Engg 255- Engineering Economics

Questions • A bank offers you a contract which pays $1000 at the end of each year for 20 years. How much are you willing to pay for this contract today? • A bank offers you a contract which pays $1000 at the end of each year forever. How much are you willing to pay for this contract today?

Engineering Design and Communication Engineering Economics

Reference: CFA Institute Program Curriculum.

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Interest Rates

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Interest Rates

• An interest rate is a rate of return that reflects the relationship between differently dated cash flows.

• r = Real risk-free interest rate + Inflation premium + Default risk premium + Liquidity premium + Maturity premium

• An opportunity cost is the value that investors forgo by choosing a particular course of action.

• The real risk-free interest rate is the single-period interest rate for a completely risk-free security if no inflation were expected.

• Interest rates are set in the marketplace by the forces of supply and demand, where investors are suppliers of funds and borrowers are demanders of funds.

• The inflation premium compensates investors for expected inflation

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• The default risk premium compensates investors for the possibility that the borrower will fail to make a promised payment at the contracted time and in the contracted amount. 4

Interest Rates

Future Value

• The liquidity premium compensates investors for the risk of loss relative to an investment's fair value if the investment needs to be converted to cash quickly.

•

• The maturity premium compensates investors for the increased sensitivity of the market value of debt to a change in market interest rates as maturity is extended

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Future Value

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Compounding

•

• With more than one compounding period per year,

• rs is the stated annual interest rate, • m is the number of compounding periods per year • N is the number of years.

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Continuous Compounding

Stated and Effective Rates

• Limiting value of the future value when mÆ ∞

• The periodic interest rate is the stated annual interest rate divided by m, where m is the number of compounding periods in one year. • With continuous compounding,

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Present Value

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Compounding •

• For a given discount rate, the farther in the future the amount to be received, the smaller that amount's present value. • Holding time constant, the larger the discount rate, the smaller the present value of a future amount.

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PV of a Series of Cash Flows: Annuity

A Series of Cash Flows: Perpetuity

• Annuity: a finite set of level sequential cash flows.

• A perpetuity is a perpetual annuity, or a set of level never-ending sequential cash flows, with the first cash flow occurring one period from now.

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Present Values at Different Times

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PV of a Series of Unequal Cash Flows

• Consider a perpetuity of $100 per year with its first payment beginning at five years from now. What is its present value today given a 5% discount rate?

•

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PV of a Series of Unequal Cash Flows

Amortization

•

• Suppose you apply for a loan (or a mortgage) in the amount of $100,000 from your bank. The bank offers you a rate of 8.75%. The loan is to be repaid in 5 years in monthly installments. • What is you monthly payment? • How much of each payment is interest payment and how much is the payment of the principle?

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Amortization

Amortization

• First calculate the monthly payment: • • • •

Present value: $100,000 Interest rate: 8.75% per year, (or 0.7291% per month) 60 payments Future value: $0

•

Monthly payment: $2063.72

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•

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Principle $100,000 $98,665 $97,321 $95,967 $94,603 $93,229 $91,845 $90,451 $89,047 $87,633 $86,208 $84,773 $83,327 $81,871 $80,404 $78,927 $77,439 $75,940 $74,430 $72,909 $71,376 $69,833 $68,279 $66 713

Monthly $2,064 $2,064 $2,064 $2,064 $2,064 $2,064 $2,064 $2,064 $2,064 $2,064 $2,064 $2,064 $2,064 $2,064 $2,064 $2,064 $2,064 $2,064 $2,064 $2,064 $2,064 $2,064 $2,064 $2 064

Interest $729.17 $719.44 $709.63 $699.76 $689.81 $679.80 $669.71 $659.54 $649.30 $638.99 $628.60 $618.13 $607.59 $596.98 $586.28 $575.51 $564.66 $553.73 $542.72 $531.63 $520.45 $509.20 $497.87 $486 45

Principle $1,334.56 $1,344.29 $1,354.09 $1,363.96 $1,373.91 $1,383.93 $1,394.02 $1,404.18 $1,414.42 $1,424.74 $1,435.12 $1,445.59 $1,456.13 $1,466.75 $1,477.44 $1,488.21 $1,499.07 $1,510.00 $1,521.01 $1,532.10 $1,543.27 $1,554.52 $1,565.86 $1 577 28

Remainder $98,665.44 $97,321.16 $95,967.07 $94,603.10 $93,229.19 $91,845.27 $90,451.25 $89,047.07 $87,632.64 $86,207.91 $84,772.78 $83,327.20 $81,871.07 $80,404.32 $78,926.88 $77,438.66 $75,939.60 $74,429.60 $72,908.59 $71,376.49 $69,833.22 $68,278.70 $66,712.84 20 $65 135 57

Questions

Capital Budgeting

• As the CEO of a company, you are offered two proposals for investment:

• • • • • •

• Project 1 requires an initial investment of $100,000 and is expected to result in cash flows of $25,000 per year for 6 years. • Project 2 requires an initial investment of $200,000 and is expected to result in cash flows of $35,000 for 10 years.

