PRINCIPLES OF FINANCE NOTES

Transcription

1 FNCE PRINCIPLES OF FINANCE NOTES Week

2 Week 3 Applicatins in Financial Mathematics I Annual lan payment is an rdinary annuity. In a lan amrtizatin schedule: Ø Interest paid = (previus perid s principal) x interest rate Ø Principal repaid = Lan payment Interest paid Ø Principal balance remaining = Previus perid s principal Principal repaid Ø Principal balance = PV f remaining lan payments Effective annual interest rate is the annualized rate that takes accunt f cmpunding within the year. r " = 1 + & ' ' 1 where r = annual percentage rate k = cmpunding frequency (1 fr annual, 2 fr semi-annual, 12 fr mnthly, etc) r/k is the per perid interest rate Prperties f the Effective Annual Interest Rate: Ø r = r " nly when cmpund frequency is ne year (k=1);; therwise r " will always exceed r. Ø The effective annual rate with cntinuus cmpunding (cmpunding frequency with k appraching infinity) is r " = e & 1. The mre frequent yu cmpund (the bigger the k), the higher the effective annual interest rate yu get. Shrt term debt instruments: Ø Matures within the year typically in 90 and 180 days Ø Issuer has the cntractual bligatin t make prmised payment at maturity 2

3 Lng term debt instruments: Ø Matures after several years Ø May r may nt prmise a regular interest (r cupn) payment Ø Issuer has the cntractual bligatin t make all prmised payments. Face (r par) value is the dllar amunt paid at maturity. Ø Usually $1000 r its multiples unless therwise mentined Cupn (r interest) rate is the interest rate prmised by the issuer. Ø It is expressed as a percentage f the face value. Cupn (r interest) payment is the peridic payment made t bndhlders. Ø Cupn payment (C) = Cupn rate x Face Value The price f discunted securities is cmputed as the present value f the face value at a market determined yield t maturity (YTM). Yield t maturity is the rate f return earned by an investr wh hlds the security until it matures assuming n default ccurs n the security. Price f a discunted security = Ø If YTM falls, price will increase Ø If YTM rises, price will fall Face value [1 + n 365 x r?] Price f a cupn paying bnd = C E r? r? F + Face Value 1 + r? F 3

4 The yield t maturity is als the interest rate that discunts the bnd s future cash flws t equal the market price tday. Ø It is als called the internal rate f return. Relatinship between Cupn Rates and Yields t Maturity: Ø When Price = Face Value The bnd is selling at par YTM = Cupn Rate Ø When Price > Face Value The bnd is selling at a premium YTM < Cupn rate Ø When Price < Face Value The bnd is selling at a discunt YTM > Cupn rate 4

5 The prices f lnger maturity bnds are mre sensitive t changes in interest rates than the prices f shrter maturity bnds. Ø This is because investrs have t bear a higher interest rate risk with lnger maturity bnds. Bnds with higher cupn rates are less sensitive t changes in the yield t maturity changes than ncupn paying bnds. 5

6 Week 4 Applicatins in Financial Mathematics II Ordinary shares represent wnership in the issuing cmpany s cash flws (earnings and dividends). Ø It is first issued and sld in the primary market typically via an initial public ffering Ø Once listed, it is traded n secndary markets such as the ASX, NYSE Ø It prvides an infinite stream f earnings and dividends Ø It has n maturity date Why dividends and nt earnings? Ø Sharehlders care abut the cash flw they receive, nt what a firm earns. Ø A firm s earnings are typically nt all paid ut as dividends, s we shuld cnsider dividends, and nt earnings, when valuing shares. The General Dividend Discunt Mdel The Cnstant Dividend Grwth Mdel Ø Dividends grw at a cnstant grwth rate (g) frever The link between earnings and dividends: D F = αe F where α is the dividend payut rati prprtin f earnings paid ut as dividends. Ø If payut rati desn t change ver time, the grwth rate in dividends will be equal t the grwth rate in earnings. 6

7 Assuming the number f shares utstanding is cnstant, a firm can increase its dividends by: Ø Increasing its earnings Ø Increasing its dividends payut rati What can a firm d with their earnings: Ø Pay them ut t sharehlders as dividends Ø Retain the earnings and reinvest them Any increase in future earnings will result frm new investments made frm earnings retained by the firm. Ø Change in earnings = New investment x Return n new investment Ø Earnings: New Investment / Retentin rate Ø Grwth = Retentin rate x Return n new investment There is an implicatin that lwering dividend t increase investment will raise the stck price, if, and nly if, the new investments have a psitive NPV. Calculating Share Price at t=0: 1. Obtain dividends up t where g becmes cnstant 2. Obtain P F (after which dividends grwth is cnstant) 3. Add the present values f dividends and the present value f the price at time t get P K Preference shares are shares which give their hlders preference ver rdinary sharehlders regarding the payment f dividends and repayment f capital in case f liquidatin. Ø Price f plain vanilla preference share P K = M N & N 7

