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FV = PV (1 + i)^n (3 find 4)
--shows interest on new earned interest from the prev. period (compounding - years)
--PV is negative bc cash outflows and FV is positive bc inflow

interest on interest



PV = FV / (1 + i))^n


Choose a dollar today vs. tomorrow bc:

1. removes RISK - not receiving a return in future
2. OPPORTUNITY - use today to invest, purchase, pay bills
3. INFLATION - causes value or purchasing power of $ to decrease


cautions with TVM

1. Make sure calc. is set to correct # payments per year
2. clear memory!!!

Check annual
--remember also that present is at TIME 0 on timeline


discounting example and meaning

If don't rec. $100 until 5 years from now
I = 6 (%), N = 5, FV = $100, Solve PV = $-74.73

means: if you were do deposit $74.73 in bank today --> at 6% interest return, would have $100 in 5 years
--OR would be indifferent btw rec. $74.73 today or $100 in 5 years - they are equal bc TVM


holding all else equal, the MORE discounting periods of a lump sum rec. in the future, the _____ the present value of the lump sum


--Present value of lump sum that will be rec. in future will be SMALLER if int. rate is larger (BUT FV will be inc. with inc. interest rate)
--FV is INC. if N inc.
--FV is inc. if PV is inc.


Variables in TVM

N = number of periods in market
I = discount rate or required rate of return

more times compound = greater return
--want a FAST compound


quarterly compound ex.

invest $10,000 (PV), 10% annual return (i), 10 year - pay quarterly

I = 10/4 = 2.5 (quarterly compound)
n = 10 * 4 = number of payments during life

FV = $26,850.64



-equally spaced sequence of equal cash flows (common in lending - car loans, mortgages, bonds, insurance)
-pmts. equally spaced - pmts. of same $ amt.
-1st PMT at TIME 1 - but annuities can start/end at any time


annuity due

an annuity that pays at the beg. of each period


compounding problem

figuring out FV of money you will invest
--when periods, pmts, and interest must be adjusted for non-annual TVM problems


Deferred annuity

an annuity that starts sometime in the future


discount rate

rate at which money is discounted or compounded


effective yield

rate that includes non-annual compounding


future value

present value

how much spending power money has at a point in the future

how much spending power money has today






--Future value investment factor (the formula for finding FV)

--Future value investment factor of an annuity

--PV investment factor (finding PV formula)


ordinary annuity

annuity that pays at the end of each period



an annuity that lasts forever
--infinite stream equally spaced cash flows
--equal value pmt. at equal intervals - infinite annuity

finding PV means discounting each cash flow for infinite # cash flows = impossible

PV = PMT x ((1/(1/1 + i)^n)/ I ) but as n approaches infinity becomes 0...so cancel and

PV = PMT / I (divide annual pmt. by discount rate)



Process of valuing money at specific points in time


Uneven cash flows

a series of cash flows that are not equal
--bc not equal size...NOT an annuity


Annuity formula

FV = PMT x (1 + i)^n - 1 (find and check on this one)

PV = PMT x ((1 - (1/ 1 + i)^n) / i)

4 find 5


Annuity ex.

Invest $1000 at END of each of 3 yrs. at 8% - how much will have after 3 yrs?
--start today at TIME 0 - so if at END then 1st pmt. at TIME 1

N = 3, I = 8 want to know present value of total cash at TIME 3

on calc. PMT = -$1000, pmt = END
solve - FV = $3246


Annuity ex. and meaning

find PV at time 0 of cash flows for 3 years at 8% discount rate
n = 3, I = 8, PMT = -$1000, FV = 0, PMT = end

PV = $2557.10

meaning: value of 3 $1000 cash flows in future is $2,557.10 - means have opp. cost of 8% s infifferent btw rec. 3 annual pmts. of $1000 at end of year or receiving $2557.10 today
--also if deposit $2557.1 in bank today at 8% interest rate...can withdraw exactly $1000 each yr. for 3 yrs. and have nothing left in bank
---also if borrow $2557.10 from bank at 8% for 3 years - in order to repay loan by end of 3rd year need to pay exactly $1000 per year


ordinary annuity

have 1 period delay btw start of annuity and time of 1st pay. - didn't start pmts. at TIME 0
--MEANS when calc. PV - remember you are calc. the PV one period before 1st cash flow (DRAW PICTURE)

