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Ib132- Foundation ofFinance1. introWe will we taking the perspective of public firms:  Market value = share price × shares outstanding Shares traded at a stock exchange  Limited liability Separation between ownership and management oo Shareholders elect board of directors, 1 share=1 voBoD elects...

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Ib132- Foundation of Finance 1.

intro

We will we taking the perspective of public firms:    

Shares traded at a stock exchange Market value = share price × shares outstanding Limited liability Separation between ownership and management o o o

Shareholders elect board of directors, 1 share=1 vote BoD elects and monitors CEO, sets executive remuneration Managers are legally distinct from owners, pay their own taxes

The objective of the management team is to make decisions that maximize the market value of the firm Capital Budgeting: How can firms choose the best investments when they have multiple investment opportunities? Capital Structure: After deciding which investments to undertake, how should the firm pay for the investments? Use a bank loan, or sell a share to partners? Pay-out decision: After the investments have generated cash, how should we return the cash to the shareholders? Project: A set of cash flows. Usually projects entail an immediate outflow (investment/expense/cost) and are followed by a series of cash inflows in the future (payoffs/revenues/returns).

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Valuation:

1. Valuation means to assign a unique number to a project that reflects its value today.

2. Valuation is forward looking. 3. All financial valuation is relative to alternatives. This is The Law of One Price (LOP), which states that the same items should cost the same (or similar items should cost similarly).

4. A central notion in LOP-pricing is the opportunity cost of capital, i.e., foregone possibilities.

Finance uses tools from other disciplines 1. Economics: The science of tradeoffs given limited resources  

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we have a limited budget and have different options so which options do we choose?

2. Statistics: The science of dealing with uncertainty.  

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cash flows not known with certainty so stats help us make predictions.

3. Accounting: The science of keeping financial accounts

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from total revenues the firm generate what cash flows are returned to investors?

2: 

Present value

Interest rate denoted as r

time value of money

The notion that £10 today is worth more than £10 in the future because it can be invested and earn interest r. Notation 􏰀

How we represent different time periods:    

􏰀

􏰀 􏰀 􏰀

0=today 1=next period (day, year, etc) t=some future time period T=denotes the final time period (if finite duration)

How we represent cash flows (-ve or +ve):   

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􏰀

􏰀 􏰀 􏰀

C0 cash amount today C1 cash amount in the next period So the subscript denotes the period that the cash flow applies

Simplifying Assumptions: 1. Perfect capital markets, which means: we pay no taxes, no transaction costs, etc. we have no information asymmetries

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No arbitrage

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2. No risk or uncertainty, which means:  

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we have perfect foresight about the future we know for certain the cash flows of each investment

Future Value of Money Since money can be invested and appreciate in value in the future according to some rate of return, we would like to know how much a certain amount will be worth in the future. Assume, for example, that the rate of return is 20% for one year and we invest £100 today. To find the future value of this amount we can work with the return’s formula:

So, substitute here what we know:

Our £100 will thus be worth £120 in one year’s time.

Compounding and Future Values of Money

The general formula for future 4

values of money for T periods is,

Example

If we are given the interest at the end of t periods, and want to find the constant rate over 1 period, the formula is :

Manipulations of the formula example:

Discounting and Present Value of Money Due to time value of money the same amount tomorrow is worth less today, because our money today can earn interest. Using discounting we can get present values from future values.

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Here, r reflects the time value of money and is commonly called the opportunity cost of capital (OCC) — that is, the rate at which our money can grow if we invest in other similar projects.

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With no uncertainty the OCC r is the risk-free rate

Discount Factor

Translates future cash flows into money today.

Example: for r1 = 10%

Opportunity cost of capital: Bonds

CT = (1 + r)C0

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Zero-coupon government bonds cost C0 today, pays the future.

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The economy wide interest rate is the cost of capital for bonds. So the prices of bonds vary inversely with the interest rate.

Net Present Value The present value of cash inflows minus the present value of cash outflows (/discounted by the opportunity cost of capital) To apply the NPV rule follow these steps: 1. translate cash flows into today’s £’s and sum to PV(inflows) 2. translate costs into today’s £’s and sum to PV(outflows) 3. subtract PV(outflows) from PV(inflows)

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In

Note: Most commonly projects have one initial outflow at time 0, i.e., the cost of the investment

Net present Value Example 1

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Net Present Value example 2

Discount Factor

= Ct * Discount Factor

NPV tells us how much a specific project is worth today.

Capital Budgeting Rule In a perfect market we should take all positive NPV projects. A positive NPV project increases your wealth by more than what other similar projects in the economy can.

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3.

Perpetuities and Annuities

perpetuity A perpetuity is an asset that pays C per period forever. With interest rate r, cash flow C, and

present value:

The general formulae for perpetuities:

Time subscript "1" reminds you that the stream of cash flows start from the next period, not the current one. Note that we’ve assumed C and r are the same every period.

How can an infinite sum be worth less than infinity? Since we have to discount each cash inflow, the extra amount added to the cumulative sum gets smaller and smaller, and therefore the overall sum converges towards a number< infinity.

