CORPORATE FINANCE II ESCP-EAP - European Executive MBA 23 Nov p.m. London Various Guises of Interest Rates and Present Values in Finance I. Ertürk Senior Fellow in Banking

TWO RULES FOR ACCEPTING OR REJECTING PROJECTS 1. INVEST IN PROJECTS WITH POSITIVE NPV –Net Present Value 2. INVEST IN PROJECTS OFFERING RETURN GREATER THAN OPPORTUNITY COST OF CAPITAL

= NPV 0 = DISCOUNTED CASH FLOW (DCF) EQUATION INCLUDING INITIAL NEGATIVE CASH FLOWS AT THE START OF THE PROJECT, C 0 ETC.

Present Values Example - continued Assume that the cash flows from the construction and sale of an office building is as follows. Given a 7% required rate of return, create a present value worksheet and show the net present value.

INTERNAL RATE OF RETURN, IRR NPV= IRR IS THE DISCOUNT RATE FOR WHICH NPV=0

IRR = 28% DISCOUNT RATE (%) NPV NET PRESENT VALUE PROFILE C 0 = - 4 C 1 = +2 C 2 = +4

IRR vs. NPV  IRR IS AN INTENSIVE MEASURE OF PROFITABILITY  NPV IS AN EXTENSIVE MEASURE OF PROFITABILITY  AT THE END OF THE DAY, WE ARE INTERESTED IN A MONETARY (EXTENSIVE) MEASURE OF PROFITABILITY –NOT IN THE PROFITABILITY PER EURO OF INVESTMENT  BOTH GIVE SAME RESULT IF WE’RE NOT DEALING WITH MUTUALLY EXCLUSIVE PROJECTS OR PROJECTS WITH NONCONVENTIONAL CASH FLOWS

COMPOUND INTEREST INTEREST EARNED ON PRINCIPAL AND REINVESTED INTEREST OF PRIOR PERIODS

SIMPLE INTEREST INTEREST EARNED ON THE ORIGINAL PRINCIPAL ONLY

Compound Interest i ii iii iv v Periods Interest Value Annually per per APR after compounded year period (i x ii) one year interest rate 1 6% 6% % = = = = =

INTEREST RATE DOES NOT HAVE TO BE FOR A YEAR  INTEREST RATE per period WHERE THE PERIOD IS ALWAYS SPECIFIED  THE EQUATION FV=PV(1+r) GIVES THE FV at the end of the period,WHEN I INVEST P AT AN INTEREST RATE OF r PER PERIOD

INVESTING FOR MORE THAN ONE PERIOD  I INVEST P=€100 FOR 2 YEARS AT r =.1 PER YEAR.  AT END OF YEAR 1, I HAVE FV 1 =100X1.1=110 IN MY ACCOUNT, WHICH IS MY BEGINNING PRINCIPAL FOR YEAR 2.  AT THE END OF YEAR 2, I WILL HAVE FV 2 =FV 1 (1+r) =P(1+r)(1+r) =P(1+r) 2 =121  I WILL EARN €10 INTEREST IN YEAR 1, €11 INTEREST IN YEAR 2, –ALTHOUGH r =.1 IN BOTH YEARS.  WHY?

FUTURE VALUE OF €121 HAS FOUR PARTS FV 2 =P(1+r) 2 =P+2rP+Pr 2  P=100  RETURN OF PRINCIPAL  2rP=20  SIMPLE INTEREST ON PRINCIPAL FOR 2 YEARS AT 10% PER YEAR  Pr 2 =1  INTEREST EARNED IN YEAR 2 ON €10 INTEREST PAID IN YEAR 1  AMOUNT OF SIMPLE INTEREST CONSTANT EACH YEAR  AMOUNT OF COMPOUND INTEREST INCREASES EACH YEAR

FV OF PRINCIPAL, P, AT END OF n YEARS IS FV n =PV(1+R) n

FUTURE VALUE Year % % % FUTURE VALUE OF €1

PRESENT VALUE PRESENT VALUE OF €1 r = 5% r = 10% r = 15% PRESENT VALUE Year 5%10%15% YEARS

FUTURE VALUE  COMPOUND PRINCIPAL AMOUNT FORWARD INTO THE FUTURE PRESENT VALUE  DISCOUNT A FUTURE VALUE BACK TO THE PRESENT

BASIC RELATIONSHIP BETWEEN PV AND FV

ANNUAL PERCENTAGE RATE (APR)  EXAMPLE: CAR LOAN CHARGES INTEREST AT 1% PER MONTH –APR OF 12% PER YEAR BUT –EAR=(1+.01) 12 -1= % PER YEAR –THIS IS THE RATE YOU ACTUALLY PAY

6% INTEREST RATE COMPOUNDING EAR APR... FREQUENCY YEAR % 6.000% QUARTER % 6.000% MONTH % 6.000% DAY % 6.000% MINUTE 525, % 6.000% CONTINUOUSLY % 6.000%

GENERAL RESULT EAR = = e r -1 = e r -1 AS m INCREASES WITHOUT LIMIT €1 INVESTED CONTINUOUSLY AT AN INTEREST RATE r FOR t YEARS BECOMES e r t

10% PER YEAR CONTINUOUSLY COMPOUNDED EAR = e = % e=2.718

Short Cuts  Sometimes there are shortcuts that make it very easy to calculate the present value of an asset that pays off in different periods. These tolls allow us to cut through the calculations quickly.  Annuities – bonds, amortized loans, pensions, etc.  Perpetuities – company valuation (terminal value)

SHORTCUTS FOR VALUING LEVEL CASH FLOWS PERPETUITIES  SERIES OF EQUAL CASH FLOWS AT END OF SUCCESSIVE PERIODS CONTINUING FOREVER