Ex. $10,000 car loan for 3 years at 12% - solve for annual pmts. like normal - but usually monthly
--adjust int. rate and n (12% annual rate --- 1% monthly)
--3 years = (3 * 12) = 36 months (periods)
n = 36, I = 1, PV = $10,000, FV = 0, PMT = END, solve PMT: -$332.14 (pmt. per/mo...period)

the mo. pmt. amt. $332.14 is NOT equal to annual pmt. (4163.49 / 12) (346.96) bc when pay 1st mo. some is principal and some is interest - bc loan balance dec. with each pmt., interest amt. also decreases so mo. pmt .is LESS THAN annual pmt.



Annual percentage rate


CFO borrowed $10,000,000 at 8% annual rate) - quarterly pmts. are $365,557.48 - how many yrs. oil paid off?

PV = 10,000,000, I = (8/4) = 2%, FV = 0, PMT = -$365,557.48 solve N = 40 periods (divided by 4) = 10 years!!



Annual percentage yield (APY)

Rate that includes non-annual compounding

(1 + ( I / m)^m - 1

I = ANNUALLY stated int. rate
m = # of compounds per year


If Invest for one yr. at 8% compounded quarterly, what is the effective rate?

(1 + (I / m)^m. - 1

(1 + (.08/4)^4. - 1 = .0824 = 8.24%

in Calc. n = 4, I = 2 (8/4), PV = 100, PMT = 0, PMT = END
SOLVE FV = $108.24 ---> effective yield (108.24 / 100) = 8.24%

meat that if you were offered 8%compounded quarterly or 8.24% compounded annually you would be indifferent bc same yield
--the FV $108.24 shows that one yr. from now the effective yield (annually compounded rate which causes $100 to grow to $108.24 in one period)

effective yield = (108.24 / 100) - 1 = 8.24%


Einstein - 8th wonder of the world is...

compound interest


Ex. Annuity car pmt.

$200 per month for 5 yr. loan - 10% rate paid off at end of month

PMT = 200
I = 10/12
N = 5 * 12
FV = 0
PV = $9413
Nominal rate - includes inflation

if in BEG. mode
PV = 9491



celestial annuity

PV = PMT / I

$70/yr. 6% = 70/.06 = $1166.67
used in preferred stock


Effective yield formula and names

EY - effective yield
APY = annual percentage yield
EAR = Effective annual rate

(1 + (I / m))^m. - 1
m = # compounds in a year
APR = 10% (annual percentage. rate - so I = 10%)
compound monthly
(in formula above, I = APR!!! Solving for APY, EAR, or EY)

Annual percentage rate of 10% compounded monthly
EY = (1 + (.1 / 12))^12. - 1
= 10.47% = EAR, APY, EY

APR = does NOT include compounding




---If invest will show APY bc looks like bigger return
--if loaning (Car, house) will show APR bc smaller


Delayed Annuity problem
--Tuition paid at beg. of 19 years - tuition costs $5000 at beg. of yr. and cont. for 4 years
--opp. cost is 2% comp. annually
--how much is PV of total tuition?

FV = 0
I = 2%
n = 4
PMT = -5000
PV = 19,419.42
--BUT THAT IS VALUE IN 19 YEARS - what is PV today if tuition costs if discount rate is 4% comp. monthly?
--if beg. mode - don't subtract 1, if end mode do subtract 1

PV = 9093.29, n = (19*12) = 228, END


Ex. perpetuity
what should you be willing to pay in order to rec. 10,000 annually forever, if require 8% per yr. on investment

