Level II
2017 Level II Formulas www.ift.world [email protected] Graphs, charts, tables, examples, and figures are copyright 2014, CFA Institute. Reproduced and republished with permission from CFA Institute. All rights reserved. www.ift.world
1
Comments and Instructions This document is a compilation of what I believe are the most important formulas for Level II. It does not list the facts which you must know in order to clear the exam. These facts are covered in our Level II crash course which is available for sale. No formula sheet can be 100% comprehensive. This formula sheet is no exception. However, it can serve as a good starting point. Print this document and add your notes/comments. Specifically, after every practice test look at this sheet and add formulas or comments which you think will be helpful. This document might start out as an IFT formula sheet but it should end up with many of your notes. If you have suggestions for improving this document please write to us at: [email protected] Do a lot of practice over the last few days and good luck on your exam.
Regards, Arif Irfanullah, CFA
www.ift.world
2
Quant (1/3) Correlation
Regression
H0: ρ = 0 versus Ha: ρ ≠ 0
Confidence interval for regression coefficients
Yi = b0 + b1Xi + εi, i = 1, ..., n
Prediction interval for regression equation:
sf = Standard deviation of prediction error
www.ift.world
3
Quant (2/3) SEE = Square root of mean square error.
Use the F-test to test: H0 : b1 = b2 = … = bk = 0 against the alternative hypothesis that at least one slope coefficient is not equal to 0.
Test for serial correlation: DW ≈ 2 (1 – r)
www.ift.world
4
Quant (3/3) Test whether the autocorrelations of the error term (error autocorrelations) differ significantly from 0. H0: error autocorrelation at a specified lag equals 0
Standard error of the residual correlation = 1 / 𝑇 where T is the number of observations Test-stat = residual autocorrelation / standard error
Consider an AR(1) model: xt = b0 + b1xt-1 + εt
Mean-reverting level is given by: xt =
𝑏0 (1 −𝑏1)
Example of first differencing:
www.ift.world
5
Econ (1/2) Ask = 1 / Bid
Covered interest rate parity
Forward points represent the difference between the forward rate and the spot rate.
Uncovered interest rate parity: Expected % change in spot rate (P/B) ≈ ip - iB Ex ante purchasing power parity: Expected % change in spot rate (P/B) ≈ ∏p - ∏B International Fisher relationship: ip - iB = ∏p - ∏B
www.ift.world
6
Econ (2/2)
P represented aggregate price (value) of stocks; E represents aggregate earnings
A is the total factor productivity
Labor productivity: Growth accounting: Growth rate in potential GDP = Long-term growth rate of labor force + Long-term growth rate in labor productivity ϴ is growth rate of TFP α is the share of GDP paid out to the suppliers of capital n is the growth rate of labor
Neo-classical model:
www.ift.world
7
FRA (1/2) Equity method: Value of investment = beginning value + share of profits – share of dividends Acquisition method: Partial goodwill = fair value of the acquisition - acquirer’s share of the fair value of acquiree’s net assets Full goodwill = fair value of the entity as a whole - the fair value of all acquiree’s assets and liabilities Funded status (net liability) = present value of defined benefit obligation – fair value of plan assets Total periodic pension cost = contributions + (ending funded status – beginning funded status) Under US GAAP actuarial gains and losses have two components: 1. Actual return – (plan assets × expected return) 2. Changes in a company’s pension obligation arising from changes in actuarial assumptions Under IFRS: Net return on plan assets = actual return – (plan assets × interest rate) Actuarial gains and losses = changes in a company’s pension obligation arising from changes in actuarial assumptions www.ift.world
8
FRA (2/2) Category
Measures
Example
Activity Ratios
Numerator / Denominator
Activity ratios
Efficiency
Revenue / Assets
Inventory turnover
Cost of good sold / Average inventory
Days of inventory on Hand
Number of days in period / Inventory turnover
Liquidity ratios Ability to meet its short term obligations
Current Assets / Current Liabilities
Solvency ratios
Ability to meet long term debt obligations
Assets / Equity
Profitability ratios
Profitability
Net Income / Assets
Valuation ratios
Quantity of an asset or flow per share
Earnings / Number of Shares
1) Name tells you balance sheet item 2) Balance sheet item income statement item 3) Income statement item in the numerator 4) Average value of balance sheet number in denominator
ROE = Return on assets × Leverage = Net profit margin × Asset turnover × Leverage
ROE = EBIT margin × Tax burden × Interest burden × Asset turnover × Leverage Balance sheet accruals ratio for time t = (NOAt - NOAt -1) / [(NOAt + NOAt-1)/2]
Cash flow accruals ratio for time t = [NIt - (CFOt + CFIt)]/[(NOAt + NOAt-1)/2] www.ift.world
9
Corporate Finance (1/3) Comparing projects with unequal lives: LCM and EAA
Capital Budgeting Initial outlay for new investment Outlay = FCInv + NWCInv
NPV of project with real option = NPV based on DCF along + value of option – cost of option
Initial outlay for replacement project Outlay = FCInv + NWCInv – Sal0 + T(Sal0 – B0)
Economic Income = Cash Flow + Change in Market Value Economic Profit = NOPAT - $WACC
Annual after-tax operating cash flow CF = (S – C)(1 – T) + TD
MVA = PV of economic profit
Terminal year after-tax non-operating cash flow TNOCF = SalT + NWCInv – T(SalT – BT)
Residual Income = Net Income – Equity Charge
Link between nominal rate, real rate and inflation: 1 + n = (1 + r) (1 + Inflation)
www.ift.world
10
Corporate Finance (2/3) Capital Structure
Dividend and Share Repurchase
rWACC = wdrd + were
Impact on share price of different tax rates:
Prop 1 and 2 with NO taxes: VL = V U re = r0 + (r0 –rd) D/E
Double taxation: Effective Tax Rate = Corporate Tax Rate + (1 – corporate tax rate)(individual marginal tax rate on dividends)