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Growing perpetuity: CF1,

CF2 = CF1(1+g), CF3 =

CF1(1+g)2 , ...

How can a growing Infinite sum be less than infinity?

The Gordon Growth Model (1 / 3): Share Price Valuation

Since, in principle firms can exist forever, we use the growing perpetuity formulae to derive a firms share price.

The Gordon Growth Model (2/3): Firm Valuation We can use the growing perpetuity formula to value a firm.

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The Gordon Growth Model (3/3): Cost of Capital The GGM can also be used to back out a firm’s cost of capital.

Annuities

This term is the value of the perpetuity. And the second An annuity is an asset that makes the same payment CF for T y term represents the (FINITE NO. OF PAYMENTS) discount The present value of an annuity with initial payment in t = 1:

v

Annuities: Application to Mortgages Example

Growing Annuities An annuity pays an initial cash flow of C1, which then grows at rate g every year for T years. Capital costs r. The PV of the annuity is

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4.

Capital Budgeting Rules

Capital Budgeting Rules

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Internal Rate of Return

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Profitability Index

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Payback Rule

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Net Present Value

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Only accept projects with positive NPV

Rate of Return: IRR at which the NPV=0 Internal Rate of Return With one outflow C0 and one inflow C1, the rate of return is:

Example for Internal Rate of Return

NOTE: With only two cash inflows we cannot compute the rate of return in the usual way

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Capital Budgeting Rule Take projects with IRR above a "hurdle rate" rhurdle— i.e. the cost of capital estimated by the manager. 

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Generally no algebraic expression for IRR exists o o o

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IRR is a zero-point of a higher-order polynomial. Multiple solutions possible: depends on degree of polynomial. With three or more cash flows, solving is messy or impossible.

We can solve for the IRRs using manual iteration, i.e. trial-and-error, but this is painful

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Internal Rate of Return: Advantages

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Calculating the correct cost of capital r non-trivial, but r doesn’t not enter into the IRR calculation. In most cases IRR gives the same recommendation as NPV, but it has some important pitfalls. IRR allows for a "margin for error" in capital budgeting:

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Suppose the chosen hurdle rate K= true cost of capital. . . Nevertheless, IRR rule may still lead to the right decision.

Internal Rate of Return: Disadvantages 􏰀

IfC0 =40,C1 =−80,andC2 =104,what’stheIRR? IRR doesn’t exist!

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IfC0 =−100,C1 =360,C2 =−431,andC3 =−171.6? IRR is not unique!

IfC0 =−13.16,C1 =7,C2 =8,r0,1 =8%,andr1,2 =10%? IRR can’t handle different interest rates! 􏰀

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Some projects have no IRR and others have multiple IRR’s 􏰀 Such cases usually occur when the cash flows alternate sign. 􏰀

Sign m

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Unique

Using the new figures calculate the new IRR 18

The Profitability Index: An Inferior Alternative

Capital Budgeting Rule: Invest if PI > 1, otherwise reject. 

PI often gives the same recommendation as NPV , but. . .



Lacks the IRR advantage of avoiding cost of capital

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Same disadvantages as IRR, specifically scale invariance

The Payback Rule: How Long to retrieve the initial Investment ??? Payback Time (PT) simply counts the periods until cash inflows exceed initial cash outflow. It can lead to silly decisions:

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Capital Budgeting Rule: Take projects with PT below a chosen cutoff time. Cutoff = 1 ⇒ take project A but not B (silly!) 􏰀 Ignores all cash flows beyond the cutoff time — not sensible 􏰀

Equally weights cash flows before cutoff (ignores time value)

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The cutoff time is often arbitrarily chosen

Capital Budgeting Rules: The Bottom Line Best Rule: Accept projects with positive net present value.

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NPV maximizes the PV of wealth and is therefore optimal Amongst alternatives, IRR is most sensible but has pitfalls PI and PT can be used informally for background information

Why don’t all firms use NPV ?

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Managers may feel uncomfortable estimating the cost of capital

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IRR does not require cost of capital, and the hurdle rate can be ad hoc

Frequent use of the payback rule suggests that managers make suboptimal decisions 􏰀

Note: regarding first point, plot NPV as a function of cost of capital to check how sensitive NPV is to cost of capital estimates.

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5.

Bonds

Bond: financial security that governments and firms sell today in exchange for promised future payments. Bonds are issued by companies/government to fund themselves/the economy. Bonds issued by companies (sell to investors), they have to pay something upfront and then the bonds will give you payments from maturity and then a final amount called face value. Payments occur until the bond’s maturity date, which may be months (shortterm) or decades (long-term) into the future. Bonds make two types of payment:   􏰀

Coupon payments: periodic interest payments Face value payment: one-time payment at maturity

A bond’s term equals its time remaining until maturity

􏰀 Bonds are traded over-the-counter, i.e. by dealers who make markets by buying and selling from inventory

+Bonds are used way more than stocks in general

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Zero Coupon Bonds: A Special Type of Bond A zero coupon bond pays no coupon payments but pays its face value at maturity.