PV = 10,000 / .08 = 125,000
infinite cash stream worth $125,000 today

common in charities - perpetual gifts for universities or hospitals


Annuity due vs. ordinary annuity

value annuity due > value ordinary annuity - bc w/ annuity due, we rec. pmts. sooner than would with ordinary - bc inflation...$ today is worth more than tomorrow


uneven cash flows

when all cash flows are diff. we have to discount/compound each ind. cash flow separately and add together

at time zero, pmt = PV
then each time going fwd. the N changes, so period 1, n = 1 and period 2, n = 2 (ALL IN END MODE)

-Then add together


deferred annuity

start later, start at time 4 to 8 for ex.

to solve, solve as an ordinary annuity (end mode) at time 3 (pretending it is time 0)

pmt = -40 n = 5 (45678), I = 20, FV= 0, PV3 = 119.62

THEN discount back to PV0
N = 3, I = 20, FV = 119.62




Loan - main conduit by which funds get from household savers into hands of entrepreneurs and business people - also known as FIXED INCOME
--A fixed income financing instrument

1. bond market is one of most imp. sources of financing in nation
--bond market - firms raise debt fin. directly from investors - huge role in corporate finance (one of primary sources of external financing)

2. plays role in majority of savers personal inv. portfolios and retirement plans - steady stream of fixed income from inv.
--bonds are "BACKBONE" of worlds pension funds

fixed income securities - vehicles by which corp. raise debt capital - IOU from corp. to holder of bond


"In bond market firms raise debt financing directly from INVESTORS"
bond variable

bond promises to pay an annuity every x # mo.
PV = price of bond - when invest in is a NEG. outflow (down in pic.)

Pmt. called coupons and are upwards until FV (which last coupon also paid on FV)

last payment is the MATURITY DATE

--big lump sup at end of pmt.

n = # pmts. until maturity


I/Y bonds

--OR called market rate of interest (MRR - Market rate of return)
--OR called promised yield

size of fixed interest pmts. depends on interest rate and amt. of principal

yield to maturity - req. rate of return or a bond - give ANNUAL rate of return that inv. expect to rec. on a bond if HOLD to maturity
--also called PROMISED MATURITY - return we are promised if buy bond today and hold to maturity


Corp. bond pmts.

not like car loans where each mo. pay part principal and part interest
--corp. bonds are ALL interest during life of bond - at end, firm pays final interest and returns principal amt. originally borrowed (FACE/PAR/MATURITY VALUE)



= Par value - sum of $ firm promises to pay at bond's expiration
--most cases it is the same amt. borrowed at PV - if not stated assume it is $1000

the lump sum amt. paid on a bond's maturity date


coupon rate

interest rate of bond - also called COUPON YIELD
--rate bond issuer promises to pay its investors
--set when bond is issued and CANNOT be changed at ANY TIME
--Multiply coupon rate by Par value and gives amt. of bonds yearly coupon or interest pmt. (for whole year)

$1000 par value at 9% coupon rate = $1000 * .09 = $90 interest ANNUALLY ($45 set-annually)


make sure remember it is ANNUAL then solve for the number it is looking for



vibes are finate-term securities - ending/maturity date
-# yrs. from when bond is issued to when it expires is maturity

maturity date - date bond issuer pays FV and bond expires


bond indenture

--the legal document detailing a bond
--contract btw bond issuer (corp.) and bond holder (investor) - describes features: coupon rate, par value, maturity


affirmative and negative

covenants - rules set forth in bond indenture to protect bond inv.

affirmative - bond covenant that req. firm to do something (pay taxes on time, maintain certain level of working capital, maintain min. debt value)

negative - bond covenant that prohibits the firm from doing something (do not sell certain assets, not pay large dividends, not issue new debt with superior claim)

if don't comply - comp. stands in default - S & D force it to work


Bond ratings
--investment grade
--junk bonds

rating assigned to a firm to measure the probability of default by a company like S&P or Moody's

investment grade - bonds rated BBB or above

junk bonds - speculative bonds rated BB or below

AAA - least risky, D = most


Convertible bonds

bonds that can be converted into equity at owner's request (some PS)
-inv. has right to trade bond for set # of shares of common stock when he chooses
--benefits either way bc if firm does well can convert, if does poorly rec. fixed income from bond
--but downside is they have lower yields