Prop 1 and 2 with taxes: VL = VU + tD re = r0 + (r0 – rd) (1 – t) D/E
Stable dividend policy: Expected increase in dividends = Increase in earnings × Target payout ratio × Adjustment factor Residual dividend policy: Dividend = Earnings – (Capital budget × Equity percent in capital structure)
Static tradeoff theory VL = VU + TD - PV (Cost of Financial Distress)
FCFE coverage ratio = FCFE / (dividends + share repurchases) www.ift.world
11
Corporate Finance (3/3) Mergers and Acquisitions Securities Offering Exchange ratio: number of shares of acquirers stock per target share; Cost = exchange ratio x number of shares of target x value of stock given to target shareholders HHI: If post-merger HHI is between 1,000 and 1,800 and change in HHI is 100+ then government might challenge merger; If post-merger HHI is greater than 1,800 and change in HHI is 50+ then government will challenge merger. Comparable company analysis: PRM = (DP – SP)/SP Bid valuation: Target Shareholder’s Gain = Premium = PT – VT Acquirer’s Gain = Synergies – Premium VA* = VA + VT + S – C
www.ift.world
12
Equity (1/4) VE – P = VE – P + V – V = (V – P) + (VE – V)
Macroeconomic multifactor model ri = T-bill rate
E(Ri) = RF + βi [E(RM) – RF]
+ (Sensitivity to confidence risk × 2.59%)
Adjusted beta = (2/3) (Unadjusted beta) + (1/3) (1.0)
− (Sensitivity to time horizon risk × 0.66%)
ri = RF + βimkt RMRF + βisize SMB + βivalue HML
− (Sensitivity to inflation risk × 4.32%)
ri = RF + βimkt RMRF + βisize SMB + βivalue HML+ βiliq LLQ
+ (Sensitivity to business-cycle risk × 1.49%) + (Sensitivity to market-timing risk × 3.61%)
www.ift.world
13
Equity (2/4) FCFF = NI + Dep + Int(1 – Tax rate) – FCInv – WCInv WCInv = Change in working capital, excluding cash and short term debt FCInv = Change in gross fixed assets FCFF = CFO + Int(1 – Tax rate) – FCInv
If interest is not categorized in CFO then do not add back.
NI
= (EBIT – Int) (1 – Tax rate) = EBIT(1 – Tax rate) – Int(1 – Tax rate)
FCFF
= EBIT(1 – Tax rate) + Dep – FCInv – WCInv
FCFF
= (EBITDA – Dep)(1 – Tax rate) + Dep – FCInv – WCInv
FCFF
= EBITDA (1 – Tax rate) + Dep (Tax rate) – FCInv – WCInv
FCFE
= FCFF - Int(1 – Tax rate) + Net Borrowing
FCFE
= NI + NCC – FCInv – WCInv + Net Borrowing
Value of firm = Value of operating assets + Value of non-operating assets Equity value = Firm value – Market value of debt Enterprise Value / EBITDA www.ift.world
14
Equity (3/4) Residual income = Net income – (Equity capital x Cost of equity) Residual income = EBIT (1 – Tax rate) – (Total capital x WACC) EVA = NOPAT – (C% x TC) NOPAT and TC are adjusted MVA = Market Value – Book Value
www.ift.world
15
Equity (4/4) Leading Price / Earnings
P0 = E0 (1+I) / (r−I) P0 = E0 (1+λI) / (r − λI) = E1 / (r − λI)
Trailing Price / Earnings
P0 = E1 / (ρ + (1−λ)I) and P0/E1 = 1 / (ρ + (1−λ)I) Price / Book
Price / Sales
Discount due to lack of control:
Trailing D/P = 4 x most quarterly divided / Price
DLOC = 1 − *1 / (1 + Control premium)]
www.ift.world
16
FIS (1/5) Forward Pricing Model: P(T* + T) = P(T*)F(T*,T) Relationship between spot rate and forward rates: (1+xS0)x = (1 + 1s0)(1 + 1f1)(1 + 1f2)…(1+1fx-1) Forward Rate model: Cox-Ingersoll-Ross (CIR) Model: Vasicek model: dr = a(b – r)dt + ςdz
Value of callable bond = Value of straight bond – Value of issuer call option Value of issuer call option = Value of straight bond – Value of callable bond Value of putable bond = Value of straight bond + Value of investor put option Value of investor put option = Value of putable bond – Value of straight bond www.ift.world
17
FIS (2/5)
Conversion value = Underlying share price × Conversion ratio Market conversion premium per share = Market conversion price – Underlying share price Market conversion price = convertible bond price / conversion ratio Premium over straight value = (convertible bond price / straight value) - 1 Value of convertible bond = Value of straight bond + Value of call option on the issuer’s stock Value of callable convertible bond = Value of straight bond + Value of call option on the issuer’s stock – Value of issuer call option Value of callable putable convertible bond = Value of straight bond + Value of call option on the issuer’s stock – Value of issuer call option + Value of investor put option
www.ift.world
18
FIS (3/5) Expected loss = Probability of default x Loss given default Holding equity is economically equivalent to owning a European call option on the company’s assets with a strike price of K. Owning the company’s debt is economically equivalent to owning a riskless bond that pays K dollars with certainty at time T, and simultaneously selling a European put option on the assets of the company with strike price K and maturity T.
www.ift.world
19
FIS (4/5)
Credit Spread = yD(t,T) – yP(t,T) = E(Percentage loss) + Liquidity premium
www.ift.world
20
FIS (5/5) Payout ratio = 1 – Recovery rate (%) Payout amount = Payout ratio x Notional Amount Expected Loss = PD x LGD Upfront payment = present value of protection leg – present value of premium leg Present value of credit spread = Upfront premium + Present value of fixed coupon Upfront premium ≈ (Credit spread – Fixed coupon) x Duration Credit spread ≈ (Upfront premium/Duration) + Fixed coupon Price of CDS in currency per 100 par = 100 – Upfront premium % Upfront premium % = 100 – price of CDS in currency per 100 par
Profit for the protection buyer = Change in spread in bps x Duration x Notional amount Percentage change in CDS price = change in spread in bps x Duration
www.ift.world
21
Derivatives (1/6) If underlying has no cash flows: F = S × (1 + r)T
If underlying has cash flows: F = S (1 + r)T + Future Value of Costs – Future Value Benefits F = Future value of underlying adjusted for carry cash flows = FV (S0 + θ0 - γ0) where θ0 represents present value of costs at time 0 and γ0 represents present value of benefits at time 0.