PV =

PERPETUITIES CASH FLOWS LAST FOREVER PV = AS n GETS VERY LARGE

C r EXAMPLE:  SUPPOSE YOU WANT TO ENDOW A CHAIR AT YOUR OLD UNIVERSITY, WHICH WILL PROVIDE $100,000 EACH YEAR FOREVER. THE INTEREST RATE IS 10% €100,000 PV = = €1,000,  A DONATION OF €1,000,000 WILL PROVIDE AN ANNUAL INCOME OF.10 X €1,000,000 = €100,000 FOREVER. PV = VALUING PERPETUITIES

PV GROWING PERPETUITIES

 SUPPOSE YOU WISH TO ENDOW A CHAIR AT YOUR OLD UNIVERSITY WHICH WILL PROVIDE €100,000 PER YEAR GROWING AT 4% PER YEAR TO TAKE INTO ACCOUNT INFLATION. THE INTEREST RATE IS 10% PER YEAR.

SHORTCUTS FOR VALUING LEVEL CASH FLOWS : ORDINARY ANNUITY: SERIES OF EQUAL CASH FLOWS AT END OF SUCCESSIVE PERIODS ANNUITIES

PV = FOUR VARIABLES, PV, r, n, C IF WE KNOW ANY THREE, SOLVE FOR THE FOURTH

Asset Year of payment Present Value t+1.. Perpetuity (first payment year 1) Perpetuity (first payment year t + 1) Annuity from year 1 to year t (1+r) 1 t ) C r ( - C r ) r C1 ( t C r PRICE AN ANNUITY AS EQUAL TO THE DIFFERENCE BETWEEN TWO PERPETUITIES

CALCULATING PV WHEN I KNOW C, r, N OR HOW MUCH AM I PAYING FOR MY CAR? EXAMPLE:  I BUY A CAR WITH THREE END-OF-YEAR PAYMENTS OF €4,000  THE INTEREST RATE IS 10% A YEAR 1 1 PV = €4,000 x - = €4,000 x = €9, (1.10) 3 ANNUITY TABLE NUMBER INTEREST RATE OF YEARS 5% 8% 10%

LOANS  IN SOME LOANS, PRINCIPAL IS REPAID OR AMORTIZED OVER LIFE OF LOAN  BORROWER MAKES FIXED PAYMENT EACH PERIOD –EXAMPLES: MORTGAGES, CONSUMER LOANS.

LOANS  EXAMPLE: AMORTIZATION SCHEDULE FOR 5-YEAR, €5,000 LOAN, 9% INTEREST RATE, ANNUAL PAYMENTS IN ARREARS.  SOLVE FOR PMT AS ORDINARY ANNUITY PMT=€1,  WE KNOW THE TOTAL PAYMENT, WE CALCULATE THE INTEREST DUE IN EACH PERIOD AND BACK CALCULATE THE AMORTIZATION OF PRINCIPAL

AMORTIZATION SCHEDULE YEAR BEGINNING TOTAL INTEREST PRINCIPAL ENDING BALANCE PAYMENT PAID PAID BALANCE 15,000 1, , ,165 1, , ,254 1, , ,261 1, , , ,179 1, , INTEREST DECLINES EACH PERIOD AMORTIZATION OF PRINCIPAL INCREASES OVER TIME

AMORTIZING LOAN Year $ AMORTIZATION  INTEREST 30

NOMINAL AND REAL RATES OF INTEREST  A BANK OFFERING AN INTEREST RATE OF 10% PER YEAR ON A €1,000 DEPOSIT WILL PAY €1,100 AT THE END OF THE YEAR  BANK MAKES NO PROMISE ABOUT WHAT THE €1,100 WILL BUY AT THE END OF THE YEAR –DEPENDS ON RATE OF INFLATION OVER THE YEAR  CPI MEASURES INFLATION IN PURCHASES OF A TYPICAL FAMILY

NOMINAL AND REAL RATES OF INTEREST  NOMINAL CASH FLOW FROM BANK Deposit IS €1,100  IF INFLATION IS 6% OVER THE YEAR, REAL CASH FLOW IS  REAL CASH FLOW = MEASURED IN CONSTANT POUNDS (EUROS OF CONSTANT PURCHASING POWER)

NOMINAL RETURNS NOT ADJUSTED FOR INFLATION  REAL RETURNS –RETURNS ADJUSTED FOR INFLATION –PERCENTAGE CHANGE IN HOW MUCH I CAN BUY AS A RESULT OF THE CHANGE IN VALUE OF MY INVESTMENT –PERCENTAGE CHANGE IN THE VALUE OF MY INVESTMENT MEASURED IN UNITS OF CONSTANT PURCHASING POWER

NOMINAL AND REAL RATES OF INTEREST  20-YEAR Deposit –€1,000 INVESTMENT –10% PER YEAR INTEREST RATE –EXPECTED AVERAGE FUTURE INFLATION 6% PER YEAR  FUTURE NOMINAL CASH FLOW = €1,000x = €6,  FUTURE REAL CASH FLOW

NOMINAL RATE OF RETURN 10%  REAL RATE OF RETURN FISHER EQUATION (1+ r nominal ) = (1+ r real )(1+EXPECTED INFLATION RATE) = 1 + r real + EXPECTED INFLATION RATE + r real (EXPECTED INFLATION RATE) APPROXIMATELY, r nominal = r real +EXPECTED INFLATION RATE

When I earn a real return, r real, my nominal return has to be increased by the expected inflation rate, to compensate me for the effect of inflation on my original principal, and an additional r real times the expected inflation rate, to compensate me for the effect of inflation on the interest.