the curvature of the price-riel relationship not captured by duration



a bond that is not secured by collateral or guarantees
- RISKY...but bc risky have higher potential return
subordinated debenture - lower-ranked bond that is not secured by collateral or guarantees
--lower claim than normal debentures to assets of firm in liquidation

ex. subordinated - IBM issues bonds in 2012 and 2013 - older bonds (2012) have 1st claim to assets over newer (2013) - if IBM liquidates, 2012 holders paid on full before 2013


debt capital

firm financings that appears in the debt section of the balance sheet

why not borrow from banks? bc they are intermediaries - pay interest owed to banks and pay of int. owed to public = a lot higher


which are secured loans?

car, mortgage - not credit card
--bonds ARE unsecured loans bc bondholders cannot "repossess" a comp. like a bank can repossess a car of house

fixed income securities - securities that pay an equal pmt. on fixed periods like a bond

modified duration - an adjusted version of Macaulay duration which makes it a little more accurate



measure of interest rate sensitivity of a bond
--if duration = 2.2%, then means that for every 1% change in int. rate...price of bond moves opp. direction by 2.2%

--Macaulay duration - a specific type of duration, a measure of interest rate sensitivity

Price inc. as yields decrease, and price dec. as yields inc.
--ex. discount rate 10% worth $1000, but rate 12% same bond = $951.96 - less than par value
--bc no one will buy same price bond at lower yield - need to lower price
--dec. I means PV inc.



bonds issued in a country not in that country's currency
--Ex. Am. bond issued in Europe - called EUROdollar (bc payable in dollar)


foreign bonds

bonds issued in a domestic market by a foreign firm, but in domestic currency
--kangaroo bond, sushi bond
--China floors debt in U.S. payable in US $


inverse-price yield relationship

When market yields of up, bond prices go down and vice versa


mortgage bonds

bonds that are secured by real property - real estate
--similar with loan - if defaults - holders get set
--less risk bc guaranteed pmt.
--but greater risk = greater reward....debentures usually have a higher yield return...but more risky


munical bond

bond issued by local municipality such as a city or county (short is muni)
--used for raids, buildings


security valuation

process of valuing assets


trade at discount/ premium

trade at a discount - when bond's market price is less than its face value

Trade at premium - when bond's market price is greater than face value


Treasury bill/bonds

bond issued by US federal gov. to support deficit spending
--risk free bc backed by faith/allegiance and taxing power of U.S. fed gov.


zero coupon bond

another name for zeros

zeros - bonds that do not pay interest payments - NO coupons (CPN rate = 0%), but high discounts
--ex. - FV = $1000 but will sell today for $500, no int. pmt.s during life


value of any asset

the PV of the stream of expected flows discounted at an appropriate required rate of return
1. determine future cash flows from asset
2. discount all future cash flows at appropriate discount rate

discouting each cash flow and adding it up - to know its value and spending power today

FOR BONDS - sum over ALL periods in bonds life
--means sum present value of int. pmts. and add discounted value of the face value


two types of cash flows

1. coupon (annuity)
--pmt. = coupon rate * par value
2. Par (fave) value - lump sum

remember!!! PV must be OPPOSITE sign of PMT and FV


2 types of bond questions

1. given discount rate, find market price

2. given market price, find discount rate
--find bond yield
PV = -951.96, FV = 1000, pmt = 100, n=3
I/yr. = 12


YTM vs. current yield

current yield - an approx. of yield that does not incorporate TVM

ANNUAL coupon / current market value (price of bond)

ESTIMATE of YTM but ignores TVM so diff.
--YTM is return we will actually rec. if. hold to maturity and no default


inverse price/yield relationship

when bond sells at discount - its YTM > Coupon yield bc. inv. not only rec. cpn. pmts. but also inc. in price over time

YTM > CPN. yield bc you buy bond @ 952 but get 1000 in future in addition to 10% pmts.

means as YTM inc. the Value of EXISTING bonds decreases (if it was new would not adjust) - coupon DOED NOT CHANGE

"bond issued at 7% at par - means YTM = 7%)
--so MRR (YTM) is 7% in 1st. year but 10 years later, the YTM has dropped to 4% (bc maturity date is closer) and the CPN = 7% still...everyone will pay more for a higher return - so require a premium for higher return and as YTM dec. the price inc.