Vt = Present value of difference in in forward prices Using continuous compounding: 1+ 𝐿0 ℎ+𝑚 ×𝑡ℎ+𝑚 1+ 𝐿0 ℎ ×𝑡ℎ
FRA(0,h,m) = (
F0 T S0e c
r T
− 1)/𝑡𝑚
FRA value per unit of notional principal is 𝑉𝑔 0, , 𝑚 =
𝐹𝑅𝐴 𝑔,ℎ−𝑔,𝑚 −𝐹𝑅𝐴 0,ℎ,𝑚 𝑡𝑚 1+𝐷𝑔 ℎ+𝑚−𝑔 𝑡ℎ+𝑚−𝑔
www.ift.world
22
Derivatives (2/6) Fixed-income forward or futures price : 𝐹0 𝑇 = 𝐹𝑉0,𝑇 𝐵0 𝑇 + 𝑌 + 𝐴𝐼0 ) − 𝐴𝐼𝑇 − 𝐹𝑉𝐶𝐼0,𝑇 The quoted price which includes the conversion factor is: 𝑄𝐹0 𝑇 = 𝐹0 (𝑇)/𝐶𝐹 𝑇 𝐵0 𝑇 + 𝑌 is the quoted price observed at Time 0 for a fixed-rate bond that matures at time T + Y, 𝐴𝐼0 is the accrued interest at Time 0, 𝐴𝐼𝑇 is the accrued interest at Time T and 𝐹𝑉𝐶𝐼0,𝑇 is the coupons paid over the life of the futures contract.
𝑉𝑡 𝑇 = 𝑃𝑉𝑡,𝑇 (𝐹𝑡 𝑇 − 𝐹0 𝑇 ) Currency contracts: 𝐹0 𝑇 = 𝑆0 (1 + 𝑟𝑝 )𝑇 /(1 + 𝑟𝑏 )𝑇 where, 𝑟𝑝 is the interest rate in the price currency and 𝑟𝑏 is the interest rate in the base currency 𝑉𝑡 𝑇 = 𝑃𝑉𝑡,𝑇 (𝐹𝑡 𝑇 − 𝐹0 𝑇 )
www.ift.world
23
Derivatives (3/6) Interest rate swap: Swap fixed rate: 𝑟𝐹𝐼𝑋 =
1 − 𝑑𝑛 𝑑1 +𝑑2 +𝑑3+ …+𝑑𝑛
where, 𝑑𝑖 is the discount factor of the given period.
Swap value: 𝑉 = 𝑁𝐴 𝐹𝑆0 − 𝐹𝑆𝑡 𝑑1 + 𝑑2 + 𝑑3 + ⋯ + 𝑑𝑛 Currency swaps: 𝐹𝐵𝑘 = 𝑁𝐴𝑘 (𝑟𝐹𝐼𝑋,𝑘 𝑑1 + 𝑑2 + 𝑑3 + ⋯ 𝑑𝑛 + 𝑑𝑛 ) where, k represents the currency. 𝑉𝑎 = 𝐹𝐵𝑎 − 𝑆𝑡 𝐹𝐵𝑏 where, FB is the fixed-rate bond value in its own currency and 𝑆𝑡 is the exchange rate at Time t. Equity swaps: Vt = FBt(C0) – (St/St–)NAE – PV(Par – NAE) where FBt(C0) denotes the Time t value of a fixed-rate bond initiated with coupon C0 at Time 0, St denotes the current equity price, St– denotes the equity price observed at the last reset date, NA denotes the notional amount and PV() denotes the present value function from Time t to the swap maturity time. www.ift.world
24
Derivatives (4/6) Exercise value: cT = Max(0, ST – X) pT = Max(0, X – ST) For call options, hedge ratio, h =
𝑐+
− 𝑐−
𝑆+ − 𝑆−
c = hS + PV(–hS– + c–) = hS + PV(–hS+ + c+) 𝑝+ − 𝑝− 𝑆+− 𝑆−
For put options, hedge ratio, h = ≤0 p = hS + PV(–hS– + p–) = hS + PV(–hS+ + p+) Using the expectations approach: c = PV*πc+ + (1 – π)c–] and p = PV*πp+ + (1 – π)p–] π = the risk-neutral probability of an up move = (1 + r – d) / (u – d)
BSM Model (no carry benefit): c = SN(d1) – e–rTXN(d2) p = e–rTXN(–d2) – SN(–d1)
BSM Model (with carry benefit): c = Se–γTN(d1) – e–rTXN(d2) p = e–rTXN(–d2) – Se–γTN(–d1) Currency call option: c = Se− rf TN(d1 ) - e–rTXN(d2)
S is the currency exchange rate, rf is risk-free rate in the base currency and r is the risk-free rate in the price currency.
