CORPORATE FINANCEII

CORPORATE FINANCEII

ESCP-EAP - European Executive MBA

23 Nov. 2005 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

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

NPV0 == CC

r

C

r

C

r0

1

1

2

2

2

t

t

t(1 ) (1 ).......

(1 )

DISCOUNTED CASH FLOW (DCF) EQUATION

INCLUDING INITIAL NEGATIVE CASH FLOWS AT THE START OF THE PROJECT, C0 ETC.

Present ValuesPresent 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.

400,18€

900,261000,300873.2

500,93000,100935.1

000,150000,1500.10Value

Present

Flow

Cash

Factor

DiscountPeriod

207.11

07.11

TotalNPV

INTERNAL RATE OF RETURN, IRR

INTERNAL RATE OF RETURN, IRR

CC C C

01 2

2T

T(1 IRR) (1 IRR)....

(1 IRR)0

NPV

=

IRR IS THE DISCOUNT RATE FOR WHICH NPV=0

IRR = 28%

+2

0

-1

50DISCOUNTRATE (%)

NPV

NET PRESENT VALUE PROFILE

C0 = - 4

C1 = +2

C2 = +4

IRR vs. NPVIRR 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 INTERESTCOMPOUND INTEREST

INTEREST EARNED ON PRINCIPAL AND

REINVESTED INTEREST OF PRIOR PERIODS

SIMPLE INTERESTSIMPLE INTEREST

INTEREST EARNED ON THE ORIGINAL

PRINCIPAL ONLY

Compound InterestCompound Interest

i ii iii iv vPeriods Interest Value Annuallyper per APR after compoundedyear period (i x ii) one year interest rate

1 6% 6% 1.06 6.000%

2 3 6 1.032 = 1.0609 6.090

4 1.5 6 1.0154 = 1.06136 6.136

12 .5 6 1.00512 = 1.06168 6.168

52 .1154 6 1.00115452 = 1.06180 6.180

365 .0164 6 1.000164365 = 1.06183 6.183

INTEREST RATE DOES NOT HAVE TO BE FOR A YEAR

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

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 FV1=100X1.1=110 IN MY

ACCOUNT, WHICH IS MY BEGINNING PRINCIPAL FOR YEAR 2.

AT THE END OF YEAR 2, I WILL HAVE

FV2 =FV1(1+r) =P(1+r)(1+r) =P(1+r)2

=121I 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

FUTURE VALUE OF €121 HAS FOUR PARTS

FV2=P(1+r)2=P+2rP+Pr2

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

FVn=PV(1+R)n

FV OF PRINCIPAL, P,

AT END OF n YEARS IS

FVn=PV(1+R)n

Future value of £1 after t years = (1+r)t

interest rate per period1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14%

no. ofperiods

1 1.010 1.020 1.030 1.040 1.050 1.060 1.070 1.080 1.090 1.100 1.110 1.120 1.130 1.1402 1.020 1.040 1.061 1.082 1.103 1.124 1.145 1.166 1.188 1.210 1.232 1.254 1.277 1.3003 1.030 1.061 1.093 1.125 1.158 1.191 1.225 1.260 1.295 1.331 1.368 1.405 1.443 1.4824 1.041 1.082 1.126 1.170 1.216 1.262 1.311 1.360 1.412 1.464 1.518 1.574 1.630 1.6895 1.051 1.104 1.159 1.217 1.276 1.338 1.403 1.469 1.539 1.611 1.685 1.762 1.842 1.9256 1.062 1.126 1.194 1.265 1.340 1.419 1.501 1.587 1.677 1.772 1.870 1.974 2.082 2.1957 1.072 1.149 1.230 1.316 1.407 1.504 1.606 1.714 1.828 1.949 2.076 2.211 2.353 2.5028 1.083 1.172 1.267 1.369 1.477 1.594 1.718 1.851 1.993 2.144 2.305 2.476 2.658 2.8539 1.094 1.195 1.305 1.423 1.551 1.689 1.838 1.999 2.172 2.358 2.558 2.773 3.004 3.252