Bond has yield to maturity of 13.9%, 9.5% annual coupon, FV. = 1000 what is current yield?

1st PMT. = 1000 * .095 = $95
--now have to find the PV
n = 5, I = 13.9, FV = 1000, PMT = 95, PV = -848.58

current yield. = 95 / 848.58 = 11.2%


CPN rate and Discount rate

CPN rate = CPN. yield
discount rate - rate$ can be discounted or compounded at

CR = DR --- sell par value (CR = YTM)
CR > DR = sell premium
CR < DR = Sell for discount


3 types of yields (bonds)

1. actual yield (Return) if held to maturity - what you will earn - ANNUAL

2. Coupon rate - Fixed pmt. by company (ANNUAL) - multiply by par value to find the ANNUAL CPN pmt. - then must divide by number CPNs paid per yr. to find (ex. quarterly = divide by 4)

3. Current yield - an estimate of the YTM (CPN / PV)


2 ways to invest

1. Buy and hold - YTM
2. Time the market
--buy (long) - low
--sell (short) - high


Ex. FV = 1000, CPM = 7% PV = 1010, what is current yield?

ALWAYS ANNUAL (in this case doesn't remember but remember on test)
--PV is greater than FV so must be sold at a premium...means YTM is less than CPN rate
--1000 * .07 = $70 coupon

70/1010 = 6.93%



another name for stock
--ownership of an asset

privileges - RESIDUAL CLAIM on earnings and assets of a company
--each yr. after comp. pays for operations and pays creditors, any RESIDUAL (remaining earnings) belong to shareholders
--claim earnings in proportionate % of shares they own
--ex. if had claim to 100 shares of stock outstanding = claim 1/5 (1/100) of residual earnings

residual given after ALL assets have been sold and debts paid
--stockholders have residual claim on firm earnings and assets


Why is stock important?

1. equity - likely large portion of ind. inv. portfolio and retirement fund
2. help friend/relative
3. compensation packages in corp. Am. include stock options in that company - calc. real value
4. stock market is one of vehicles through which capitalism functions

if future of comp. seems bright = stock value inc.,

if seems bad = stock value drops

stock price shows current value of that company


Common stock

Variable return security (securities that are not fixed income - CS is 90%)
--variable income - dividends inc. or dec. by mgmt. decision and market conditions
--grants equity/ownership
-VOTING rights - main vote on election of board of directors (who hire comp. mgmt. team)
--common stock has lowest position and last claim to earnings and assets
--most costs are fixed...so residual earnings can inc. if comp. does well or dec. to zero
--NO maturity - NO expiration date (sell on market or back to comp._
-unlimited earnings potential


corporate governance

the structure, rules, and regulations for owners and managers of a firm


preferred stock

HYBRID SECURITY (characteristics livestocks and bonds)

DEBT char.
--almost FIXED pmts. (called dividends) - pays same dividend each year regardless of comp. progress
STOCK char.
--Not fixed (dividends, not coupons)
--ownership and claim to comp. assets
--on equity side in accounting
--no fixed maturity or expiration date


Reasons for Debt or Equity financing

--1. FIXED INCOME - have to make periodic pmts. called annuity
--3. CREDITORS - if loan money to a comp. called creditor
--4. only creditors can drive a comp. into BANKRUPTCY - default

--1. % of firm OWNERSHIP
--2. pay DIVIDENDS
--3. RESIDUAL INCOME (could get more than loan if firm does well, 600% return)
--4. "UPSIDE" PAYOFF - all $ leftover after pay creditors given to shareholders - could be high or low

goal of firm: to maximize owner wealth - optimal capital structure is a mix of debt and equity (to max. owner wealth - makes it optimal)


valuing asset - 2 ways to make ratio big

1. increase CF - inc. your number of cash flows
2. decrease r = dec. discount rate to max. wealth minimize discount rate


dividends in arrears

PS characteristic where common stock dividends cannot be paid until the preferred dividends are paid