Replicating strategy cost = nSS + nBB
Expectations approach and 2-period model: c = PV*π2c++ + 2π(1 – π)c+– + (1 – π)2c– –] p = PV*π2p++ + 2π(1 – π)p+– + (1 – π)2p– –]
For calls: nS = N(d1) > 0, and nB = –N(d2) < 0
For puts: nS = –N(–d1) < 0 and nB = N(–d2) > 0 The price of the zero-coupon bond is B = PV(X) = e–rTX www.ift.world
25
Derivatives (5/6) When underlying instrument is a futures contract: c = e–rT[F0(T)N(d1) – XN(d2)]; p = e–rT[XN(–d2) – F0(T)N(–d1)] F0(T) = the futures price at Time 0 that expires at Time T. Futures option put–call parity: c = e–rT[F0(T) – X] + p Interest rate options: c = (AP)e−rT [FRA(0, tj−1, tm) N(d1) − RXN(d2)]. AP = Accrual period; for 90-day Libor the AP is 90/360. FRA(0,1,0.25) = FRA rate at Time 0 that expires at Time 1 and is based 0.25 year Libor, RX= Exercise rate Payer swaption for $1 notional amount: PAYSWN = (AP)PVA[RFIXN(d1) – RXN(d2)] Receiver swaption for $1 notional amount: RECSWN = (AP)PVA[RXN(–d2) – RFIXN(–d1)] AP stands for the accrual period. PVA refers to the present value of an annuity. RFIX is the market swap fixed rate at expiration of the option. RX is the exercise rate. Option deltas for calls: Deltac = e–δTN(d1). For puts: Deltap = – e–δTN(–d1); δ is the dividend yield on the underlying. New value of call ≅ c + Deltac (Ŝ − S) ; new value of put ≅ p Deltap (Ŝ − S). Ŝ represent new value the stock. Number of units of the hedging instruments, NH = Gammac = Gammap =
− Original Portfolio delta 𝐷𝑒𝑙𝑡𝑎𝐻
𝑒 −𝛿𝑇 𝑛(𝑑1 ) 𝑆𝜎 𝑇 www.ift.world
26
Derivatives (6/6) Covered call: short call + stock • • • • •
Maximum gain = (X – S0) + c0 Maximum loss = S0 – c0 Breakeven point = S0 – c0 Expiration value = ST – Max[(ST – X),0] Profit at expiration = ST – Max[(ST – X),0] + c0 – S0
Protective put: stock + long put • • • • •
Maximum profit = ST – S0 – p0 = Unlimited Maximum loss = S0 – X + p0 Breakeven point = S0 + p0 Expiration value = Max(ST,X) Profit at expiration = Max(ST,X) – S0 – p0
Long N shares and short N ATM call options = long N shares and short forward position on N/2 shares
Bull spread: buy call option with low exercise price and sell call option with high exercise price. Max profit: XH - XL - (cL – cH); Breakeven: XL + (cL – cH) Bear spread: buy put option with high exercise price and sell put option with a low exercise price. Max profit: XH - XL – cost; Breakeven: XH - (pH – pL)
Long calendar spread: sell near-dated call, buy long-dated call Short calendar spread: sell long-dated call, buy near-dated call Long straddle: buy put, buy call; Short straddle: sell put, sell call Long straddle strategy has two breakeven point: • Exercise price + cost of buying the call and put options • Exercise price - cost of buying the call and put options
www.ift.world
27
Alternative Investments (1/2) Value = NOI / Cap rate “All risk yield” = cap rate Cap rate = Discount rate - Growth rate
Value = GIM x Gross Income Debt service coverage ratio = first year NOI / debt service Loan to value ratio = loan amount / value
V5 = NOI6 / (r – g)
FFO is accounting net earnings excluding: 1. Depreciation charges on real estate 2. Deferred tax charges (deferred portion of tax expenses) 3. Gains/losses from sale of property and debt Restructuring
NAV = Estimated value of operating real estate + cash and A/R – debt and other liabilities
AFFO = FFO - straight line adjustment – recurring maintenance type capital expenditures and leasing commissions
NAVPS = AFFO/share x P/AFFO multiple
NAVPS = NAV / # of shares outstanding NAVPS = FFO/share x P/FFO multiple
Can value REIT share www.ift.world
28
Alternative Investments (2/2) Venture capital valuation method: 1. Post-Money Valuation
POST = V/(1 + r)t
2. Pre-Money Valuation
PRE = POST−I
3. Ownership Fraction
F = I/POST
4. Number of Shares
y = x * F / (1−F) +
5. Price of Shares
P1 = I/y
Hedge fund return = Alpha + Risk free rate +
𝑖 𝐵𝑒𝑡𝑎𝑖
𝑥 𝐹𝑎𝑐𝑡𝑜𝑟𝑖
Futures price = spot price + direct storage costs - convenience yield
Total return on a commodity futures position = price return + roll return + collateral return
www.ift.world
29
Portfolio Management (1/3) E R p = R F + λ1 βp ,1 + ⋯ + λK βp ,K E R p = R F + βp ,1 RMRF + βp ,2 SMB + βp ,3 HML + βp ,4 WML
Active Return: RA = RP – RB
Active return =
K j=1
RA = ∑ΔwiRAi
Active Weight: Δwi = wP,i – wB,i
Portfolio sensitivity j − Benchmark sensitivity
j
∗ Factor return j + Asset selection
R A = (Δwstocks R B ,stocks + Δwbonds R B ,bonds ) + (wP ,stocks R A ,stocks + wP ,bonds R A ,bonds )
www.ift.world
30
Portfolio Management (2/3) IR = Active return / Active risk =
STD R A = STD R B SR2P = SR2B + TC
E RA
∗
= IC
2
Rp −RB s Rp −RB
R Ai μi IC = COR , σi σi
IR ∗ SR B IR∗
2
IR = IC
BR σA
E R A ICR = TC ICR
μi TC = COR σi , Δwi σi
BR
BR σA
Risk premium = Yield on the nominal bond – yield on real bond – inflation expectation
www.ift.world
31
Portfolio Management (3/3) Using the parametric method: 5% VaR = (Expected Return – 1.65 ς) (-1) (Notional Amount) 1% VaR = (Expected Return – 2.33 ς) (-1) (Notional Amount) Bond price (B) sensitivity to changes in yield (y) can be expressed in terms of duration (D) and convexity (C): ∆𝐵 ∆𝑦 1 ∆𝑦 2 ≈ −𝐷 + 𝐶 𝐵 1+𝑦 2 1+𝑦
2
Option price sensitivity to changes in value of underlying: 𝐶𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑙𝑎𝑢𝑒 𝑜𝑓 𝑜𝑝𝑡𝑖𝑜𝑛 ∆𝑐 𝐷𝑒𝑙𝑡𝑎 = 𝐶𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑢𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔
𝑉𝑒𝑔𝑎 =
𝐶𝑎𝑛𝑔𝑒 𝑖𝑛 𝑑𝑒𝑙𝑡𝑎 𝐺𝑎𝑚𝑚𝑎 = 𝐶𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑢𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔
𝐶𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑜𝑝𝑡𝑖𝑜𝑛 𝐶𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑢𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔
www.ift.world
32
Practice, Practice, Practice.
www.ift.world
33
1
Comments and Instructions This document is a compilation of what I believe are the most important formulas for Level II. It does not list the facts which you must know in order to clear the exam. These facts are covered in our Level II crash course which is available for sale. No formula sheet can be 100% comprehensive. This formula sheet is no exception. However, it can serve as a good starting point. Print this document and add your notes/comments. Specifically, after every practice test look at this sheet and add formulas or comments which you think will be helpful. This document might start out as an IFT formula sheet but it should end up with many of your notes. If you have suggestions for improving this document please write to us at: [email protected] Do a lot of practice over the last few days and good luck on your exam.
