interest, simple interest, compounding, annuity, ordinary annuity, annuity due, PVA factor, PVF factor, stated rate, nominal rate, coupon rate, effective rate, discounting, compounding frequency, ACC 472, intermediate accounting, Chapter 6