Cumulative dividends
--PS char. where if a firm fails to pay preferred dividends one year, it must catch up and Pau all preferred dividends before any common stock dividend can be paid


2 types of firms that use PS

1. Utilities - steady and predictable cash flow
2. start-up ventures who seek equity capital from venture capital to CS if startup is big


finding fixed annual dividend

Preferred stock usually sold per share par value
--amt. of fixed dividend usually expressed as % of stocks par value

Fixed ANNUAL dividend = PV (par value) * % of stocks PV

ex. Xerox issued $75 mil of 8.25% PS at $50 per share par value
--calc. fixed annual dividend = (.0825 * 50) = $4.125 per share
---will pay over of share of stock $4.125 per share annually


Residual claim

right of shareholder to profit after all comp. obligations have been paid
--PS - non voting, non participating (Except VC)


Value of a firm is....

the sum of the discounted value of its future cash flows

2 ways to inc. value
1. inc. # cash flows
2. dec. denominator of DCF model, or cost of capital firm uses to discount future cash flows


intrinsic value

estimate - of any asset is PV of a stream of expected future cash flows discounted at an appropriate rate of return

stocks = value of any stock is equal to present value of its future expected cash flows - intrinsic value

look at intrinsic value to know where to invest - good comp. but many people invest and drive up the stock price until it meets or exceeds the company's intrinsic value of future cash flows
---BUT if secret comp. sells stocks for $10 per share but has intrinsic value of $25 per share...should invest 1st and wait for rest of world to see
--not about finding good comp. but looking for stocks hat are undervalued relative to their intrinsic value


3 ways to estimate value of future cash flows of share of stock

1. single period model for both PS and CS
2. constant growth (Gordon Growth) for mature firms
3. 2 stage model representing large class of multi-stage models


Valuing pref. stock

constant annuity is perpetuity

Vps = D / Kps --->
--D = ANNUAL fixed dividend
--Kps = Discount rate or req. rate of return

to find price of stock today that would yield at LEAST 9.5% return = divide ANNUAL dividend by our req. rate of return

Vps = $4.125 / .095 = $43.42 (remember 9.5% is NOT dividend percent...it is the I - discount rate!!)
--willing to pay MAX of $43.24 for one share - any price higher would yield rate of return lower than 9.5%


1. single holding period model (common stock)

Assumption: Inv. buys today (Stock) and holds on year and sells
--find value of stock by estimating FCF (future CF) from 2 sources and discounting back to PV

ex. stock pay dividends at $5.50 - end of yr. pmt. - that time we expect stock price to be $120
--req. 15% rate of return
-how much willing to pay for stock?

so FV is adding together the dividend and stock price
--$5.50 + $120 = $125.50, n = 1(start at time 0 and end at time 1), I = 15%, PV at Vo = $109.13
--109.13 is the price of stock that yields req. rate of return

--BUT bc CS has infinite life, FV not easy to estimate to infinity (bc formula on note card req. value at future time...which is an estimate and never certain) - need to estimate future dividends forever then discount back to present


Gordon Growth Model

expressing infinite series in non-infinite equation
--closed form solution called GORDON GROWTH or CONSTANT DIVIDEND GROWTH model

V(0) = (D(0) * (1 + g) / (k - g). numerator is finding D1 (D0 times the growth to find next period's dividend amt)

simplify to... V(0) = D1 / (k - g)
--this assumes dividends grow at a constant rate each year


multiple holding period models - constant dividend growth model

assumes inv. will hold stock for more than one period
--assumes firms and their div. grow at constant rate forever (utility or high growth firms)
--basically a GROWING PERPETUITY - stream of cash flows that grows at a CONSTANT rate forever
--g will be close to growth of economy - about 3-4%, never greater than 6%
--good in stable, mature firms

constant growth model common in real estate - inflation % = g, Net operation income = div.