Regards, Arif Irfanullah, CFA
www.ift.world
2
Quant (1/3) Correlation
Regression
H0: ρ = 0 versus Ha: ρ ≠ 0
Confidence interval for regression coefficients
Yi = b0 + b1Xi + εi, i = 1, ..., n
Prediction interval for regression equation:
sf = Standard deviation of prediction error
www.ift.world
3
Quant (2/3) SEE = Square root of mean square error.
Use the F-test to test: H0 : b1 = b2 = … = bk = 0 against the alternative hypothesis that at least one slope coefficient is not equal to 0.
Test for serial correlation: DW ≈ 2 (1 – r)
www.ift.world
4
Quant (3/3) Test whether the autocorrelations of the error term (error autocorrelations) differ significantly from 0. H0: error autocorrelation at a specified lag equals 0
Standard error of the residual correlation = 1 / 𝑇 where T is the number of observations Test-stat = residual autocorrelation / standard error
Consider an AR(1) model: xt = b0 + b1xt-1 + εt
Mean-reverting level is given by: xt =
𝑏0 (1 −𝑏1)
Example of first differencing:
www.ift.world
5
Econ (1/2) Ask = 1 / Bid
Covered interest rate parity
Forward points represent the difference between the forward rate and the spot rate.
Uncovered interest rate parity: Expected % change in spot rate (P/B) ≈ ip - iB Ex ante purchasing power parity: Expected % change in spot rate (P/B) ≈ ∏p - ∏B International Fisher relationship: ip - iB = ∏p - ∏B
www.ift.world
6
Econ (2/2)
P represented aggregate price (value) of stocks; E represents aggregate earnings
A is the total factor productivity
Labor productivity: Growth accounting: Growth rate in potential GDP = Long-term growth rate of labor force + Long-term growth rate in labor productivity ϴ is growth rate of TFP α is the share of GDP paid out to the suppliers of capital n is the growth rate of labor
Neo-classical model:
www.ift.world
7
FRA (1/2) Equity method: Value of investment = beginning value + share of profits – share of dividends Acquisition method: Partial goodwill = fair value of the acquisition - acquirer’s share of the fair value of acquiree’s net assets Full goodwill = fair value of the entity as a whole - the fair value of all acquiree’s assets and liabilities Funded status (net liability) = present value of defined benefit obligation – fair value of plan assets Total periodic pension cost = contributions + (ending funded status – beginning funded status) Under US GAAP actuarial gains and losses have two components: 1. Actual return – (plan assets × expected return) 2. Changes in a company’s pension obligation arising from changes in actuarial assumptions Under IFRS: Net return on plan assets = actual return – (plan assets × interest rate) Actuarial gains and losses = changes in a company’s pension obligation arising from changes in actuarial assumptions www.ift.world
8
FRA (2/2) Category
Measures
Example
Activity Ratios
Numerator / Denominator
Activity ratios
Efficiency
Revenue / Assets
Inventory turnover
Cost of good sold / Average inventory
Days of inventory on Hand
Number of days in period / Inventory turnover
Liquidity ratios Ability to meet its short term obligations
Current Assets / Current Liabilities
Solvency ratios
Ability to meet long term debt obligations
Assets / Equity
Profitability ratios
Profitability
Net Income / Assets
Valuation ratios
Quantity of an asset or flow per share
Earnings / Number of Shares
1) Name tells you balance sheet item 2) Balance sheet item income statement item 3) Income statement item in the numerator 4) Average value of balance sheet number in denominator
ROE = Return on assets × Leverage = Net profit margin × Asset turnover × Leverage
ROE = EBIT margin × Tax burden × Interest burden × Asset turnover × Leverage Balance sheet accruals ratio for time t = (NOAt - NOAt -1) / [(NOAt + NOAt-1)/2]
Cash flow accruals ratio for time t = [NIt - (CFOt + CFIt)]/[(NOAt + NOAt-1)/2] www.ift.world
9
Corporate Finance (1/3) Comparing projects with unequal lives: LCM and EAA
Capital Budgeting Initial outlay for new investment Outlay = FCInv + NWCInv
NPV of project with real option = NPV based on DCF along + value of option – cost of option
Initial outlay for replacement project Outlay = FCInv + NWCInv – Sal0 + T(Sal0 – B0)
Economic Income = Cash Flow + Change in Market Value Economic Profit = NOPAT - $WACC
Annual after-tax operating cash flow CF = (S – C)(1 – T) + TD
MVA = PV of economic profit
Terminal year after-tax non-operating cash flow TNOCF = SalT + NWCInv – T(SalT – BT)
Residual Income = Net Income – Equity Charge
Link between nominal rate, real rate and inflation: 1 + n = (1 + r) (1 + Inflation)
www.ift.world
10
Corporate Finance (2/3) Capital Structure
Dividend and Share Repurchase
rWACC = wdrd + were
Impact on share price of different tax rates:
Prop 1 and 2 with NO taxes: VL = V U re = r0 + (r0 –rd) D/E
Double taxation: Effective Tax Rate = Corporate Tax Rate + (1 – corporate tax rate)(individual marginal tax rate on dividends)
Prop 1 and 2 with taxes: VL = VU + tD re = r0 + (r0 – rd) (1 – t) D/E
Stable dividend policy: Expected increase in dividends = Increase in earnings × Target payout ratio × Adjustment factor Residual dividend policy: Dividend = Earnings – (Capital budget × Equity percent in capital structure)
Static tradeoff theory VL = VU + TD - PV (Cost of Financial Distress)
FCFE coverage ratio = FCFE / (dividends + share repurchases) www.ift.world
11
