COMM 309 — Business Finance
CHAPTER 5: Time Value of Money
Comprehensive Exam Preparation Notes
5.3 Compound Interest
Compound interest is interest that is earned on the principal amount invested AND on any
accrued interest. This is the dominant form of interest in all real-world financial applications.
The key distinction from simple interest is that interest earned in each period itself earns interest
in subsequent periods — the foundation of exponential growth in finance.
Compounding (Computing Future Values)
Compounding is the process of calculating a future value from a present value by applying an
interest rate repeatedly over multiple periods. Each period, interest is earned not only on the
original principal but also on all previously accumulated interest.
Deriving the Future Value Formula:
Year 1: PV₀ + (PV₀ × k) = PV₀(1 + k)
Year 2: PV₀(1 + k) × (1 + k) = PV₀(1 + k)²
Year n: FVₙ = PV₀(1 + k)ⁿ
Eq. 5-2 | Future Value (Compound Interest)
FVₙ = Future value at end of period n
0
FVₙ=P V × ( 1+k ) ⁿ PV₀ = Present value today (time 0)
k = Market interest rate (per period)
n = Number of periods
(1 + k)ⁿ = Future Value Interest Factor (FVIF)
The FVIF tells you what $1 grows to after n periods at
rate k.
📌 Simple vs. Compound Interest — Key Comparison
Simple Interest: Interest only on the original principal each period.
→ FV = P + (n × P × k) [linear growth — straight line]
Compound Interest: Interest on principal PLUS all accumulated interest.
→ FV = PV₀(1 + k)ⁿ [exponential growth — curved line]
Example: $1,000 at 10% for 5 years:
Simple: $1,000 + (5 × $1,000 × 0.10) = $1,500
, Compound: $1,000 × (1.10)⁵ = $1,610.51
The difference ($110.51) is called 'interest on interest.'
Over long periods (e.g., 50 years), this gap becomes enormous:
Simple: $6,000 | Compound: $117,390.85
Reinvestment: Compounding assumes all interest is reinvested — i.e., neither spent nor
withdrawn. This is the critical assumption that allows future interest to be earned on past
interest.
Power of Compounding — Real Data (Table 5-1):
$1,000 invested in 1938 and held for 77 years (to end of 2014):
• Government of Canada T-Bills (4.73% avg return) → $35,114
• Government of Canada Bonds (5.95%) → $85,659
• Canadian Stocks (9.91%) → $1,444,991
• U.S. Stocks (11.15%) → $3,427,906
Takeaway: Even a seemingly small difference in rate (9.91% vs 11.15%) produces ~$2 million
in additional wealth over 77 years due to compounding. This is why finance professionals track
returns down to 1/100 of 1% (a basis point). Compound return is also called the geometric
return and differs from a simple arithmetic average.
Discounting (Computing Present Values)
Discounting is the reverse of compounding — it finds the present value (PV) of a future cash
flow. Because future dollars are worth less than today's dollars (due to opportunity cost), we
'discount' them back. Discounting allows us to compare cash flows occurring at different points
in time using a single reference point: today.
Eq. 5-3 | Present Value (Discounting)
FVₙ FVₙ∗1 PV₀ = Present value today
PV 0= =
( 1+ k ) ⁿ (1+ k ) ⁿ FVₙ = Future value at period n
k = Discount rate (opportunity cost / required rate of
return)
n = Number of periods
1/(1 + k)ⁿ = Present Value Interest Factor (PVIF)
Note: PVIF = 1 / FVIF (they are reciprocals)
📌 Key Properties of Discount Factors (PVIF)
1. PVIF is ALWAYS < 1 when k > 0 (future dollars are worth less today).
2. Higher k → smaller PVIF → lower present value.
3. Longer n → smaller PVIF → lower present value.
, 4. PVIF = 1 / FVIF (reciprocal relationship).
5. Exception: negative interest rates can make PVIF > 1 (rare; occurred in eurozone crisis).
Pension fund implication: When interest rates fall, the PV of future pension liabilities RISES,
creating funding deficits — because the discount rate used to value future obligations decreases.
Example (Discounting): How much must you invest today at 10% to have $1,000,000 in 40
years?
PV₀ = $1,000,000 × [1/(1.10)⁴⁰] = $1,000,000 × 0.02209 = $22,094.93
Interpretation: $22,094.93 today is economically equivalent to $1,000,000 in 40 years at k =
10%.
Determining Rates of Return or Holding Periods
The same equation FVₙ = PV₀(1 + k)ⁿ can be rearranged to solve for any one of the four
variables when the other three are known:
◦ Solve for FV: How much will I have in n years at k% if I invest PV today?
◦ Solve for PV: What is the value today of receiving FV in n years at k%?
◦ Solve for k (IRR): What rate of return will I earn if I invest PV today and get FV in n
years?
◦ Solve for n: How long must I wait to get FV if I invest PV today at k%?
Solving for k — Internal Rate of Return (IRR)
(1n )−1
( )
FVₙ The IRR is the interest rate internal to the problem.
k= Example: PV = $20,000, FV = $32,000, n = 5
PV 0
k = (32,000/20,000)^(1/5) − 1
k = (1.6)^0.2 − 1 = 9.856%
In finance, IRR problems arise when comparing
investments
based on rates of return rather than dollar amounts.
Solving for n — Holding Period
FV Uses natural logarithms (ln) to solve for the time
n=ln ÷ l n (1+ k ) period.
PV
Example: PV = $20,000, FV = $32,000, k = 10%
n = ln(32,000/20,000) ÷ ln(1.10)
n = ln(1.60) ÷ ln(1.10)
n = 0.47 ÷ 0.09531 = 4.93 years
Use a financial calculator (TI BA II Plus) for efficiency.