8 The P/E rati is the rati f the current market price t expected (r current earnings) per share. Ø Expected (frward) P/E rati = O P Q R Ø Current (trailing) P/E rati = O P Q P The expected P/E rati is the amunt investrs are willing t pay nw fr $1 f future expected earnings. This implies as expected P/E rati rises: Ø The payut rati (α) rises Ø The grwth rate f dividends (g) rises Ø The required return n equity (r " ) falls If a firm s earnings and dividends d nt grw (g=0) and it pays ut all earnings (α = 1) as dividends, the price is P K = Q R & T. 8

9 WEEK 5 Mdern Prtfli Thery and Asset Pricing I Ordinary Shares Returns Bnd Returns Arithmetic average return measures the return earned frm a single, ne perid investment ver a specific time hrizn. Gemetric average return measure the return earned per perid frm an investment ver an investr s entire time hrizn. If returns are cnstant (example: returns are always 10%), gemetric return = average return. The bserved (realized) risk f a security is measured by the variability in its realized returns arund the arithmetic average return. 9

10 Least t Mst Risky Investments: Ø Treasury bills Ø Crprate bnds Ø Wrld Prtfli Ø Standard & Pr s 500 Ø Small stcks The expected return is the expected utcme measured as the weighted average f the individual utcmes. The variance r standard deviatin f returns is the measure f dispersin arund the expected return. Ø The greater the dispersin, the higher the uncertainty and risk. Ø Variance and standard deviatin take int accunt returns abve and belw the expected return. Investrs are typically cncerned with returns belw the expected return (dwnside risk). 10

11 Investrs are assumed t be risk averse. Ø The higher the variance r standard deviatin f returns the wrse ff the investr Ø Their bjective is t minimize the risk f prtfli f investments given a desired level f expected return;; r t maximize the expected return f prtfli f investments given a desired level f risk. Prtfli risk falls as the number f securities in the prtfli increases. Ø Hwever, prtfli risk can t be entirely eliminated Ø The risk that can t be eliminated is called systematic risk A prtfli s expected return is the weighted average f the cmpnent securities expected returns. Ø Weights are percentages f the investr s riginal wealth invested in each security. Ø It is assumed all funds are invested in tw securities. A prtfli s variance is the weighted average f the variance f its cmpnent securities and the cvariance between the securities returns. Ø The first and secnd term can nly be psitive Ø The last term can be negative, psitive, r zer 11

12 Psitive cvariance (σ EW > 0) Ø Abve (belw) average returns n security 1 tend t cincide with abve (belw) average returns n security 2. Negative cvariance (σ EW < 0) Ø Abve (belw) average returns n security 1 tend t cincide with belw (abve) average returns n security 2. Cvariance = 0 (σ EW = 0) Ø Security 1 s return tends t mve independently f security 2 s return. Ø i.e. security 2 is a risk-free security. 12

13 The risk-return trade-ff in a tw-security prtfli depends n the level f c-mvement (r crrelatin) between their returns. Ø Return crrelatin f +1 (perfect psitive crrelatin) ρ EW = +1 There are n gains frm diversificatin in this case as the prtfli s risk (standard deviatin) is a weighted-average f the risks (standard deviatins) f the tw securities Ø Return crrelatin f 1 (perfect negative crrelatin) ρ EW = 1 There will be maximum gains frm diversificatin It is pssible t cnstruct a zer-risk prtfli Ø Return crrelatin between +1 and 1 (general case) 1 < ρ EW < +1 Sme diversificatin benefits always exist The diversificatin benefit depends n the crrelatin f returns;; The lwer the crrelatin, the higher the diversificatin benefits 13

14 In the case where there is a perfect negative crrelatin and the minimum variance (r standard deviatin) f the prtfli is 0 (σ \ = 0), the standard deviatin expressin is simplified int: Ø 0 = σ p = x 1 σ 1 (1 x 1 )(σ 2 ) Prtfli leveraging is the strategy where an investr brrws funds at the risk-free rate f return and invests all the available funds in a risky security (r prtfli). Ø The mre we brrw, the riskier the investment gets. Shrt selling refers t brrwing shares, and selling them nw with a cntractual prmise t buy them back later at (an expected) lwer price. Ø This methd f leveraging increases prtfli risk! Ø Brrwing and shrt selling security A and investing prceeds in security B implies x c < 0% and x e > 100% such that x c + x e = 1. Prtfli leveraging and shrt selling increases the expected return f a prtfli and the risk assciated with that prtfli! 14