10 1.105 1.219 1.344 1.480 1.629 1.791 1.967 2.159 2.367 2.594 2.839 3.106 3.395 3.70711 1.116 1.243 1.384 1.539 1.710 1.898 2.105 2.332 2.580 2.853 3.152 3.479 3.836 4.22612 1.127 1.268 1.426 1.601 1.796 2.012 2.252 2.518 2.813 3.138 3.498 3.896 4.335 4.81813 1.138 1.294 1.469 1.665 1.886 2.133 2.410 2.720 3.066 3.452 3.883 4.363 4.898 5.49214 1.149 1.319 1.513 1.732 1.980 2.261 2.579 2.937 3.342 3.797 4.310 4.887 5.535 6.26115 1.161 1.346 1.558 1.801 2.079 2.397 2.759 3.172 3.642 4.177 4.785 5.474 6.254 7.13816 1.173 1.373 1.605 1.873 2.183 2.540 2.952 3.426 3.970 4.595 5.311 6.130 7.067 8.13717 1.184 1.400 1.653 1.948 2.292 2.693 3.159 3.700 4.328 5.054 5.895 6.866 7.986 9.27618 1.196 1.428 1.702 2.026 2.407 2.854 3.380 3.996 4.717 5.560 6.544 7.690 9.024 10.57519 1.208 1.457 1.754 2.107 2.527 3.026 3.617 4.316 5.142 6.116 7.263 8.613 10.197 12.05620 1.220 1.486 1.806 2.191 2.653 3.207 3.870 4.661 5.604 6.727 8.062 9.646 11.523 13.74321 1.232 1.516 1.860 2.279 2.786 3.400 4.141 5.034 6.109 7.400 8.949 10.804 13.021 15.66822 1.245 1.546 1.916 2.370 2.925 3.604 4.430 5.437 6.659 8.140 9.934 12.100 14.714 17.86123 1.257 1.577 1.974 2.465 3.072 3.820 4.741 5.871 7.258 8.954 11.026 13.552 16.627 20.36224 1.270 1.608 2.033 2.563 3.225 4.049 5.072 6.341 7.911 9.850 12.239 15.179 18.788 23.21225 1.282 1.641 2.094 2.666 3.386 4.292 5.427 6.848 8.623 10.835 13.585 17.000 21.231 26.46226 1.295 1.673 2.157 2.772 3.556 4.549 5.807 7.396 9.399 11.918 15.080 19.040 23.991 30.16727 1.308 1.707 2.221 2.883 3.733 4.822 6.214 7.988 10.245 13.110 16.739 21.325 27.109 34.39028 1.321 1.741 2.288 2.999 3.920 5.112 6.649 8.627 11.167 14.421 18.580 23.884 30.633 39.20429 1.335 1.776 2.357 3.119 4.116 5.418 7.114 9.317 12.172 15.863 20.624 26.750 34.616 44.69330 1.348 1.811 2.427 3.243 4.322 5.743 7.612 10.063 13.268 17.449 22.892 29.960 39.116 50.950

0

2

4

6

8

10

12

14

16

18

0 5 10 15 20 25

Year

r = 5%

r = 10%

r = 15%

FUTURE VALUEFUTURE VALUE

FUTURE VALUEYear

125

1020

5%1.0501.1031.2761.6292.653

10%1.1001.2101.3312.5946.727

15%1.1501.3232.0114.04616.37

FUTURE VALUE OF €1

PRESENT VALUEPRESENT VALUEPRESENT VALUE OF €1

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12 14 16 18 20

r = 5%

r = 10%

r = 15%

PRESENT VALUE

Year 5% 10% 15% 1 .952 .909 .870 2 .907 .826 .756 5 .784 .621 .497 10 .614 .386 .247 20 .377 .149 .061