CHAPTER 5: Time Value of Money
Comprehensive Exam Preparation Notes
5.3 Compound Interest
Compound interest is interest that is earned on the principal amount invested AND on any
accrued interest. This is the dominant form of interest in all real-world financial applications.
The key distinction from simple interest is that interest earned in each period itself earns interest
in subsequent periods — the foundation of exponential growth in finance.
Compounding (Computing Future Values)
Compounding is the process of calculating a future value from a present value by applying an
interest rate repeatedly over multiple periods. Each period, interest is earned not only on the
original principal but also on all previously accumulated interest.
Deriving the Future Value Formula:
Year 1: PV₀ + (PV₀ × k) = PV₀(1 + k)
Year 2: PV₀(1 + k) × (1 + k) = PV₀(1 + k)²
Year n: FVₙ = PV₀(1 + k)ⁿ
Eq. 5-2 | Future Value (Compound Interest)
FVₙ = Future value at end of period n
0
FVₙ=P V × ( 1+k ) ⁿ PV₀ = Present value today (time 0)
k = Market interest rate (per period)
n = Number of periods
(1 + k)ⁿ = Future Value Interest Factor (FVIF)
The FVIF tells you what $1 grows to after n periods at
rate k.
📌 Simple vs. Compound Interest — Key Comparison
Simple Interest: Interest only on the original principal each period.
→ FV = P + (n × P × k) [linear growth — straight line]
Compound Interest: Interest on principal PLUS all accumulated interest.
→ FV = PV₀(1 + k)ⁿ [exponential growth — curved line]
Example: $1,000 at 10% for 5 years:
Simple: $1,000 + (5 × $1,000 × 0.10) = $1,500
, Compound: $1,000 × (1.10)⁵ = $1,610.51
The difference ($110.51) is called 'interest on interest.'
Over long periods (e.g., 50 years), this gap becomes enormous:
Simple: $6,000 | Compound: $117,390.85
Reinvestment: Compounding assumes all interest is reinvested — i.e., neither spent nor
withdrawn. This is the critical assumption that allows future interest to be earned on past
interest.
Power of Compounding — Real Data (Table 5-1):
$1,000 invested in 1938 and held for 77 years (to end of 2014):
• Government of Canada T-Bills (4.73% avg return) → $35,114
• Government of Canada Bonds (5.95%) → $85,659
• Canadian Stocks (9.91%) → $1,444,991
• U.S. Stocks (11.15%) → $3,427,906
Takeaway: Even a seemingly small difference in rate (9.91% vs 11.15%) produces ~$2 million
in additional wealth over 77 years due to compounding. This is why finance professionals track
returns down to 1/100 of 1% (a basis point). Compound return is also called the geometric
return and differs from a simple arithmetic average.
Discounting (Computing Present Values)
Discounting is the reverse of compounding — it finds the present value (PV) of a future cash
flow. Because future dollars are worth less than today's dollars (due to opportunity cost), we
'discount' them back. Discounting allows us to compare cash flows occurring at different points
in time using a single reference point: today.
Eq. 5-3 | Present Value (Discounting)
FVₙ FVₙ∗1 PV₀ = Present value today
PV 0= =
( 1+ k ) ⁿ (1+ k ) ⁿ FVₙ = Future value at period n
k = Discount rate (opportunity cost / required rate of
return)
n = Number of periods
1/(1 + k)ⁿ = Present Value Interest Factor (PVIF)
Note: PVIF = 1 / FVIF (they are reciprocals)
📌 Key Properties of Discount Factors (PVIF)
1. PVIF is ALWAYS < 1 when k > 0 (future dollars are worth less today).
2. Higher k → smaller PVIF → lower present value.
3. Longer n → smaller PVIF → lower present value.
, 4. PVIF = 1 / FVIF (reciprocal relationship).
5. Exception: negative interest rates can make PVIF > 1 (rare; occurred in eurozone crisis).
Pension fund implication: When interest rates fall, the PV of future pension liabilities RISES,
creating funding deficits — because the discount rate used to value future obligations decreases.
Example (Discounting): How much must you invest today at 10% to have $1,000,000 in 40
years?
PV₀ = $1,000,000 × [1/(1.10)⁴⁰] = $1,000,000 × 0.02209 = $22,094.93
Interpretation: $22,094.93 today is economically equivalent to $1,000,000 in 40 years at k =
10%.
Determining Rates of Return or Holding Periods
The same equation FVₙ = PV₀(1 + k)ⁿ can be rearranged to solve for any one of the four
variables when the other three are known:
◦ Solve for FV: How much will I have in n years at k% if I invest PV today?
◦ Solve for PV: What is the value today of receiving FV in n years at k%?
◦ Solve for k (IRR): What rate of return will I earn if I invest PV today and get FV in n
years?
◦ Solve for n: How long must I wait to get FV if I invest PV today at k%?
Solving for k — Internal Rate of Return (IRR)
(1n )−1
( )
FVₙ The IRR is the interest rate internal to the problem.
k= Example: PV = $20,000, FV = $32,000, n = 5
PV 0
k = (32,000/20,000)^(1/5) − 1
k = (1.6)^0.2 − 1 = 9.856%
In finance, IRR problems arise when comparing
investments
based on rates of return rather than dollar amounts.
Solving for n — Holding Period
FV Uses natural logarithms (ln) to solve for the time
n=ln ÷ l n (1+ k ) period.
PV
Example: PV = $20,000, FV = $32,000, k = 10%
n = ln(32,000/20,000) ÷ ln(1.10)
n = ln(1.60) ÷ ln(1.10)
n = 0.47 ÷ 0.09531 = 4.93 years
Use a financial calculator (TI BA II Plus) for efficiency.
