Chapter 3
Time Value of Money and Financial Mathematics
Bachelors Program in Accounting and Finance
Financial Management
•
Bachelor
Time Value of Money and Financial Mathematics
Introduction
A dollar available today is normally worth more than a dollar received in the future. The current dollar can be invested, may lose purchasing power through inflation, and avoids the uncertainty associated with delayed receipt. The time value of money (TVM) converts cash flows occurring at different dates into comparable values at a common date.
TVM underpins investment appraisal, bond and share valuation, lease accounting, retirement planning, loan pricing, impairment measurement and business valuation. Its correct use requires more than substituting numbers into formulas: the interest-rate period, cash-flow timing, compounding convention and risk assumptions must all be consistent.
Learning Objectives
After studying this chapter, students should be able to:
Calculate present and future values of single cash flows.
Convert nominal rates into periodic and effective annual rates.
Value ordinary annuities, annuities due, perpetuities and growing perpetuities.
Discount uneven cash-flow streams and calculate net present value.
calculate loan payments and construct a complete amortization schedule.
Apply TVM functions using formulas, spreadsheets and financial calculators.
Evaluate practical limitations involving inflation, taxation, risk and contractual conventions.
1. The Economic Logic of TVM
The required return on money generally compensates the provider of funds for:
Pure time preference: preference for current consumption over future consumption.
Expected inflation: compensation for declining purchasing power.
Risk: compensation for uncertainty about the amount or timing of repayment.
Liquidity: compensation for restricting access to funds.
Opportunity cost: the return forgone on alternative investments.
A cash-flow timeline helps prevent timing errors.
Cash-flow timeline showing present and future values
Figure: Cash-flow timeline showing present and future values
Figure: A cash-flow timeline and the direction of compounding and discounting.
Time 0 is the valuation date. Positive cash flows normally represent receipts and negative cash flows represent payments. A consistent sign convention is essential when using calculators and spreadsheets.
2. Simple and Compound Interest
2.1 Simple interest
Simple interest is calculated only on the original principal:
FVn = PV(1 + in)
where:
PV = present principal, in currency units;
FVn = value after n periods;
i = simple interest rate per period, expressed as a decimal;
n = number of periods.
The formula applies only where interest is not added to the interest-bearing balance.
2.2 Compound interest
Under compound interest, each period's interest is added to principal and earns interest thereafter. After one period, FV1 = PV(1+i). Repeated substitution produces:
FVn = PV(1+i)n
Rearranging gives the present-value formula:
PV = \frac{FVn}{(1+i)n} = FVn(1+i)-n
Here, i must be the effective rate for each period and n must count periods of the same length. For example, a monthly rate requires a number of months.
Example 1: Single-sum compounding and discounting
An investment of $5,000 earns 8% annually for three years:
FV3 = 5{,}000(1.08)3 = \$6{,}298.56
If $6,298.56 is required in three years and the relevant annual rate is 8%, its present value is:
PV = \frac{6{,}298.56}{(1.08)3} = \$5{,}000.00
Compounding and discounting are inverse operations.
Example 2: Future value using an effective annual rate
Suppose $7,000 is deposited for four years at an effective annual rate of 5.959%:
FV4 = 7{,}000(1.05959)4 = \$8{,}823.67
The rate is annual and the exponent therefore counts years.
3. Interest-Rate Conventions
3.1 Nominal and periodic rates
A nominal annual rate compounded m times per year has periodic rate:
ip = \frac{j}{m}
where j is the nominal annual rate, m is the number of compounding periods per year and ip is the rate per compounding period. This relationship applies to a conventional quoted nominal rate convertible m times annually.
3.2 Effective annual rate
The effective annual rate (EAR) includes within-year compounding:
EAR provides a common basis for comparing loans or investments with different compounding frequencies, provided fees and other contractual costs are treated consistently.
3.3 Converting between effective rates
Equivalent rates produce the same accumulation over the same total time. If ia is an effective annual rate, the equivalent monthly rate is:
im = (1+ia)1/12-1
Dividing an effective annual rate by 12 is generally incorrect because it ignores compounding.
3.4 Continuous compounding
If a nominal annual rate r is compounded continuously for t years:
FV = PV ert
and:
PV = FV e-rt
Continuous compounding is most common in financial theory and derivative pricing. It should not be used when a contract specifies discrete compounding.
4. Annuities
An annuity is a finite series of equal payments at regular intervals. An ordinary annuity pays at the end of each period; an annuity due pays at the beginning.
