Equity
Holding Period Return
For a stock of price S, with dividend D
Dy +S —S
E(y) = 7%
0
Dividend Yield
Ta = Di/S0
Gordon model
‘Assume constant growth of dividends g, req rate of return r
(Last year’s dividend Do)
Next year’s dividend D, = Do(1 +g)
Present value of future cash payments in perpetuity
Dị
g
= 1 + BX (tm -7%)
4g = Plowback ratio x Return on equity
Dividend per share
Payout ratio = vy Earnings per share -
Plowback ratio = 1 — Payout ratio
Earnings per share
Book value per share
— — Netincome
~ Book value of equity
Return on equity =
Net income
Earnings per share = 57 sber of outstanding shares
Book value = Common stock + Retained earnings
Common stock = Face value of shares
No-Growth Value
Dividend @ 100% payout
nov = FPS: r
Present Value of Growth Opportunities
PVGO =P—NGV
D1 +g)_ Bị
mg
PV60 =
PE Ratios with growth
Dị E,(1~b)
=g r-@XxR0E)
(b x ROE)
(b = plowback ratio)
Residual dividend approach
Project returns: 7p
Shareholder cost of equity: re
If > re, company creates value for shareholders
If % = 7%, shareholders indifferent
If % < 7%, company destroys value for shareholders
*Stipulate whether average or year-end equity is used
Gordon two-stage
(Last year’s dividend Do) e.g 3 years’ variable growth g,, g; and gs, after which growth becomes constant g,
Dividends are
D3 = D2(1 + gs)
Dg = D3(1 + ga) (Caution!]
Present value of future constant growth at t = 3 is
Ds
Py =
ONT os
‘Stage 1: Sum of discounted D4, D; and D; to t=0
S, = Ltr) De :
Stage 2: Discount P3 to t=O
52h = Pa + 98)
PV = Stage 1+ Stage 2 EBITDA multiples
EBITDA Multiple = pha “z—g
+ multiple => 7 r (risk) or + g (growth)
Calculate multiples for comparable companies, to work out
enterprise value
Enterprise value = EBITDA X Multiple
Fair val of Equity = EV + excess cash — long term debt
Equity
Fail ue = ———————
alrvalue = ¥ shares outstanding
Residual income model
PV of stock price in terms of ROE and book value of firm equity (per share) projected PV increases when ROE>re:
(ROE) — r,)BVạ (ROE; — r,)BVị
PV = BY) + on Ore) (1+?)
‘ROE; — r,)BV;
(+n)
Dupont
Ror = ——Net income
~ Shareholder equity
How effectively has the frm been run to benefit shareholders
‘Return fomopsforshareholiersvcredton | How well did the firm use debt financing to
NOPAi iI + int exp.x ) | benefit shareholders
R0A= Average assets
‘Assettumover Sales | Operatingetficiency | Interest efficiency | Leverage ‘NoPAT Wel Av assets
Fwassets* | sales * “i Nopar * Av shequity
vowmuchrevene | “reemeiietior | “Mhetgmuamafaea | yma mats beyond Tomssanr | cetonvsantoisen | MGOAESkkAshe | samydaMeoneov
Tư toan | "Sowing inet na
Trang 2
Debt
Effective annual rate
With a compounding period T (measured in years e.g 1 month
T = 1/12), return over that period r7(T)
FAR = [L+r()]/”—1
= (+APRxT)1T—1
= exp(Tec) — 1
Annual percentage rate: APR = r;(T) x (1/T)
Continuously compounding rate: rec = In(1 + EAR)
Real and nominal interest rates
isinflation
Thom Treat = TT Thơm — TreaL
Treat +1
Treat © Tom — i
Forward rates
po
Tr
(1+m)"
year forward rate = TY đặc
Forward rates give expectation of interest rates, as short-rates
based on yields to maturity of different-duration assets Discount
future cashflows by product of relevant forward rates (taken from
t=0, these products are just the compounded yields)
Liquidity preference theory
This theory asserts that forward rates reflect expectations about
future interest rates plus a liquidity premium that increases with
maturity:
Interest rate = Forward Rate — Liquidity Premium
Deferred loans
Implied annually compounded forward rate for a deferred loan of
length X beginning in year Y, based on the yields ris
p= [Otte oe #
Bond equivalent ytm
Convention: work out semi-annual yield to maturity, (ie 2x
number of periods) then double it
Realised compounding yield to maturity
= yield with reinvestment of coupons at a given interest rate Add
up total proceeds, including T-compounded value of all coupons
Total proceeds) iB 4 Reatiytinis ( Current price
Equivalent annual costs (Loan repayments)
Take out a loan of NPV=SL, paid over a period of n with interest
rate r n-year annuity factor is
AR,= 1 r r1+p)" 1
Equivalent per-period costs (repayments) is
_ NPV
= AR,
NPV
ssid + et
NPV = Chota yt Garett Gant
IRR: Expected rate of return Use IRR({Cashflows}) in excel
IRR =r>NPV =0
IRR > discount = accept IRR < discount => reject
Bonds: YTM = IRR (BBB+ investor, BBB- junk)
YTM = coupon = ‘Par’
YTM < coupon = ‘Premium!
