ECON1003 - Quantitative Methods in
Economics
Lecture 6
Chapter 5 Bradley
ECON1003 Lecture 6 1
ECON1003 Lecture 6 2
OUTLINE:
• Sequences
• Series
• Applications to Finance
• Simple interest
• Compound interest
• Depreciation
• Net Present Value and Internal Rate of Return
• Annuities
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1. Sequence
• A sequence is a list of numbers that follow a definite pattern or rule
• Two types of sequences:
- arithmetic sequence: obtained by adding a constant to each
subsequent term (d is common difference)
- geometric sequence multiply the previous term by a constant r
(r is common ratio)
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2. Series
• A series is the sum of the terms in a sequence
• Finite series is the sum of a finite number of terms of a sequence
• Infinite series is the sum of an infinite number of terms of a sequence
• Two types of series:
- Arithmetic series
- Geometric series
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2.1. Arithmetic series (sum of the first n terms)
Proof:
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2.2. Geometric series (sum of the first n terms)
Proof:
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2.3. Geometric series (sum of an infinite number of terms)
• If r < 1, when n →∞, lim Sn = a/(1-r)
• If r ≥ 1, when n →∞, the series does not converge but just keeps getting
bigger and bigger.
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2.4 Examples
Example 1: Find the sum of the first 15 terms of the series: 20 + 18 + 16+ 14+ …
Example 2: Find the sum of the first 12 terms of the series: 4 + 2 + 1 + ½ + ¼ + …
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3. Financial Applications
3.1. Simple interest
The amount of interest I received with a simple interest rate of i% per year
paid on principal P0 over t years is
I = P0 i t
The total value (future value) after t years Pt is
Pt = P0(1 + i t )
The principal (present value) is
0 =
(1+)
(present value formula)
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3.2. Compound interest
• Total value Pt, of a principal P0, when interest is compounded at i%
per annum after t years is
Pt = P0(1 + i)
t
• Present value formula:
0 =
(1+)
• If interest is compounded twice (2 times) a year at i% per annum
Pt = P0(1 + i/2)
2t
• If interest is compounded n times a year at i% per annum
Pt = P0(1 + i/n)
nt (t is the number of years )
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3.3 Continuous compounding interest
• If interest is compounded continuously, n →∞
= lim
→∞
0(1 +
)
Define
=
1
=
= lim
→∞
0(1 +
1
)
= lim
→∞
0(1 +
1
)= 0 lim
→∞
(1 +
1
)
= 0
• Present value formula:
0 =
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Example 3: Find future value of $1000 at 8% per annum compounded
annually and compounded continuously for 3 years.
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Example 4: How much do you have deposit in the bank now to get $15000
in 3 years at 8% per annum compounded annually and compounded
continuously?
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3.4. Depreciation
Reducing-balance depreciation at an annual rate of depreciation i
At = A0(1 - i)
t
Example 5: A machine with initial value of $30000 depreciates by 15% per
year. What is the machine’s value in 5 years?
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3.5. Net present value and internal rate of return
1. Net present value
The net present value is the present value of several future sums discounted back
to the present.
Can be used to appraise the profitability of investment undertaken by a firm
NPV = present value of cash inflows – the present value of cash outflows
A decision rule is
NPV > 0 Invest in the project
NPV < 0 Don’t invest in the project
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Example 6: Calculate NPV of the below project assuming 8% discount rate
compounded annually.
Year Cash flow
0 -400
1 120
2 130
3 140
4 150
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2. Internal rate of return (IRR)
• The IRR is the interest rate for which the NPV is zero
• Accept the project if the IRR is greater than the market rate of interest
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3.6. Annuities
• Series of equal deposits (or payments) made at equal intervals of time.
• Future value (FV) of an annuity at the end of t years with a deposit of
A0 each year, compounded annually at an interest rate of i% per
annum:
= 0
(1+)−1
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Example 7: Calculate the value of a fund after 10 years when a single deposit
of $2000 made annually.
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• Present value (PV) of an annuity
How much should be invested now (PV) for t years at a given interest rate
of i% per annum to cover a series of t equal annual payments?
PV = 0
1−(1+)−
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Example 8: Find the PV of an annual payment of $1000 for 20 years at an
interest rate of 6% per annum.