4.1 Present value of an ordinary annuity
Consider n payments of C, with the first payment one period from today. The present value is:
A perpetuity pays a constant amount indefinitely. If the first payment occurs one period from the valuation date:
PVperpetuity = \frac{C}{i}
The formula requires a positive constant discount rate and equal end-of-period payments continuing indefinitely. A perpetuity paying $600 annually with an 8% required return is worth 600/0.08 = \$7{,}500.
5.2 Growing perpetuity
If the next payment is C1 and payments grow at constant rate g:
PVgrowing = \frac{C1}{i-g}
This formula requires i>g, constant rates and a first payment one period ahead. If i \le g, the infinite discounted series does not converge to a finite value under these assumptions.
5.3 Growing annuity
For n payments beginning with C1 one period from today:
This expression assumes constant i and g. When i=g, the displayed formula is undefined; the special-case value is PV = nC1/(1+i).
6. Uneven Cash Flows and Net Present Value
When cash flows differ by period, each amount must be discounted separately:
PV = \sumt=1n\frac{CFt}{(1+i)t}
Including an initial cash flow gives net present value:
NPV = CF0 + \sumt=1n\frac{CFt}{(1+i)t}
Here, CFt is the signed cash flow at time t, CF0 is usually an immediate investment and i is the appropriate risk-adjusted effective rate per period. A positive NPV indicates that the investment exceeds the required return, assuming the forecast cash flows and rate are reliable.
Example 5: Uneven project cash flows
A project requires $10,000 immediately and generates $2,500, $3,500, $4,000 and $5,000 at the ends of years 1–4. At 8%:
Year
Cash flow ($)
Present value ($)
0
−10,000
−10,000.00
1
2,500
2,314.81
2
3,500
3,000.69
3
4,000
3,175.33
4
5,000
3,675.15
NPV = -10{,}000 + 12{,}165.98 = \$2{,}165.98
On the stated assumptions, the project creates value relative to an 8% required return.
7. Loan Payments and Amortization
An amortizing loan is repaid through payments containing both interest and principal. For a fully amortizing loan with equal end-of-period payments:
L = PMT\left[\frac{1-(1+i)-n}{i}\right]
Solving for the payment:
PMT = L\left[\frac{i}{1-(1+i)-n}\right]
where L is the initial loan balance, PMT is the regular payment, i is the effective borrowing rate per payment period and n is the number of payments. Payments must occur at period-end unless the formula is adjusted for an annuity due.
Interest and balances are rounded to cents. The final payment is adjusted by one cent to eliminate the residual created by rounding.
Month
Opening balance ($)
Interest ($)
Payment ($)
Principal ($)
Closing balance ($)
1
6,241.73
62.42
554.57
492.15
5,749.58
2
5,749.58
57.50
554.57
497.07
5,252.51
3
5,252.51
52.53
554.57
502.04
4,750.47
4
4,750.47
47.50
554.57
507.07
4,243.40
5
4,243.40
42.43
554.57
512.14
3,731.26
6
3,731.26
37.31
554.57
517.26
3,214.00
7
3,214.00
32.14
554.57
522.43
2,691.57
8
2,691.57
26.92
554.57
527.65
2,163.92
9
2,163.92
21.64
554.57
532.93
1,630.99
10
1,630.99
16.31
554.57
538.26
1,092.73
11
1,092.73
10.93
554.57
543.64
549.09
12
549.09
5.49
554.58
549.09
0.00
Total payments are $6,654.85, comprising $6,241.73 of principal and $413.12 of interest. Interest declines because it is calculated on a falling balance, while the principal component increases.
8. Applied Case: Selecting Production Equipment
Background
This is an illustrative case. Meridian Components must choose between two machines that provide equivalent output for four years. Because their operating capability is equivalent, management applies the present-value-of-cost approach.
Problem and cash flows
The required return is 9% annually. All maintenance payments and salvage receipts occur at year-end.
Year
Machine A net cash flow ($)
Machine B net cash flow ($)
0
−45,000
−50,000
1
−6,000
−4,000
2
−6,500
−4,000
3
−7,000
−4,000
4
−3,000
3,000
Machine A's year-4 amount is an $8,000 maintenance cost less $5,000 salvage value, producing a $3,000 net outflow. Machine B's year-4 amount is $4,000 maintenance less $7,000 salvage, producing a $3,000 net receipt.
Analysis and solution
For clarity, costs are shown as positive present-value costs:
Machine B has the lower present-value cost by approximately $5,506.17.