Decisions are based on cash flows. Timing of cash flows is crucial. Cash flows are based on opportunity costs. Cash flows are analyzed on an after-tax basis. Financing costs are ignored. Capital budgeting cash flows are not accounting net income. Accounting net income is reduced by non-cash charges such as - accounting- depreciation.

• Sunk Costs are ignored. – A sunk cost is one that has already been incurred. You cannot change a sunk cost.

• Are these projects worth investing in? If so, which one is more suitable for investment? 21

Capital Budgeting

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Capital Budgeting

• An externality the effect of an investment on other things besides the investment itself.

• An incremental cash flow is the cash flow that is realized because of a decision: the flow with a decision minus the cash flow without that decision.

• An opportunity cost is what a resource is worth in its next-best use. – If a company uses some idle property, what should it record as the investment outlay: the purchase price several years ago, the current market value, or nothing? – If you replace an old machine with a new one, what is the opportunity cost? – If you invest $10 million, what is the opportunity cost? •

The current market value, the cash flows the old machine would generate, and $10 million (which you could invest elsewhere).

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Net Present Value

Internal Rate of Return

•

• IRR is the discount rate that makes the present value of the future after-tax cash flows equal that investment outlay.

• Algebraically, this equation would be very difficult to solve. Use trial and error!

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Internal Rate of Return Outlay

0

Outlay =

CF1

CF2

1

2

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Internal Rate of Return • Example

CF3

3

0 ‐1000

CF3 CF1 CF2 + + 2 (1 + IRR) (1 + IRR) (1 + IRR) 3

1000.00

Outlay × (1 + IRR)3 = CF1 × (1 + IRR) 2 + CF2 × (1 + IRR) + CF3 • IRR is based on the assumption that we can reinvest a cash flow at that same discount rate.

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1 500

1440.71

2 800

3 800

2075.64

2990.38

500 720.3535 800

1037.81 1152.57 800 2990.38

Discount Rate 10%

IRR 44.07%

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Net Present Value and IRR

Ranking Conflicts between NPV and IRR

• NPV

• For a single conventional project, the NPV and IRR will agree on whether to invest or to not invest. • For independent, conventional projects, no conflict exists between the decision rules for the NPV and IRR. • What is we have to select one of two projects and there is a conflict between NPV and IRR?

• IRR

• Example: Cash Flows Year Project A Project B

0 -200 -200

1 80 0

2 80 0

3 80 0

4 80 400

NPV 53.59 73.21

IRR 21.86% 18.92%

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Ranking Conflicts between NPV and IRR

Ranking Conflicts between NPV and IRR

• Whenever the NPV and IRR rank two mutually exclusive projects differently, we choose the project based on the NPV.

• Would you rather have a small project with a higher rate of return or a large project with a lower rate of return? Sometimes, the larger, low rate of return project has the better NPV.

• Whenever you discount a cash flow at a particular discount rate, you are implicitly assuming that you can reinvest a cash flow at that same discount rate.

Cash Flows Year Project A Project B

– In the NPV calculation, you use the same discount rate for both projects. – In the IRR calculation, you use a discount rate equal to the IRR.

0 ‐100 ‐400

1 50 170

2 50 170

3 50 170

4 50 170

NPV 58.49 138.88

IRR 34.9 25.21%

– The fact that you earned an IRR rate in one project in does not mean that you can reinvest future cash flows at those rates.

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The Multiple IRR Problem

No IRR Problem

• This equation has no solution. • There are two solutions: IRR=100% and IRR=200%

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Payback Period

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Discounted Payback Period

• The payback period is the number of years required to recover the original investment in a project.

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Profitability Index

Capital Budgeting • Independent versus mutually exclusive projects-

• The profitability index (PI) is the present value of a project's future cash flows divided by the initial investment.

– Independent projects are projects whose cash flows are independent of each other. Mutually exclusive projects compete directly with each other.

• Project sequencing– Many projects are sequenced through time, so that investing in a project creates the option to invest in future projects.

