COMM1180 T3 2024
Final Exam
Formula Sheet
Week 3: Value from Pricing
Pricing and Cost-Volume Profit Analysis
Single Product:
Multiple Products
Margin of Safety
Margin of Safety = Actual or expected units of activity – Units at BEP
Total Sales
Quantity Breakeven =>
point =
Total Sales
Quantity Target =>
Profit
=
+
Total Sales
Quantity Breakeven =>
point =
ℎ
Total Sales
Quantity Target =>
Profit
=
+
ℎ
Week 5: Time Value of Money
“Time Travel Formula” – Single Cash Flows
Here are the equations that describe how present value PV, future value FV, interest rate r and time
horizon n are connected:
= (1 + )
= (1 + )
Required rate to achieve a savings goal:
= �
� 1 � − 1
Required number of periods to achieve a savings goal:
= lnln(1 + )
Switching between APR and EARs
For cases that involve an annual percentage rate (APR) with m compounding periods per year and a
time horizon of t years, we can compute the EAR as follows:
= �1 +
�
− 1
And vice versa, given the EAR, the APR with m-times compounding is:
=
�(1 + )1 � − 1�
Annuities
Every period for the next n periods (e.g., years, quarters, months), a payment C is made. r is the per- period rate.
Ordinary annuity (first payment at the end of the first period):
=
�1 − (1 + )−
�
=
�(1 + ) − 1
�
Solving for cash flow amount
=
� 1 − (1 + )−
�
Annuity due (first payment at the beginning of the first period):
=
�1 − (1 + )−
� (1 + )
=
�(1 + ) − 1
� (1 + )
Deferred annuity:
Begin of payments
delayed by k periods relative to ordinary annuity:
=
�1 − (1 + )−
� (1 + )−
Perpetuities
Ordinary:
=
Due:
=
(1 + )
Deferred:
=
(1 + )−
Optional: Paying off a loan
Time n required to pay off a current balance of PV with periodic payments C while the loan compounds
at rate:
= −
�1 −
×
� (1 + )
Students may find this decision flow chart useful in the finance weeks of the course as well as for the
final exam.
Week 7: Investment Decision Rules
Net Present Value
The present value of a project is the present value of all future cash flows including initial cash
flows at time 0:
= 0 + 1(1 + )1 + 2(1 + )2 + 3(1 + )3 + ⋯
Internal Rate of Return
The IRR is that rate ̃ for which the NPV is equal to zero:
0 = 0 + 1(1 + ̃)1 + 2(1 + ̃)2 + 3(1 + ̃)3 + ⋯
Profitability Index
The profitability index or Benefit-Cost Ratio of project
can be computed as
=
Or sometimes,
� =
Note: The ranking will be the same, but the latter can be useful when costs and benefits are
in different units.
Equivalent Annual Annuity
The EAA is that level cash flow from time 1 to time n (no cash flow at time 0!), whose present
value is equal to the NPV of an investment project (or purchase) with a lifetime of n years
(including the upfront cost):
=
= 1 − (1 + )−
Accounting Rate of Return
The accounting rate of return (ARR) divides average incremental accounting profits by upfront
investment cost:
= avg. acc. profitUpfront Cost
Week 8: Value for Investors
Simple bond pricing formula
The fair value of a bond with exactly n 6-month periods remaining to maturity, face value ,
and a coupon rate c% (i.e., a coupon amount of
= %/2 × ) and a per period discount
rate of
is
0 =
1 − (1 + )− + (1 + )
Constant ordinary perpetuity (e.g. perpetual preference shares):
0 =
Growing perpetuities (applicable to growing dividends or growing FCF)
Assume that payments grow at constant rate g from one period to the next (typically
< ),
e.g., 2 = (1 + )1 ⋯
= (1 + )−11.
Growing perpetuity
If 1 is the first payment at the end of the first period, then:
0 = 1 −
= 0(1 + ) −
Growing Perpetuity due
If 0 is the first payment at the beginning of the first period, then:
0 = 0 −
(1 + )
Return decomposition
Given current price 0, future price 1, end of period dividend 1, the expected return on
equity
can be decomposed into 2 parts, the forward dividend yield and expected capital
gains:
= 10 + 1 − 00
An alternative representation where g is the earnings (dividend) growth rate is:
= 10 +
Re-invest to grow
The growth rate of earnings in year t
= Retention Rate%,−1 × Return on new investments%,
= −1 ×
Week 9: Firm Value & Cost of Capital
Holding Period Return
(Holding) Period return for one period based on prices and dividend payment:
1 = Div1 + P1 − P0P0
Systematic Risk
The beta coefficient for asset
is a function of the correlation of asset
with the market portfolio,
adjusted by the ratio of volatilities of the asset and the market:
= ,
Capital Asset Pricing Model (CAPM):
The expected return for asset i can be computed from the risk-free return , the asset’s beta , and
the expected market return in excess of the risk-free rate, E[] − :
E[] =
+ �E[] − �
Required return on equity
Under dividend growth model: based on return decomposition (as discussed in Week 8):
= 10 +
= 0(1 + )0 +
Weighted average cost of capital
The after-tax weighted average cost of capital (WACC) for a 3-piece capital structure comprised of
equity (Market Value of E), preferred shares (Market Value of P) and (net) debt (Market Value of Net
Debt) with tax rate
can be computed as:
RWACC =
RE +
RP +
RD(1 − ) = wERE + wPRP + wDRD(1 − )
Where firm value is
=
+
+
Net debt means debt less cash:
=
− ℎ
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