YTM > coupon = ‘Discount’
YTM ~ RFR + Default risk + Interest rate risk
+ Premium for embedded options Interest rates T = bond price J, Interest rates | => bond price T E.g with coupons C, YTM r, num periods T, and face value F
PV(coupon + FV) = PV(r,7,C, FV) PV(coupon only) = PV(r,7,C) PV(FV only) = PV(r,T, 0, FV) YTM = RATE(tperiods, payments, PV, FV) or IRR({Cashflows})
*NOT THE DISCOUNTED CASHFLOWS, and with -ve PV [Check semi-annual, for rates + yields PV must be negative.) Bid = sell $ to broker, Ask = buy $ from broker
Liquidity « 1/(Ask — Bid) Duration
= weighted average of the times when the bond's cash payments are received
„_ _1xPV(;), 2xPV(Œ;) T x PV(Cr)
Duration = PV PV pee PV
Portfolio duration
Total equity in portfolio = > Assets — 5 Liabilities Portfolio duration = Multiply component durations by their values (liabilities negative), divide by total equity
Ho duration = L4S8¢tX Durat.~ 5 Liab.x Durat
Portfolio duration = Total equity in portfolio
Modified duration
= percentage change in bond price for a 1 percentage-point change in yield (adjusts for accuracy) Approximates % change in
bond price for 1% change in yield Units is ‘years’, so %A(price) =
lyr] * [%Arate/year] = %
Duration
Mod dur = volatility (%) = 1+yield
Trade discounts
Discount rg for payment at ¢ days rather than usual tz day payment terms
Trang 3Capital structure
Cost of debt
Tq = 17 + credit spread
Credit spread = default risk for firms of similar interest coverage
Interest expenses
Cost of equity (From CAPM):
Te = 17 + BX Market risk premium
Weighted Average Cost of Capital (WACC)
Financial + systematic risk: V = D + E
WACC = (L—T,) Xỹ X† +,
Levered/unlevered return (concentration of risk)
rf is return on unlevered equity = return on assets, leverage D/E,
gives rf as cost of levered equity, and:
rẻ =rÿ +(g)(1~T2Œ# =m)
Levered/unlevered beta
'When determining cost of equity, consider current firm leverage
D./E¢ vs target leverage D;/Et Levered beta = business risk +
financial risk
D
8, = 8u x[L+ yx Œ~T2]
Bu
u—p-——n
[t+#x-T|
So, to determine target beta:
+R)
Be = Be X —y
1+ Re -T) Tax shields
When value of debt Dis permanent, tax shields with deductions at
tax rate T increases firm value:
Value of levered firm = Value if all equity-financed + T, x D
Leverage => add financial distress (measure as PV, -ve) Optimise!
Value of government claim on firm
Tax rate Tạ, PV of taxes paid is
PV(T) = (E+T)XTe
1-T
Sustainable Growth Rate (SGR)
Rate at which firm can grow in without further equity raise,
assuming new debt can be taken on (Uses book value of equity)
SER om Retained earnings
Beginning equity
RE NI Sales D+E
W.I * Sales“ N.A.* E
D
Plowback x N.Margin x Asset util.x (: + ?)
Net assets = BV.E + Liabilities(Loans)
Buybacks
ADebt = excess cash used + new debt raised
PV (Tax shields) = AD xT
Debt shield Value to share sellers = Premium x Volume
% to sellers = Value to sellers / PV (Tax) Enew = Eoig — AD + AD xT,
Total Enterprise Value (TEV)
TEV = D + E —non-operating cashflows
During buybacks, incorporate changes including — cost of buyback + PV tax shields + cash value of new debt
Equate to PV of sustainable cash flows
EBIT x (1~T.)