• Unlimited funds versus capital rationing – An unlimited funds environment assumes that the company can raise the funds it wants for all profitable projects simply by paying the required rate of return. – Capital rationing exists when the company has a fixed amount of funds to invest. 37

Project Sequencing

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Cost of Capital

• Investing in one project may provide an opportunity to invest in other projects.

• Suppose that, to undertake a new project, a company needs new funds.

• Example

• It can raise new funds by either borrowing money or issuing new stocks.

0 ‐100,000

1   40,000

2  40,000

3  40,000

Case 1 NPV‐1 Case 2 NPV‐2

‐20,000 12,622 0 0

 10,000

 30,000

0

0

0 ‐100,000

1   46,311

2  40,000

3  40,000

r 10%

NPV ‐478

IRR 9.70%

Average=0.5*12822+0.5*0=631

r 10%

NPV 4,738

IRR 13%

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• Or if the company uses the NPV method, what rate should it use to discount the cash flows? • If the company uses the IRR method, what return should the company expect from the new investment to make it worthwhile?

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Cost of Capital

Cost of Capital

• Sources of capital: • Stocks

• Typically companies have a target debt-equity ratio that they try to maintain as part of their capital structure.

– Investors pay for company’s stock hoping to receive a dividend, or sell the company’s stock at a higher price. – How much is a company’s stock worth?

• For example, they may choose that 30% of their financing is obtained through debt and the rest through equity.

• Long term debt – Cost of debt is the rate of return the investors require when they buy the company’s bonds. – Interest paid on debt is subtracted from income. (Therefore, it is tax-deductible)

• This ratio is generally maintained when new capital is to be raised.

• What is the cost of raising funds through debt and equity? 41

Cost of Equity: Example

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Cost of Equity: Example

• Suppose that, starting from next year, a company pays $2 per year per stock as dividend. This dividend will remain unchanged in the future. • Suppose that the investors require a 10% return on their investment in the stock of this company.

• What if the dividend is to grow at a rate of g% per year?1 3 4 2 D1 D1 (1 + g ) D1 (1 + g ) 2 D1 (1 + g ) 3 K PV =

• How much should one stock of this company be worth now?

=

D1 D1 (1 + g ) D1 (1 + g ) 2 + + + ... 1+ r (1 + r )2 (1 + r )3 ⎞ D1 ⎛ (1 + g ) (1 + g ) 2 D ⎜⎜1 + + + ... ⎟⎟ = 1 2 (1 + r ) (1 + r ) 1+ r ⎝ ⎠ r−g

P=

D1 r−g

• Solution: • Example:

– This is a perpetuity! – The present value :

D1=$2; Dividend grows at a rate of 5% per year. 2 2 P= = = $40 0.1 − 0.05 0.05

A 2 = = $20 r 0.1 43

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Cost of Equity

Cost of Debt

• Given the price of the stock in the market, cost of equity can be calculated from the previous equation: P=

D1 r−g

r=

D1 +g P

• Financing through debt – Loans from banks or other financial institutions – Sale of bonds • Bonds are tradable, and their price constantly fluctuates till the maturity date as the require rate of return could change. • Before maturity, the bond may be sold or traded below, at or above its par value.

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Elements of a Bond

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Example

• Face Value (Par Value)

• Suppose a company issues a bond with a face value of $1000 which has a coupon rate of 8% per year, and matures in 4 years. If investors require a return of 10%, what is the current market price of this bond?

– A bond has a face value (or par value). This is the amount that will be paid on maturity date.

• Coupon Rate

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– Annual rate of the interest paid, usually semi-annually – Zero-coupon bonds do not pay any interest

0

• Maturity Date

1

40

40

40+1000

2

3

4

80 80 80 80 1000 + + + + 2 3 4 (1 + 0.1) (1 + 0.1) (1 + 0.1) (1 + 0.1) (1 + 0.1) 4 = $936.60

Current Price =

– Date when the face value of the bond (and possibly the last coupon) are paid out.

• What if the required rate of return is 7%?

• Required rate of return – Also called Yield to Maturity (YTM)

Current Price = $1033.87 47

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Example

Cost of Debt

• Alternatively, from the market price of a bond, the rate that the investors require can be inferred.

• Cost of debt is the return rate required by lenders/investors.

• Suppose a 4-year $1000 bond, with a coupon payment of 8% is being traded at $890. What is the implied cost of debt?

• In the US, interests paid out on debt are tax-deductible. This reduces the cost of debt.