Corporate Finance (3/3) Mergers and Acquisitions Securities Offering Exchange ratio: number of shares of acquirers stock per target share; Cost = exchange ratio x number of shares of target x value of stock given to target shareholders HHI: If post-merger HHI is between 1,000 and 1,800 and change in HHI is 100+ then government might challenge merger; If post-merger HHI is greater than 1,800 and change in HHI is 50+ then government will challenge merger. Comparable company analysis: PRM = (DP – SP)/SP Bid valuation: Target Shareholder’s Gain = Premium = PT – VT Acquirer’s Gain = Synergies – Premium VA* = VA + VT + S – C
www.ift.world
12
Equity (1/4) VE – P = VE – P + V – V = (V – P) + (VE – V)
Macroeconomic multifactor model ri = T-bill rate
E(Ri) = RF + βi [E(RM) – RF]
+ (Sensitivity to confidence risk × 2.59%)
Adjusted beta = (2/3) (Unadjusted beta) + (1/3) (1.0)
− (Sensitivity to time horizon risk × 0.66%)
ri = RF + βimkt RMRF + βisize SMB + βivalue HML
− (Sensitivity to inflation risk × 4.32%)
ri = RF + βimkt RMRF + βisize SMB + βivalue HML+ βiliq LLQ
+ (Sensitivity to business-cycle risk × 1.49%) + (Sensitivity to market-timing risk × 3.61%)
www.ift.world
13
Equity (2/4) FCFF = NI + Dep + Int(1 – Tax rate) – FCInv – WCInv WCInv = Change in working capital, excluding cash and short term debt FCInv = Change in gross fixed assets FCFF = CFO + Int(1 – Tax rate) – FCInv
If interest is not categorized in CFO then do not add back.
NI
= (EBIT – Int) (1 – Tax rate) = EBIT(1 – Tax rate) – Int(1 – Tax rate)
FCFF
= EBIT(1 – Tax rate) + Dep – FCInv – WCInv
FCFF
= (EBITDA – Dep)(1 – Tax rate) + Dep – FCInv – WCInv
FCFF
= EBITDA (1 – Tax rate) + Dep (Tax rate) – FCInv – WCInv
FCFE
= FCFF - Int(1 – Tax rate) + Net Borrowing
FCFE
= NI + NCC – FCInv – WCInv + Net Borrowing
Value of firm = Value of operating assets + Value of non-operating assets Equity value = Firm value – Market value of debt Enterprise Value / EBITDA www.ift.world
14
Equity (3/4) Residual income = Net income – (Equity capital x Cost of equity) Residual income = EBIT (1 – Tax rate) – (Total capital x WACC) EVA = NOPAT – (C% x TC) NOPAT and TC are adjusted MVA = Market Value – Book Value
www.ift.world
15
Equity (4/4) Leading Price / Earnings
P0 = E0 (1+I) / (r−I) P0 = E0 (1+λI) / (r − λI) = E1 / (r − λI)
Trailing Price / Earnings
P0 = E1 / (ρ + (1−λ)I) and P0/E1 = 1 / (ρ + (1−λ)I) Price / Book
Price / Sales
Discount due to lack of control:
Trailing D/P = 4 x most quarterly divided / Price
DLOC = 1 − *1 / (1 + Control premium)]
www.ift.world
16
FIS (1/5) Forward Pricing Model: P(T* + T) = P(T*)F(T*,T) Relationship between spot rate and forward rates: (1+xS0)x = (1 + 1s0)(1 + 1f1)(1 + 1f2)…(1+1fx-1) Forward Rate model: Cox-Ingersoll-Ross (CIR) Model: Vasicek model: dr = a(b – r)dt + ςdz
Value of callable bond = Value of straight bond – Value of issuer call option Value of issuer call option = Value of straight bond – Value of callable bond Value of putable bond = Value of straight bond + Value of investor put option Value of investor put option = Value of putable bond – Value of straight bond www.ift.world
17
FIS (2/5)
Conversion value = Underlying share price × Conversion ratio Market conversion premium per share = Market conversion price – Underlying share price Market conversion price = convertible bond price / conversion ratio Premium over straight value = (convertible bond price / straight value) - 1 Value of convertible bond = Value of straight bond + Value of call option on the issuer’s stock Value of callable convertible bond = Value of straight bond + Value of call option on the issuer’s stock – Value of issuer call option Value of callable putable convertible bond = Value of straight bond + Value of call option on the issuer’s stock – Value of issuer call option + Value of investor put option
www.ift.world
18
FIS (3/5) Expected loss = Probability of default x Loss given default Holding equity is economically equivalent to owning a European call option on the company’s assets with a strike price of K. Owning the company’s debt is economically equivalent to owning a riskless bond that pays K dollars with certainty at time T, and simultaneously selling a European put option on the assets of the company with strike price K and maturity T.
www.ift.world
19
FIS (4/5)
Credit Spread = yD(t,T) – yP(t,T) = E(Percentage loss) + Liquidity premium
www.ift.world
20
FIS (5/5) Payout ratio = 1 – Recovery rate (%) Payout amount = Payout ratio x Notional Amount Expected Loss = PD x LGD Upfront payment = present value of protection leg – present value of premium leg Present value of credit spread = Upfront premium + Present value of fixed coupon Upfront premium ≈ (Credit spread – Fixed coupon) x Duration Credit spread ≈ (Upfront premium/Duration) + Fixed coupon Price of CDS in currency per 100 par = 100 – Upfront premium % Upfront premium % = 100 – price of CDS in currency per 100 par
Profit for the protection buyer = Change in spread in bps x Duration x Notional amount Percentage change in CDS price = change in spread in bps x Duration
www.ift.world
21
Derivatives (1/6) If underlying has no cash flows: F = S × (1 + r)T
If underlying has cash flows: F = S (1 + r)T + Future Value of Costs – Future Value Benefits F = Future value of underlying adjusted for carry cash flows = FV (S0 + θ0 - γ0) where θ0 represents present value of costs at time 0 and γ0 represents present value of benefits at time 0.