Metrics
Economic Value Added (Stern-Stewart)
EVA = residual income = income earned — income required
= income earned — cost of capital x investment
Economic Profit (McKinsey)
EP = (ROI —r) x capital invested Earnings per share
~ Number of shares outstanding Return on equity (net less re)
Book value of common equity
Return on assets (net less WACC)
Net operating profit after taxes (NOPAT)
ROA=
Total assets
Price-Earnings ratio
_ Share price
“EPS Personal vs Corporate tax rates for dividends Corp tax rate T;, personal tax rate T, Compare:
1 Distribute $1 today, shareholder invest for 1 year will have
$q—T,) x(1+r x(1—T,))
2 Firm invest $1 today, distribute next year, shareholder will have
$(1-T,) x (1+rx(1-T))
Te > Ty > payout 1, T; < Ty payout 1
PE
Franking credit
If T; is corp tax rate and DIV is franked credit distributed, then franking credit is
Te
FC = DIV x
idend policy
Tproj < Te = poor projects, FCFE to shareholders: Dividends /
buybacks, review policies/management
Tproj > Te = ood projects, invest where cash available
Working capital management
AWorking capital = AAcc rec + A Inv - A Acc payable Work out PV cash flow perpetuity after tax for r = WACC
AEBIT
EV=eTDS
NPV = -AWC - PV
Trang 4Markets
Synthetic bonds / law of one price
Consider three cash flows C,, C2 and C3 If
Cy =AXC, + (I-A) x Cp,
then
Œ q—G"
A=
with
PV; = Ax PV, +(1—A) x PV,
e.g C,=$100, C = $0 and C; = $50, Then
$50 — $0
4E §100= s0 — 50%
Two-asset portfolio
Mean returns {14 g, risk (StDev) đa g, correlation
— £00 (Ha He)
risk-free rate 7 Weights «4 and Wp = 1— wa
Mean portfolio return
Mp = 0à X HẠ + 0g XMg Variance of portfolio
Op = whox + wh of + 2 wp (0p COV(Tạ, Tp)
= Wh ox + wh of + 2 0A (0g DApØÀg
Minimum-variance portfolio
đỗ — c0v(a,Ta)
KT an
^ ` gà + độ — 2 cov(a,rạ)
Optimal (tangent) portfolio is given by A-stock weighting:
(ua — rr)ØÄ — (Mạ — r/)PAnØÀp
®ˆ (wạ = rr)gÄ + (wg —rr)øÄ — (MA + tp — 2r/)PasÀp
Sharpe ratio / ‘reward to risk’ is
tp
Sharpe ratio =
p
* Sharpe is max along Tangent line/Capital Market Line (‘CML’)
Mp = Sharpe X oy +Tr
For CML portfolio with risky asset allocation Wp = 1— wy
Mẹ = (1.— @p)ry ‡ @gMp
Øc = 0pớp
To achieve a target return pr,
Mr —Tr
wy =f
Mp~Tr
Utility
U =E() —0.5Aø2 A>0: Risk aversion, preference for lower risk
A=0: Risk neutral, highest return sought
A<0: Pathological risk seeking, lower returns ok with † risk
With risk-free rate ry, and risky asset 7), 0, maximum level of risk
aversion for which risky asset is preferred:
A=2?Π=7)
oD
For the CML portfolio, utility is
1
U=(1 =p) 17 + Opty — 2 Aojøj
So to maximise utility,
da, 777 + He — Apo} — 0
= tp ~Tr
> On =" Age
Capital Asset Pricing Model / CAPM
Market risk premium = tm — Ty
Required return on risky asset (systematic risk only!)