Gordon Growth ex.
-$5 div. paid today expected to grow 10% per year to infinity. What would we be willing to pay for one share of stock if req. rate of return is 15%?

find value of stock today at V0
--If div. are "Recently paid," "currently being paid" or "paid today" - they are time 0 div. and need to solve for D1 by multiplying by (1 + g)

D1 = (5) * *1.10) = 5.50
5.50 / (.15 - .10) = $110.00

One share of stock is worth $110.00 today


2 stage growth model

2 stage growth - so value a firm growing at diff. rates at diff. times - allows firms to grow at an above ave. rate for initial period of time w/o assuming it will grow at that rate forever

Final stock/firm value is PV of stage 1 + PV of stage 2

--super normal period - high variable growth and extreme cash flows - find CF and discount back to present

--begins after super-normal stage 1 - assumes comp. has now matured and stabilized and will cont. to grow at industry ave. rate from BEG of stage 2 on - growing perpetuity


Multi-stage growth modes

chart future CF and discount to present
terminal value - the final $ amt. from a project - use Gordon model to find value of all future cash flows from today to infinity


Example!! Multi stage
--Alpha inc. has earnings and div. expected to grow at 15% next 3 years - after firm expected to grow at industry ave. of 5% infinitely - recently paid $1 in div. req. rate - 10%

what is the most you should pay for alpha??

STAGE 1 ( = 3.28)
g = 15% D0 = $1, D1 = $1.15 K = .10

yr. 1 = FV (1.15), PV (1.05)
yr. 2 = FV (1.32), PV (1.09)
yr. 3 = FV (1.52), PV (1.14)
total PV = 3.28 (PV of super-normal stage

--just solved and find D3 = 1.52
--Stage 2 starts in D4 which would be (1.52 * 1.15) = 1.60
V0 = D1 / (k-g) ---> 1.60 / (.1 - .05) = $32
---BUT $32 is the FV in D4 - Gordon growth doesn't give us value at time 1 when plug in D1, but time 0 - always gives us one previous
--so discount 3 years back not 4!!!
FV = 32, n = 3, I = 10, PV = SOLVE = $24.04

now add stage 1 and 2 together
3.28 + 24.04 = 27.32 per share!! highest price


return on equity and net income problem in quiz ex.

Just paid div. of $1.75. comp. is expected to grow at rate equal to sustainable growth rate
--comp. recently reported a return on equity of 15% and paid out 25% of net income in div.
--if req. rate of return by shareholders is 20%, what is price of stock today??

1st need to find the growth rate
g = sustainable growth rate
g = ROE * b (b = portion of net income retained by the company - so if paid out 25% of NI, then b = 75%)
g = .15 * (1 - .25)
g - 11.25%

now plug into Gordon model
D1 = 1.75 * (1 + .1125) = 1.95

1.95 / (.20 - .1125) = $22.25


stock XYZ will pay div. of $2 within next year. price of stock will be $100 one year from today - req. rate of return is 12%
--what is the value of stock today?

r = (P1 - P0 + D) / P0

plug in
.12 = (100 - X + 2) / X
.12x = 100 - x + 2
1.12x = 102
x = $91.07


1. 52 week high

2. 52 week Low

-highest value that stock attained in past 52 weeks (updated daily and 52 weeks counted back from current day)

--lowest value stock had over 52 weeks


3. ticker symbol

each stock listed and identified by its own unique symbol - called ticker symbol-- usually 3-4 char. long with either numbers and or letters

3 char. tickers usually associated w/ New York stock exchange and am. stock exchange

4 char. usually associated with Nasdaq


4. dividend (DIV) on stock page

4th column
--lists most recent quarterly div. annualized by x by 4 - listed on ANNUAL basis - but not necessarily actual pmt. made - bc many firms do not pay equally sized div. per quarter
--div. listed may be last div. paid x $4 = $80 per share - but really comp. paid less for 1st 3 Q - listed estimate of div. coming in year


5. DIV yield percentage = (YLD %)

divide div. in 4th column by closing price of stock (10th) - gives yld. %
--it is the return on inv. that investor will rec. this year through div. paid
--in IBM = .80 / 105 --- .8% = REALLY LOW return - remaining returns comes from capital gains - when inv. originally purchased stock, sold for $90/95 - today closing price is 105 - means 10/15 inc. in P gives inc. in return that they miss in div. return
--shows div. are really small and insignificant compared to capital gains

returns in 2 ways: div. and capital gains


6. PE ratio

price to earnings
--when set price or value of stock, are really showing earning power of stock/company. PE ratio compares stock price to comp. earnings