Yield-to-Maturity on Zero Coupon Bonds A bond’s IRR has a special name: yield to maturity (YTM). Now instead of cash-flow notation, let’s use bond notation: YTM yield to maturity ( ) P0 price at time of purchase FV face value at time of maturity

Coupon Bonds: A More General Case

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A coupon bond pays periodic coupons and then pays its face value at maturity. The general expression for a coupon bond’s YTM is:

Bond Price

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As the maturity of the bond increases, essentially the price converges towards the face value of the bond.

Bond Yields, Bond Prices, and Opportunity Cost of Capital If you hold a bond to maturity, your return equals the bond’s YTM because CPN and FV are fixed, and we assume no default. What if you sell early?   

Then you care about the market price of the bond when you sell. This market price (and your return) moves up and down. 􏰀 Why does the market price move? Because buyers constantly compare your bond to other assets and adjust their demand. 􏰀 If other assets earn higher returns, buyers demand your bond less and your bond’s price falls along with your return. 􏰀

Remember: In equilibrium, your bond’s market price adjusts until YTM for the next buyer equals the opportunity cost of capital r.

Bond Prices and Risk Free Rates

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With no uncertainty, bond yields are free of default risk — i.e. there is no risk that bond issuers won’t pay. So bonds, and especially short-term bonds, must have an opportunity cost of capital that lies close to the risk-free rate. The risk-free rate is tightly controlled by the central bank, who uses the risk-free rate to conduct monetary policy: 􏰀

they lower the risk-free rate to boost economic activity

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they raise the risk-free rate to dampen economic activity

Bond Prices and Risk Free Rates: An Inverse Relationship

Interest Rate Sensitivity: Short vs. Long Term Bonds

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The Yield Curve: Compensating for Interest Sensitivity

6.

Uncertainty, Default and Risk Attitude

Expected PV : Gives information about the average attractiveness of the project Variance PV: Gives information about the risk of the project

Risk neutral Investors – evaluate risk and gains symmetrically

Risk Adverse Investors- Feel losses very strongly

Risk AdverseThe utility they get from 500 with certainty is more than 1000 with risk.

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In this case u (E(W)) > E(u(W))

Risk Attitudes and Risk Premiums Risk averse individuals require a risk premium to take on risk. 

with phones, I could entice you by offering my phone plus £X

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the more risk averse, the higher £X must be to entice you

Risk neutral individuals require no risk premium to take on risk. 

with phones, you would willingly flip the coin even if X = 0

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you view gains and losses symmetrically in terms of utility

Risk seeking individuals are willing to pay to take on risk. 

with phones, I could offer you my phone minus £X

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you probably also go to casinos and buy scratch lottery tickets

Changing Our Basic Assumptions: Allowing for Risk So far our capital budgeting calculations involved no risk; future cash flows were certain, risk attitudes were irrelevant. 􏰀

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Now we let future cash flows be risky but assume that investors are risk

neutral and rank projects by E(NPV) Assuming risk neutrality is unrealistic in most cases, but it lets us easily study capital budgeting under uncertainty. 􏰀

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(210*0.99) +(50*0.1) ------------------------------------ = 198.48 1.05

The greater the gradient of the curve, the more risk adverse the investor is and consequently the greater the difference between the risk adverse price and risk neutral price Expected Interest Rate is the rate after taking the risk and uncertainty into account

Expected interest rate also includes the premium for investing the money today (takes into account not being able to use the money at a later time )

rf is also equal; to the expected return

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Uncertainty, Bonds and Stocks

Debt may be in the form of a bank loan/ bond.

May also be the risk that the firms make no profit or just enough to cover costs.

Unlevered equity levered equity

Appropriate price = Present Value (Because if the price = PV then the NPV=0 and this is what we would expect in competitive markets.)

;

Default Premium = Promised Return – Expected Return = 2.14% TYPO: Equity should equal 31 in all 3

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This table tells us: -

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Levered equity risker than unlevered Any type of equity is riskier than bond (Since equity investors are residual claimants to profits so they only get what is left over after paying the bond owners so this type will always be risker. Expected return is always the same regardless of the type of financing

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8.

Risk and Reward

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This would be the ideal place for risk adverse investors, as SD is low (low risk) but expected returns are high.

The rounded shape between assets X and Y represent the benefit of diversifying assets as the point at the end pf the bow has a lower sd (risk)

Anything below this line can be beaten by the point horizontally above as it has the same level of risk but higher E(r).

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9.

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CAPM

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Real World Examples: - coronavirus

Real World Examples: - Airbus fines

Price of Risk

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ßM = 1

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This is the extra return that a company will have to offer in order to make investors indifferent between investing in this lending activity or some alternative asset, with the knowledge that this project may fail.

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The more you go back in history the more data you will acquire typically, but the less informative your estimates will be about the future of the company. So this Instead of annual returns use

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10.

Capital Budgeting with Market

Imperfections

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Transaction costs compensate intermediaries for their services: 􏰀

banks charge interest margins: lending rates > borrowing rates

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real estate agents charge commissions to property buyers and sells

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market-makers in financial markets earns bid-ask spreads

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