Vt = Present value of difference in in forward prices Using continuous compounding: 1+ 𝐿0 ℎ+𝑚 ×𝑡ℎ+𝑚 1+ 𝐿0 ℎ ×𝑡ℎ
FRA(0,h,m) = (
F0 T S0e c
r T
− 1)/𝑡𝑚
FRA value per unit of notional principal is 𝑉𝑔 0, , 𝑚 =
𝐹𝑅𝐴 𝑔,ℎ−𝑔,𝑚 −𝐹𝑅𝐴 0,ℎ,𝑚 𝑡𝑚 1+𝐷𝑔 ℎ+𝑚−𝑔 𝑡ℎ+𝑚−𝑔
www.ift.world
22
Derivatives (2/6) Fixed-income forward or futures price : 𝐹0 𝑇 = 𝐹𝑉0,𝑇 𝐵0 𝑇 + 𝑌 + 𝐴𝐼0 ) − 𝐴𝐼𝑇 − 𝐹𝑉𝐶𝐼0,𝑇 The quoted price which includes the conversion factor is: 𝑄𝐹0 𝑇 = 𝐹0 (𝑇)/𝐶𝐹 𝑇 𝐵0 𝑇 + 𝑌 is the quoted price observed at Time 0 for a fixed-rate bond that matures at time T + Y, 𝐴𝐼0 is the accrued interest at Time 0, 𝐴𝐼𝑇 is the accrued interest at Time T and 𝐹𝑉𝐶𝐼0,𝑇 is the coupons paid over the life of the futures contract.
𝑉𝑡 𝑇 = 𝑃𝑉𝑡,𝑇 (𝐹𝑡 𝑇 − 𝐹0 𝑇 ) Currency contracts: 𝐹0 𝑇 = 𝑆0 (1 + 𝑟𝑝 )𝑇 /(1 + 𝑟𝑏 )𝑇 where, 𝑟𝑝 is the interest rate in the price currency and 𝑟𝑏 is the interest rate in the base currency 𝑉𝑡 𝑇 = 𝑃𝑉𝑡,𝑇 (𝐹𝑡 𝑇 − 𝐹0 𝑇 )
www.ift.world
23
Derivatives (3/6) Interest rate swap: Swap fixed rate: 𝑟𝐹𝐼𝑋 =
1 − 𝑑𝑛 𝑑1 +𝑑2 +𝑑3+ …+𝑑𝑛
where, 𝑑𝑖 is the discount factor of the given period.
Swap value: 𝑉 = 𝑁𝐴 𝐹𝑆0 − 𝐹𝑆𝑡 𝑑1 + 𝑑2 + 𝑑3 + ⋯ + 𝑑𝑛 Currency swaps: 𝐹𝐵𝑘 = 𝑁𝐴𝑘 (𝑟𝐹𝐼𝑋,𝑘 𝑑1 + 𝑑2 + 𝑑3 + ⋯ 𝑑𝑛 + 𝑑𝑛 ) where, k represents the currency. 𝑉𝑎 = 𝐹𝐵𝑎 − 𝑆𝑡 𝐹𝐵𝑏 where, FB is the fixed-rate bond value in its own currency and 𝑆𝑡 is the exchange rate at Time t. Equity swaps: Vt = FBt(C0) – (St/St–)NAE – PV(Par – NAE) where FBt(C0) denotes the Time t value of a fixed-rate bond initiated with coupon C0 at Time 0, St denotes the current equity price, St– denotes the equity price observed at the last reset date, NA denotes the notional amount and PV() denotes the present value function from Time t to the swap maturity time. www.ift.world
24
Derivatives (4/6) Exercise value: cT = Max(0, ST – X) pT = Max(0, X – ST) For call options, hedge ratio, h =
𝑐+
− 𝑐−
𝑆+ − 𝑆−
c = hS + PV(–hS– + c–) = hS + PV(–hS+ + c+) 𝑝+ − 𝑝− 𝑆+− 𝑆−
For put options, hedge ratio, h = ≤0 p = hS + PV(–hS– + p–) = hS + PV(–hS+ + p+) Using the expectations approach: c = PV*πc+ + (1 – π)c–] and p = PV*πp+ + (1 – π)p–] π = the risk-neutral probability of an up move = (1 + r – d) / (u – d)
BSM Model (no carry benefit): c = SN(d1) – e–rTXN(d2) p = e–rTXN(–d2) – SN(–d1)
BSM Model (with carry benefit): c = Se–γTN(d1) – e–rTXN(d2) p = e–rTXN(–d2) – Se–γTN(–d1) Currency call option: c = Se− rf TN(d1 ) - e–rTXN(d2)
S is the currency exchange rate, rf is risk-free rate in the base currency and r is the risk-free rate in the price currency.
