The Time Value of Money I: The
Basics
Dr Eoin Delaney
eoin.delaney@tcd.ie
(Slides based on those from Dr. John
McDonagh)
STU11004 Introduction to
Management Science 2025-2026
The Time Value of Money I: the basics
Part 1 of the TVOM topic will look at:
• Simple Interest
• Compound interest
• Discounting
• Present and Future Values
• Concept of Net Present Value(NPV)
• IRR and Pay back approaches
The Time Value of Money
• What do you prefer: 10k Euros Today? Or 10k
in 5 Years?
• Would you prefer 10k today or 12k next year?
• Would you prefer 10k today or 5k a year over
the next three years?
Time Value of Money – Applications
• Individuals – mortgages, pensions, car loans,
student loans
• Firms – Should we invest or not? Should we pay
off loan? Should we borrow or not?
• Govt – Should we spend/tax or not? Should we
issue bonds?
Basic Assumptions
• For the basics we will assume that we will:
– Abstract from risk – assume people are "risk
neutral" - and assume payments are certain
– Inflation is not a huge issue – either zero, or
returns are inflation adjusted
– There are no taxes
– (we will return to these assumptions but they
don't really change the main points and we can
extend the analysis)
What We Won't Cover
• What determines the interest rate.
• Yield curve and risk effects – i.e. a link
between interest rates and time horizon, risk.
• Simple definition: The interest rate is the
amount a lender charges a borrower and is a
percentage of the principal—the amount
loaned. An interest rate also applies to the
amount earned on a principal.
Simple Interest Reminder
• Where:
𝐴×𝑅×𝑛
𝐼=
100
– I = Interest accumulated
– A = Amount loaned (Principal)
– R = Interest rate per unit time (as a percentage)
– n = Number of units of time
• Principal after n units of time is then:
A + I = A(1 + Rn/100)
• Example: What is the total amount to be repaid on a loan
of €5,000 for five years at 10% simple interest?
Simple Interest Reminder
• Principal after n units of time is then:
A + I = A(1 + Rn/100)
• Total repayment:
A + I = 5000(1 + (10×5)/100) = €7500
• Simple interest has limited application in business
today.
• Important to distinguish it from compound interest.
Compound Interest
• Interest is added to the principal at the end of each
compounding period.
• Issues:
– Unclear real interest rate
– What is the compounding interval?
– Full and partial compounding
• Questions:
– What is the total interest on €1,000 compounded at 10%
per annum for five years?
– What is the monthly interest rate that compounds to 12%
per annum?
Compound Interest Example
Compound Interest - Take a five-year investment (say in a savings bond):
€1,000 @ 10% per annum starting 1/1/2025
Year
Date
Amount
0
1 January 2025
€1,000
1
1 January 2026
€1,100
2
1 January 2027
€1,210
3
1 January 2028
€1,331
4
1 January 2029
€1,461
5
1 January 2030
€1,612
What assumptions are built in to this?
Assumptions
• Interest rate is constant over the total investment time
• Time periods are all of equal length
• Payments are all based on the same method of
calculation
• Annuities (free of tax until withdrawn) are simple,
certain, discrete and ordinary
– Simple – payments happening on the same day
– Certain – fixed number of payments
– Discrete – equal intervals between payments
– Ordinary – payments made at the end of each period
Let's look at this algebraically!
• End of Year 1 figure
= €1,000 + 10% of €1,000
= €1,000 + €1,000 × 10/100
= €1,000 × (1 + 0.1)
= €1,000 × 1.1
= €1,100
• End of Year 2 figure
= €1,100 + 10% of €1,100
= €1,100 + €1,100 × 10/100
= €1,100 × (1 + 0.1)
• As we have above = (€1,000 × 1.1) × 1.1
= €1,000 × 1.1²
• Continuing like this we find by Extrapolation for year n:
= €1,000 × 1.1ⁿ
Compound Interest – Generalising
• Principal of A at R% per time period over n
time periods:
𝐴 × (1 + 𝑅/100)𝑛
• Interest over n time periods:
𝐼 = 𝐴 × (1 + 𝑅/100)𝑛 − 𝐴
𝐼 = 𝐴 [(1 + R/100)𝑛 − 1]
Inversion
What interest rate per month is equivalent to 12% per annum compound interest?