$890 =

cost of debt = rd (1 − T )

80 80 80 80 1000 + + + + 2 3 4 (1 + r ) (1 + r ) (1 + r ) (1 + r ) (1 + r ) 4

where T is the tax rate

• The solution is found by trial and error. 1 + r = 1.1159 ⇒

r = 11.59% 49

Cost of Debt

Weighted Average Cost of Capital

• What is the effect of interest payments on the net income (profit)? EBIT

50

100

100

0

10

Profit after interest

100

90

Tax (40%)

40

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Net Income

60

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Interest, 10%

• If a company’s capital structure is made up of ws percent equity and wd percent debt, then the Weighted Average Cost of Capital (WACC) is given by:

WACC = ws rs + wd rd (1 − T ) where T: tax rate rd: cost of debt rs: cost of equity

• Difference: $6 = 10 × (1 − T ) •

EBIT: earning-before-interest-and-taxes

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Example

Breakeven Economics

• A company finances 30% of its capital through debt and the rest through equity. Suppose that the cost of its debt is 9%. Also it will pay a dividend of $1.30 next year which is expected to grow by 3% each year forever. The current price of the company’s stock is $14.44 and it pays 40% of its income as tax. • What is the company’s WACC?

• Business Costs – Variable costs (VC) • Vary proportionally with sales • Utilities, hourly wages, equipment,…

– Fixed costs (FC) • Constant over a relevant range of sales • Salaries, lease payments, depreciation, …

• Operating break-even point: Revenues (Sales)= FC + VC

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Breakeven Economics

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Breakeven Economics • Operating break-even point in units:

Quantity * Price= FC + Quantity * VC/unit

Q=

Q × ( p − v) = FC

FC p−v

• (p-v) is also referred to as the contribution margin (CM)

• Where – Q is the quantity, – p is the price per unit, – v is the variable cost per unit

• This is the amount that each unit sold contributes to paying off the fixed costs.

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Breakeven Economics

Example

• Monetary operating break-even point :

• Suppose that a shoe-manufacturing company has a fixed cost of $1,000,000. Variables cost is $8 per shoe, and the sale price is $25. How many shoes should it sell to break even?

FC ×p p−v FC FC = = ( p − v ) / p CM _ P

Q× p =

• Where CM_P denotes the contribution margin as a percentage of price

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Breakeven Economics

Cash-Flow Breakeven Point

• Repeat the previous analysis but now consider that an operating profit, i.e., before the payment of interests and taxes, (Also called EBIT-earning-before-interestand-taxes) is required:

Q=

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• If we subtract the depreciation expense (a non-cash expense) from fixed cost, we can calculate the breakeven point on a cash flow basis:

FC + EBIT p−v

Q=

FC − Depreciation p−v

• Example (Cont.) – Repeat the previous example if a profit of $400,000 is expected

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Risks

Operating Leverage

• Business Risks

• Question:

– Is the profit stream steady or volatile? – Degree of operational leverage (DOL)

– How is the % change in profits affected by % change in sales?

DOL =

• Financial Risk – How much of the company’s profit is reduced due to payment of interest on debt? – Degree of financial leverage, (DFL)

%Δ EBIT %Δ Sales

– It is a measure of a company’s business risk. – It reflects the impact of the company’s fixed costs on profits. Q( p − v) Sales − VC DOL = = Q( p − v) − FC EBIT

=

EBIT + FC EBIT

• Notice that DOL is goes up if FC increases. 61

Financial Leverage DFL =

Degree of Combined Leverage • The degree of combined leverage (DCL) includes the effects of both operational and financial leverages:

%Δ EBT %Δ EBIT

DCL =

• DFL is sometimes defined as:

DFL =

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%Δ EBT %Δ EBIT %Δ EBT = × = DOL × DFL %Δ Sales %Δ Sales %Δ EBIT

EBIT EBIT = EBT EBIT - Interest

• Notice that the higher the interest payment, the higher the DFL.

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Example •

Sales Variable Costs Fixed Costs Depreciation EBIT Interest Expense EBT

Sales EBIT EBT

Base Case 1000 450 300 100 150 30 120

Sales Down 10% 900 405 300 100 95 30 65

Sales up 10% 1100 495 300 100 205 30 175

Percentage Changes Relative to the Base Case -10.000% 10.000% -36.667% 36.667% -45.833% 45.833%

Leverage Measures Using a single income statement: DOL 3.67 5.21 DFL 1.25 1.46 DCL 4.58 7.62

2.95 1.17 3.46

Using two income statements: DOL 3.67 DFL 1.25 DCL 4.58

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