Replicating strategy cost = nSS + nBB
Expectations approach and 2-period model: c = PV*π2c++ + 2π(1 – π)c+– + (1 – π)2c– –] p = PV*π2p++ + 2π(1 – π)p+– + (1 – π)2p– –]
For calls: nS = N(d1) > 0, and nB = –N(d2) < 0
For puts: nS = –N(–d1) < 0 and nB = N(–d2) > 0 The price of the zero-coupon bond is B = PV(X) = e–rTX www.ift.world
25
Derivatives (5/6) When underlying instrument is a futures contract: c = e–rT[F0(T)N(d1) – XN(d2)]; p = e–rT[XN(–d2) – F0(T)N(–d1)] F0(T) = the futures price at Time 0 that expires at Time T. Futures option put–call parity: c = e–rT[F0(T) – X] + p Interest rate options: c = (AP)e−rT [FRA(0, tj−1, tm) N(d1) − RXN(d2)]. AP = Accrual period; for 90-day Libor the AP is 90/360. FRA(0,1,0.25) = FRA rate at Time 0 that expires at Time 1 and is based 0.25 year Libor, RX= Exercise rate Payer swaption for $1 notional amount: PAYSWN = (AP)PVA[RFIXN(d1) – RXN(d2)] Receiver swaption for $1 notional amount: RECSWN = (AP)PVA[RXN(–d2) – RFIXN(–d1)] AP stands for the accrual period. PVA refers to the present value of an annuity. RFIX is the market swap fixed rate at expiration of the option. RX is the exercise rate. Option deltas for calls: Deltac = e–δTN(d1). For puts: Deltap = – e–δTN(–d1); δ is the dividend yield on the underlying. New value of call ≅ c + Deltac (Ŝ − S) ; new value of put ≅ p Deltap (Ŝ − S). Ŝ represent new value the stock. Number of units of the hedging instruments, NH = Gammac = Gammap =
− Original Portfolio delta 𝐷𝑒𝑙𝑡𝑎𝐻
𝑒 −𝛿𝑇 𝑛(𝑑1 ) 𝑆𝜎 𝑇 www.ift.world
26
Derivatives (6/6) Covered call: short call + stock • • • • •
Maximum gain = (X – S0) + c0 Maximum loss = S0 – c0 Breakeven point = S0 – c0 Expiration value = ST – Max[(ST – X),0] Profit at expiration = ST – Max[(ST – X),0] + c0 – S0
Protective put: stock + long put • • • • •
Maximum profit = ST – S0 – p0 = Unlimited Maximum loss = S0 – X + p0 Breakeven point = S0 + p0 Expiration value = Max(ST,X) Profit at expiration = Max(ST,X) – S0 – p0
Long N shares and short N ATM call options = long N shares and short forward position on N/2 shares
Bull spread: buy call option with low exercise price and sell call option with high exercise price. Max profit: XH - XL - (cL – cH); Breakeven: XL + (cL – cH) Bear spread: buy put option with high exercise price and sell put option with a low exercise price. Max profit: XH - XL – cost; Breakeven: XH - (pH – pL)
Long calendar spread: sell near-dated call, buy long-dated call Short calendar spread: sell long-dated call, buy near-dated call Long straddle: buy put, buy call; Short straddle: sell put, sell call Long straddle strategy has two breakeven point: • Exercise price + cost of buying the call and put options • Exercise price - cost of buying the call and put options
www.ift.world
27
Alternative Investments (1/2) Value = NOI / Cap rate “All risk yield” = cap rate Cap rate = Discount rate - Growth rate
Value = GIM x Gross Income Debt service coverage ratio = first year NOI / debt service Loan to value ratio = loan amount / value
V5 = NOI6 / (r – g)
FFO is accounting net earnings excluding: 1. Depreciation charges on real estate 2. Deferred tax charges (deferred portion of tax expenses) 3. Gains/losses from sale of property and debt Restructuring
NAV = Estimated value of operating real estate + cash and A/R – debt and other liabilities
AFFO = FFO - straight line adjustment – recurring maintenance type capital expenditures and leasing commissions
NAVPS = AFFO/share x P/AFFO multiple
NAVPS = NAV / # of shares outstanding NAVPS = FFO/share x P/FFO multiple
Can value REIT share www.ift.world
28
Alternative Investments (2/2) Venture capital valuation method: 1. Post-Money Valuation
POST = V/(1 + r)t
2. Pre-Money Valuation
PRE = POST−I
3. Ownership Fraction
F = I/POST
4. Number of Shares
y = x * F / (1−F) +
5. Price of Shares
P1 = I/y
Hedge fund return = Alpha + Risk free rate +
𝑖 𝐵𝑒𝑡𝑎𝑖
𝑥 𝐹𝑎𝑐𝑡𝑜𝑟𝑖
Futures price = spot price + direct storage costs - convenience yield
Total return on a commodity futures position = price return + roll return + collateral return
www.ift.world
29
Portfolio Management (1/3) E R p = R F + λ1 βp ,1 + ⋯ + λK βp ,K E R p = R F + βp ,1 RMRF + βp ,2 SMB + βp ,3 HML + βp ,4 WML
Active Return: RA = RP – RB
Active return =
K j=1
RA = ∑ΔwiRAi
Active Weight: Δwi = wP,i – wB,i
Portfolio sensitivity j − Benchmark sensitivity
j
∗ Factor return j + Asset selection
R A = (Δwstocks R B ,stocks + Δwbonds R B ,bonds ) + (wP ,stocks R A ,stocks + wP ,bonds R A ,bonds )
www.ift.world
30
Portfolio Management (2/3) IR = Active return / Active risk =
STD R A = STD R B SR2P = SR2B + TC
E RA
∗
= IC
2
Rp −RB s Rp −RB
R Ai μi IC = COR , σi σi
IR ∗ SR B IR∗
2
IR = IC
BR σA
E R A ICR = TC ICR
μi TC = COR σi , Δwi σi
BR
BR σA
Risk premium = Yield on the nominal bond – yield on real bond – inflation expectation
www.ift.world
31
Portfolio Management (3/3) Using the parametric method: 5% VaR = (Expected Return – 1.65 ς) (-1) (Notional Amount) 1% VaR = (Expected Return – 2.33 ς) (-1) (Notional Amount) Bond price (B) sensitivity to changes in yield (y) can be expressed in terms of duration (D) and convexity (C): ∆𝐵 ∆𝑦 1 ∆𝑦 2 ≈ −𝐷 + 𝐶 𝐵 1+𝑦 2 1+𝑦
2
Option price sensitivity to changes in value of underlying: 𝐶𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑙𝑎𝑢𝑒 𝑜𝑓 𝑜𝑝𝑡𝑖𝑜𝑛 ∆𝑐 𝐷𝑒𝑙𝑡𝑎 = 𝐶𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑢𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔
𝑉𝑒𝑔𝑎 =
𝐶𝑎𝑛𝑔𝑒 𝑖𝑛 𝑑𝑒𝑙𝑡𝑎 𝐺𝑎𝑚𝑚𝑎 = 𝐶𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑢𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔
𝐶𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑜𝑝𝑡𝑖𝑜𝑛 𝐶𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑢𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔
www.ift.world
32
Practice, Practice, Practice.
www.ift.world
33