We want principal of €1,000 to become €1,120 in 12 months.
Let R be the monthly interest rate. From our formula:
Principal = 1,120 = 1,000 X (1 + R/100)¹²
This gives:
(1 + R/100)¹² = 1120/1000 = 1.12
1 + R/100 = 1.12⁽1/12⁾
R = (1.12⁽1/12⁾ - 1) X 100
= 0.9489% per month
Calculation
Compound Interest Question:
What compound annual interest rate is equivalent to 10% simple interest over five years?
Recall that €5,000 became €7,500 in five years. As before we have:
7,500 = 5,000 × (1 + 𝑅/100)5
Where R is the interest rate. This gives:
(1 + 𝑅/100)5 = 7500/5000 = 1.5
1+R/100=1.5(1/5)
R=(1.5(1/5)−1)×100
=8.45% per annum.
• Therefore quoting simple interest rates can be deceptive.
Partial Compounding
• Many investment accounts and loans use partial
compounding.
• Under partial compounding, accumulated interest is
added to the principal at intervals which are greater
than the interest calculation interval. E.g. interest is
10% per annum and interest added every two years
• It is never safe to assume that interest earned on
savings or interest paid on borrowings is a straight
compound rate.
• In borrowing, lending and investment, timing is critical.
Partial Compounding
• The bank takes the nominal 17% p.a., turns it into a daily
rate rd=0.17/365, applies it for 91 days within a quarter without
compounding inside the quarter, then compounds at the quarter
end.
Quarterly factor:
1+rd×91
= 1+0.17365×91
=1.0423835616
Effective annual rate (AER):
• (1.0423835616)^4−1≈0.18062 = 18.06%For a €1,000 overdraft,
the total interest over the year would be:
• €1,000×18.062%≈€180.62.
Interest Questions
• What rate of interest will convert €1000 into
€2000 over a period of 7 years?
Discounting
• Compounding answers: “If I invest money
today, how much will it grow to in the future?”
• Discounting answers: “If I expect money in the
future, how much is it worth today?”
• In other words, discounting is the process of
finding the present value (PV) of a future sum
of money (future value, FV) by removing the
effect of interest over time.
General Formula (Discounting)
𝐹𝑉
𝑃𝑉 =
(1 + 𝑅/100)𝑛
– PV = Present Value (the value today)
– FV = Future Value (the value at some time in the future)
– R = Discount rate (as a percentage per period)
– n = Number of periods
• Discounting is essential for evaluating investments,
pricing financial instruments (like bonds), and deciding
whether a project is worthwhile.
Cash Flows
The Time Value of Money
Cash Flows
• Present value
• Valuing cash flows and projects
– Net Present Value (NPV)
– Internal Rate of Return (IRR)
– Payback approach
Cash Flows – What and Why?
• So far we have only been concerned with lump sums.
• One of the most important issues in all finance is cash
streams or cash flows.
• Cash flows occur in loans, pensions, annuities and other
financial transactions and instruments.
• They may be:
– inwards (e.g., pension)
– outwards (e.g., loan repayments)
– a mixture of both (e.g., investments of various types)
• Example:
You are offered an income of €1,000 for the next three
years. You may take a lump sum instead. What size should
the lump sum be?
Present Value (PV)
• To answer this question, we look at what we call
the present value (PV) of the future stream of payments.
• The first question is when do the payments fall due?
• Assume:
– First payment is today
– Second in a year’s time
– Third a year later
• We must select a discount rate (e.g., one of 10%).
• Calculation:
PV = €1,000 + €1,000/1.1 + €1,000/(1.1)² = €2,735
Net Present Value (NPV)
• Net Present Value (NPV)
• Example: What if period was 5 years instead
of 3?
• PV = €1,000 + €1,000/1.1 + €1,000/(1.1)² +
€1,000/(1.1)³ + €1,000/(1.1)⁴ = €4,169