T =Tr + X (im —Ty)
Asset Beta:
_cov(stock, market)
_ 0ar(market)
B > 1> Aggressive stock
<1 = Defensive stock
Realised CAPM in terms of risk premium: R=r-rf, a is anomalous
return and e; is random term
Rị = ai + B(RM +e
Risk is expressed in terms of systematic and firm specific components of variance
of = Bi ok + 07(e:) With R-square, expressing portion of variation in stock movement
that is systematic:
When there is a portfolio of stocks P with weight w, so that
Rp = ap + BpRu + ep
Bp
oF = Boon + 07(ep)
And for two stocks i and j
Cov(Ri, Rj) = BiBjCov(Ru, Ru) = BiBj om
_ Pibjok
Py
Fama-French
CAPM, but with three betas for market, firm size and book-to-
market ratios
a = % + — Brmarket X (nm — 7)
+ Bsize X (Tsmau ~ Tiarge) + Bom X (raemt — Poems)
a > 0 4ve risk-adjusted returns
@ <0 = ~ve risk-adjusted returns
Trang 5Options and risk management
Sell call: “You buy this stock in future at a set price”
Buy call: “I can buy this stock at set price in future”
Sell put: “You can sell this stock to me at a set price in future”
Buy put: “I can sell this stock to you at set price in future”
IR swaps
Lay off risk: Pay fixed and receive floating in IR swap (-ve duration)
Party that receives fixed has duration equivalent to owning the
bond (+ve) Party that pays has -ve Change in equity AP for a
change in yield is backed out from duration of bond with price P
@ a)
Dur =
Binomial pricing:
Portfolio is perfectly hedged if one holds H shares of stock for
each call option written, where C and S are the prices in future
‘outcomes:
"—-
Su-Sa
Equivalent portfolio is holding (1-H) shares and placing
remaining funds (from stock + put/call portfolio) into T-bills
Black Scholes
Value of call option [delta x share price] — [bank loan}
=_ [N()xP]— [N(đ;) x EXe""T]
where
ag, = MIP/EX] + (r+ 0? /2)T
d, = d,—oVT;
N(@) = NORMSIDST(d);
EX = Exercise price of option;
T =number of periods to expiration;
P = stock price now; and
@ = standard deviation per period of (continuously compounded)
rate of return of stock
= continuously compounded risk free rate
B-S With dividends
r= annual dividend yield
Value of call option = [N(d) x Soe~"#"] — [N(d;) x EXe“""]
Value of put = EXe™’™[1 — N(d2)] — Sue""#7[1 — N(d;)]
where
In[P/EX] + (r — rạ + ø?/2)T
=
ovT
d, = d,—ovT;
Put call parity (ex-dividend So)
Py = Cy + PV(X) — So
Factor increase Call value Put value
Stock price Increases Decreases
Volatility Increases Increases
Time to expiration Increases Increases
Interest rate Increases Decreases
Dividend rate Decreases Increases
Hedge ratio or delta
The number of stocks required to hedge against the price risk of holding one option (d; from B-S)
Call = N(d;) Put = N(d,)-1 Change in the value of the option
delta = Cannga ta the salle of tha stank
Treasury yield curve, swap curve and funding curve
AB's costo fixed rate borrowing for
Vụ three years = swap + 85 bps
NAB cost ofa cue
sw ay sos
=4.35% >
Z@ Bài Len Compenaaton IR ona to party poying feat or at commencement
absorbing IR risk 5.40-4.35 = 1.05%
‘The cost to NAB of fixing the interest rate for three years ls an extra 125 bps per year (5.40%-4.15%) NAB protection agains interest rate rises over the three years atthe time the bonds are issued The extra 125 bps buys
Collars
Natural long positions (e.g Gold miner):
© Payoff increase with increase of asset price
* Protect downside with long put
‘* Pay for put by selling call, limiting upside
naTucar ñesttzem
Natural short positions (e.g Airline in oil):
‘* Payoff decrease with increase of asset price
* Protect downside with long call
© Pay of call by selling put, limiting upside
Pay
on
ATu6AL
NRosetron
Zrfe Cost (2 A£- p-c=0
Trang 6
t-period annuity aie 4 | 4 1 1
l-period growing anny 1 1x(1+g) 1x(+g) ee 1x+ø) tất
x0
Be = > Đi For f (x,y), with covariance dy = 0x0yPxy
af\? |r| of af
—b + Vb? = 4ac
2a
Taylor
m (œx-a)"
With a sample x;, approximation o ~ s [Often] when f(x > c) = g(x >) = Oor tm,
am £2) = jim ©
PM ge) eg)
Confidence intervals
1~ CI%
Pearson product-moment correlation coefficient Gaussian PDF
Least Squares
Variables x, y have least squares slope Geometric average
% Intercept