PE = price per share / earnings per share

shows much value the market places on every dollar of comp. earnings - r amt. market will pay for each dollar of earnings
--ex. IBM has PE ratio of 18 and Microsoft as PE ratio of 54 - means that investor in skeet will pay $54 for every one dollar earned by Microsoft if chooses to invest

ex. inv. in shop that made $10,000 earnings last year - if PE ratio is 18, then willing to pay (18 * $10,000) = $180,000 or in Microsoft $540,000

small businesses usually have btw 2-5 ratios - large btw 50 and 200...
---pay more b expect comp. to grow and become more valuable in future


7. Round lots traded (VOL 100s)

show amt. of trading activity in stock of that company
--expressed in round lots (Each lot = 100 shares)

for IBM - 54,970 round lots = 5,497,000 shares traded on the exchange that day


8. Days HI

9. Days LO

10. Days closing price (close)

--highest value at which stock was sold or purchased during current day's trading

--lowest price stock was traded that day

--Price at which stock closed TODAY at end of trading - last price stock was bought and sold before market closed that day


11. Net day's change (net chg.)

calc. as diff. btw today's closing Price and yesterday's closing price
--shows cumulative effects of today's price of stock
--for IBM, day was positive: stock was up $2 from prev. day's closing price


order in case of bankruptcy

1. lawyers, employees and taxes
2. senior bond holders (collateralized - tied to an asset so ensure pmt.)
3. Jr. bond holders - debentures (no collateral)
4. preferred stock
5. common stock


Am. stock exchange
New York stock exchange

--a physical trading floor and computer network where stocks are bought and sold

--" " - largest stock exchange in the world

--" " - 2nd largest stock exchange in the world -typically technology-related companies will go public through this exchange (Microsoft)


cap rate

in real estate, the denominator of the Gordon Model (r - g) - the discount rate of a perpetuity

capitalization rate = yearly income / total value


variable return securities

securities that are not fixed income returns


how to find market share

# shares * PPS = market cap value
--measures how big the company is
--PPS = price per share

bull market
--optimistic - think market will go straight up

bear market
--think market is going to do poorly, think market will go straight down


DOW Jones industrial average

S&P 500

dow jones industrial average
--30 stocks from diff. industries

S&P 500 - 500 stocks - broader index


2 ways to make $ with stocks

1. dividends (Cash pmts.)
2. capital gains - from stock price changes
---buy stock at low price, price inc. and sell high (called LONGING - mutual funds)
---sell HIGH (hoping stock will go down) and then buy low (SHORTING - hedge funds)



HPR - holding period return

% return = cash dividends + capital gains yield (buy low sell high)

single HPR = money get out / money put in -->

div. pmt. + (P1 - P0) /P0
--P0 = money put in - amt.

P0 = $10,
P1 = $11
D = $1

HPR = 1 + (11-10) / 10
20% return
--money you got out divided by money you put in (rec. by div. and capital gains)

Is an ESTIMATE bc does not include TVM
---in these problems it will ask what value is today
--variables you need are div. pmt. for this year (or maybe will need to find it but it is D1), the price in future at time 1, and the req. rate of return


constant DIVIDEND model vs. constant GROWTH

G = Postitive means constant growth - div. pmts. will inc. at a constant rate each year

g = 0 means same value div. paid each year infinitely

so equation simplifies from

D0 ( 1 + g)/ (k - g) to

D0 / k ---(bc no G)
remember it is D0 not 1!!!!!


also TRICK question with stocks

gave inf. about past dividends
--we only care about future!!!
HAS paid constant div. of .....
--predict constant div. (means lack of growth)
--only buy stock if earn 15% rate of return)
--what is MAX willing to pay?

D0 / r (bc no growth)
= $2 / .15 = $13.33