Outcome
If the machines truly provide equivalent capacity, quality, risk and useful lives, Machine B is financially preferable. The result should be stress-tested for maintenance uncertainty, downtime, disposal value and technological obsolescence.
Learning points
Every underlying cash flow must be disclosed so that the valuation is reproducible.
Salvage receipts reduce lifecycle cost.
Equivalent-service alternatives may be compared using present-value costs.
Nonfinancial differences can invalidate a simple lowest-cost decision.
9. Spreadsheet and Financial-Calculator Applications
Common spreadsheet functions include:
Task
Spreadsheet form
Timing note
Future value
FV(rate,nper,pmt,pv,type)
type=0 for period-end payments
Present value
PV(rate,nper,pmt,fv,type)
Use consistent cash-flow signs
Loan payment
PMT(rate,nper,pv,fv,type)
Monthly inputs require monthly rate and periods
Uneven periodic NPV
NPV(rate,CF1:CFn)+CF0
Initial cash flow is added separately
Effective annual rate
EFFECT(nominal_rate,npery)
Nominal rate must follow spreadsheet convention
For Example 6, a spreadsheet expression is =-PMT(12%/12,12,6241.73). The negative sign displays the payment as a positive amount when the loan proceeds are entered as a positive inflow.
A financial calculator typically uses the registers N, I/Y, PV, PMT and FV. Before calculation, users should clear old data, set the correct payments per year, select end or beginning mode, and follow a consistent sign convention.
Spreadsheets increase efficiency but do not validate economic assumptions. Formula auditing, locked input cells, independent review and version control are important professional controls. AI tools may assist in identifying inconsistent references or generating scenarios, but their formulas and outputs require human verification.
10. Inflation, Tax and Risk Consistency
10.1 Nominal and real rates
The exact Fisher relationship is:
1+iN = (1+iR)(1+\pi)
where iN is the nominal rate, iR is the real rate and \pi is expected inflation. Therefore:
iR = \frac{1+iN}{1+\pi}-1
Nominal cash flows, which include expected inflation, should be discounted at a nominal rate. Real cash flows should be discounted at a real rate.
10.2 Tax consistency
After-tax cash flows should normally be discounted using an appropriate after-tax required return. Tax timing, deductibility, depreciation allowances and capital-gains treatment can materially affect value.
10.3 Risk-adjusted rates
A higher discount rate lowers the present value of positive future cash flows, but simply increasing the rate is not always an adequate risk model. Different cash flows may have different risk profiles. Scenario analysis, probability-adjusted cash flows or market-based valuation methods may be more defensible.
11. Challenges, Governance and Professional Practice
TVM calculations are sensitive to assumptions that may be hidden in seemingly precise answers. Important limitations include:
uncertain cash-flow forecasts and terminal values;
nonconstant or changing interest rates;
reinvestment assumptions;
day-count conventions and irregular payment dates;
fees excluded from advertised interest rates;
rounding in long amortization schedules;
default, liquidity and inflation risk;
model and spreadsheet errors.
Professional practice requires clear documentation of cash-flow dates, rate sources, compounding conventions, taxes, fees and rounding policies. Consumer-credit regulation may require disclosure of annual percentage rates or comparable effective borrowing costs. In financial reporting, effective-interest calculations are relevant to amortized-cost measurement under IFRS 9, while discounting is also central to lease liabilities under IFRS 16. Applicable standards and local regulations should always be checked rather than inferred from a generic TVM model.
Recommended controls include an independent reasonableness check, reconciliation of opening and closing balances, sensitivity analysis and separation of input, calculation and output cells. Ethical presentation requires that analysts disclose assumptions rather than selecting rates or terminal values merely to support a preferred decision.
Chapter Summary
The time value of money provides a consistent basis for comparing cash flows occurring at different dates. Compound-interest formulas move single amounts through time, while annuity and perpetuity models value structured payment streams. Uneven cash flows require individual discounting, and NPV measures value relative to a required return.
Loan payments are derived from the present value of an annuity, and amortization schedules separate each payment into interest and principal. Reliable application requires alignment among cash-flow frequency, effective periodic rates and timing conventions. Spreadsheets and financial calculators improve efficiency, but sound valuation also depends on transparent assumptions, model governance and consistency in the treatment of inflation, tax and risk.
Key Terms
Amortization: The gradual repayment of loan principal through scheduled payments.
Annuity: A finite series of equal cash flows at regular intervals.
Annuity due: An annuity whose payments occur at the beginning of each period.
Compound interest: Interest calculated on principal and accumulated interest.
Discount rate: The required return used to convert future cash flows into present value.
