S I G N A L & R U N WAY
Corporate Finance Core Pack
Time value of money, NPV and IRR, bond and stock valuation, CAPM, and WACC - with 10
practice questions and a full answer key.
Study pack · September 2026 · aligned to standard introductory corporate finance courses
Original study material by Signal & Runway. Formulas follow standard textbook conventions (annual periods, end-of-period cash
flows unless stated).
1. Time value of money
A euro today is worth more than a euro later, because today's euro can be invested to earn a return. Every
valuation in finance is TVM in disguise.
Core formulas (r = rate per period, n = number of periods):
Future value: FV = PV × (1+r)n
Present value: PV = FV / (1+r)n
Perpetuity (level payment C forever): PV = C / r
Growing perpetuity (first payment C, growth g, r > g): PV = C / (r - g)
Annuity (level payment C for n periods): PV = C × [1 - (1+r)-n] / r
Quantity Formula Quick example
FV of a lump sum PV(1+r)^n 1,000 at 5% for 3 yrs = 1,157.63
PV of a lump sum FV/(1+r)^n 1,000 in 3 yrs at 5% = 863.84
PV of an annuity C[1-(1+r)^-n]/r 100/yr for 5 yrs at 8% = 399.27
PV of a perpetuity C/r 100/yr at 8% = 1,250
PV of growing perpetuity C/(r-g) 100 growing 2%, r=8% = 1,666.67
Rate conversions: a nominal annual rate compounded m times per year gives an effective annual rate EAR =
(1 + rnom/m)m - 1. A 12% nominal rate compounded monthly is EAR = 12.68%. Always match the discount rate to
the compounding of the cash flows.
Annuity due (payments at period start, e.g. rent): multiply the ordinary annuity PV by (1+r).
2. NPV and IRR
Net present value: NPV = -Initial outlay + Σ CFt / (1+r)t. Accept projects with NPV > 0; between mutually
exclusive projects pick the higher NPV.
Internal rate of return: the discount rate that makes NPV = 0. Accept if IRR > required return (cost of
capital).
, Worked example
A project costs 10,000 and returns 4,000 at the end of each of years 1-3. At r = 10%: PV of inflows = 4,000 × [1-
(1.1)-3]/0.1 = 4,000 × 2.4869 = 9,947.41. NPV = 9,947.41 - 10,000 = -52.59: reject. The IRR is just under 10%
(about 9.7%), consistent with rejection at a 10% hurdle.
Rule Strengths Failure modes
NPV Values in money terms; correct ranking; Needs a discount rate up front
handles any cash-flow pattern
IRR Intuitive "return" number; no rate Multiple or no IRRs with sign-changing flows; misleads on scale and
needed to compute it on mutually exclusive projects; reinvestment assumption
Payback Simple; rough liquidity screen Ignores the time value of money and all cash flows after payback
When IRR and NPV disagree (different project sizes or timing), NPV wins: it measures value created, not a
percentage.
Corporate Finance Core Pack
Time value of money, NPV and IRR, bond and stock valuation, CAPM, and WACC - with 10
practice questions and a full answer key.
Study pack · September 2026 · aligned to standard introductory corporate finance courses
Original study material by Signal & Runway. Formulas follow standard textbook conventions (annual periods, end-of-period cash
flows unless stated).
1. Time value of money
A euro today is worth more than a euro later, because today's euro can be invested to earn a return. Every
valuation in finance is TVM in disguise.
Core formulas (r = rate per period, n = number of periods):
Future value: FV = PV × (1+r)n
Present value: PV = FV / (1+r)n
Perpetuity (level payment C forever): PV = C / r
Growing perpetuity (first payment C, growth g, r > g): PV = C / (r - g)
Annuity (level payment C for n periods): PV = C × [1 - (1+r)-n] / r
Quantity Formula Quick example
FV of a lump sum PV(1+r)^n 1,000 at 5% for 3 yrs = 1,157.63
PV of a lump sum FV/(1+r)^n 1,000 in 3 yrs at 5% = 863.84
PV of an annuity C[1-(1+r)^-n]/r 100/yr for 5 yrs at 8% = 399.27
PV of a perpetuity C/r 100/yr at 8% = 1,250
PV of growing perpetuity C/(r-g) 100 growing 2%, r=8% = 1,666.67
Rate conversions: a nominal annual rate compounded m times per year gives an effective annual rate EAR =
(1 + rnom/m)m - 1. A 12% nominal rate compounded monthly is EAR = 12.68%. Always match the discount rate to
the compounding of the cash flows.
Annuity due (payments at period start, e.g. rent): multiply the ordinary annuity PV by (1+r).
2. NPV and IRR
Net present value: NPV = -Initial outlay + Σ CFt / (1+r)t. Accept projects with NPV > 0; between mutually
exclusive projects pick the higher NPV.
Internal rate of return: the discount rate that makes NPV = 0. Accept if IRR > required return (cost of
capital).
, Worked example
A project costs 10,000 and returns 4,000 at the end of each of years 1-3. At r = 10%: PV of inflows = 4,000 × [1-
(1.1)-3]/0.1 = 4,000 × 2.4869 = 9,947.41. NPV = 9,947.41 - 10,000 = -52.59: reject. The IRR is just under 10%
(about 9.7%), consistent with rejection at a 10% hurdle.
Rule Strengths Failure modes
NPV Values in money terms; correct ranking; Needs a discount rate up front
handles any cash-flow pattern
IRR Intuitive "return" number; no rate Multiple or no IRRs with sign-changing flows; misleads on scale and
needed to compute it on mutually exclusive projects; reinvestment assumption
Payback Simple; rough liquidity screen Ignores the time value of money and all cash flows after payback
When IRR and NPV disagree (different project sizes or timing), NPV wins: it measures value created, not a
percentage.