Effective annual rate: The annual rate that reflects within-year compounding.
Future value: The value of a present amount at a specified future date.
Growing annuity: A finite stream of cash flows growing at a constant rate.
Growing perpetuity: An indefinitely continuing stream growing at a constant rate.
Nominal annual rate: A quoted annual rate that does not itself incorporate within-year compounding.
Net present value: Present value of all signed project cash flows, including the initial investment.
Ordinary annuity: An annuity whose payments occur at the end of each period.
Periodic rate: The effective interest rate applicable to one compounding or payment period.
Perpetuity: An equal periodic cash-flow stream continuing indefinitely.
Present value: The current equivalent of one or more future cash flows.
Principal: The outstanding amount borrowed or invested, excluding accrued interest.
Real interest rate: A rate expressed after removing the effect of inflation.
Simple interest: Interest calculated only on the original principal.
Review Questions
Short Answer Questions
Why is a dollar received today normally worth more than a dollar received in one year?
Distinguish between a nominal annual rate and an effective annual rate.
What timing difference separates an ordinary annuity from an annuity due?
Why must a growing perpetuity satisfy i>g?
What happens to the interest component of a standard fully amortizing loan payment over time?
Long Answer Questions
Derive the present-value formula for an ordinary annuity using a geometric series.
Explain how inflation, taxation and risk should be incorporated consistently into a TVM analysis.
Compare simple interest, discrete compound interest and continuous compounding.
Explain how spreadsheet controls can reduce errors in financial models.
Discuss the assumptions and limitations of using a single discount rate for a long-term investment project.
Analytical / Application Questions
A business deposits $12,000 today at a nominal annual rate of 9% compounded monthly for three years. Calculate the future value and effective annual rate.
A loan of $20,000 carries a nominal annual rate of 10.8% compounded monthly and requires 36 month-end payments. Calculate the payment and prepare the first three rows of its amortization schedule.
A project costs $50,000 and produces year-end cash flows of $12,000, $18,000, $20,000 and $22,000. Calculate its NPV at 9% and explain whether the quantitative result alone is sufficient for approval.
Multiple Choice Questions with Answers
Question 1
What is the future value of $5,000 invested for three years at an effective annual rate of 8%?
A. $5,400.00
B. $6,298.56
C. $6,500.00
D. $7,346.64
Correct Answer: B. $6,298.56
Question 2
What is the present value of $10,000 receivable in two years when the effective annual discount rate is 10%?
A. $8,264.46
B. $8,100.00
C. $9,090.91
D. $12,100.00
Correct Answer: A. $8,264.46
Question 3
A nominal annual rate of 12% is compounded monthly. What is its effective annual rate?
A. 1.00%
B. 12.00%
C. 12.68%
D. 14.40%
Correct Answer: C. 12.68%
Question 4
Relative to an otherwise identical ordinary annuity, how is the present value of an annuity due calculated?
A. Divide the ordinary-annuity value by 1+i
B. Multiply the ordinary-annuity value by 1+i
C. Add one extra payment without discounting
D. Multiply the ordinary-annuity value by n
Correct Answer: B. Multiply the ordinary-annuity value by 1+i
Question 5
A perpetuity pays $600 at each year-end. What is its value at an annual required return of 8%?
A. $4,800
B. $7,500
C. $8,100
D. $9,600
Correct Answer: B. $7,500
Question 6
A growing perpetuity pays $1,000 one year from today, grows at 4% annually and has a 9% required return. What is its present value?
A. $11,111.11
B. $13,000.00
C. $20,000.00
D. $25,000.00
Correct Answer: C. $20,000.00
Question 7
For the $6,241.73 loan in Example 6, what is the first month's interest at a monthly rate of 1%?
A. $5.55
B. $49.22
C. $55.46
D. $62.42
Correct Answer: D. $62.42
Question 8
A project costs $10,000 and the present value of its future receipts is $12,165.98. What is its NPV?
A. −$2,165.98
B. $2,165.98
C. $10,000.00
D. $22,165.98
Correct Answer: B. $2,165.98
Question 9
Which pairing is financially consistent?
A. Nominal cash flows discounted at an appropriate nominal rate
B. Monthly cash flows discounted at an unconverted annual rate
C. Real cash flows discounted at a nominal rate without adjustment
D. After-tax cash flows discounted at a pre-tax rate in every circumstance
Correct Answer: A. Nominal cash flows discounted at an appropriate nominal rate
Question 10
In the equipment case, which machine has the lower present-value cost at 9%?