YEARS

FUTURE VALUECOMPOUND

PRINCIPAL AMOUNT

FORWARD INTO THE FUTURE

PRESENT VALUEDISCOUNT

A FUTURE VALUE BACK

TO THE PRESENT

BASIC RELATIONSHIP BETWEEN PV AND FV

PVFV

(1 )0

t

t r

ANNUAL PERCENTAGE RATE (APR)

ANNUAL PERCENTAGE RATE (APR)

EXAMPLE: CAR LOAN CHARGES INTEREST AT 1% PER

MONTH – APR OF 12% PER YEAR BUT

– EAR=(1+.01)12-1=12.6825% PER YEAR

– THIS IS THE RATE YOU ACTUALLY PAY

6% INTEREST RATE COMPOUNDING EAR APR. . . FREQUENCY

6% INTEREST RATE COMPOUNDING EAR APR. . . FREQUENCY

YEAR 1 6.000% 6.000%

QUARTER 4 6.136% 6.000%

MONTH 12 6.168% 6.000%

DAY 365 6.183% 6.000%

MINUTE 525,600 6.184% 6.000%

CONTINUOUSLY - 6.184% 6.000%

GENERAL RESULT

EAR =(1m

) 1m r

= = eerr -1-1 AS m INCREASES WITHOUT LIMIT

€1 INVESTED CONTINUOUSLY

AT AN INTEREST RATE r FOR t YEARS BECOMES ert

10% PER YEAR CONTINUOUSLY COMPOUNDED

10% PER YEAR CONTINUOUSLY COMPOUNDED

EAR = e.1 - 1 = 10.51709%

e=2.718

Short CutsShort 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

SHORTCUTS FOR VALUING LEVEL CASH FLOWS

PERPETUITIES

SERIES OF EQUAL CASH FLOWS AT END OF SUCCESSIVE PERIODS CONTINUING FOREVER

PV = C

rC

rCr(1 ) (1 )

.......(1 )2 n

Cr

r

r(1 )

11

(1 )

11

(1 )

n

Cr

r

11

(1 )n

PERPETUITIES

Cr

r

11

(1 )n

CASH FLOWS LAST FOREVER

PV

Cr

== 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 = = €1,000,000 .10

A DONATION OF €1,000,000 WILL PROVIDE AN ANNUAL INCOME OF .10 X €1,000,000 = €100,000 FOREVER.

PV =

VALUING PERPETUITIES

Cr

Cr

Cr

Cr

Cr

C

r

C

rC

rr

Cr

Cr

1 2

2

3

3

4

4

1 1

2

1

2

3

1 1

1

1 (1 ) (1 ) (1 ).........

(1 )

(1 g)

(1 )

(1 g)

(1 )....

(1 )1

1(1 g)

(1 )(1 ) (1 g)

g

PV

GROWING PERPETUITIES

GROWING PERPETUITIESGROWING 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.

€1,666,667.04 .10

100,000gr

CPV 1

SHORTCUTS FOR VALUING

LEVEL CASH FLOWS:

ORDINARY ANNUITY:

SERIES OF EQUAL CASH FLOWS AT END OF SUCCESSIVE PERIODS

ANNUITIES

PV = C

rCr

Cr(1 ) (1 )

.......(1 )2 n

Cr

r

11

(1 )n

FOUR VARIABLES, PV, r, n, C

IF WE KNOW ANY THREE, SOLVE FOR THE FOURTH

Asset Year of payment Present Value

1 2 . . t+1 . .

Perpetuity (first payment year 1)

Perpetuity (first payment year t + 1)

Annuity from year 1 to year t

(1+r)1

t)Cr(-

Cr

(1+r))rC 1

( t

Cr

PRICE AN ANNUITY AS EQUAL TO THE DIFFERENCE BETWEEN TWO PERPETUITIES

PRICE AN ANNUITY AS EQUAL TO THE DIFFERENCE BETWEEN TWO PERPETUITIES

Annuity table: Present value of €1 per period for each of t periods = 1/r - 1/[r(1+r)t]

interest rate per period1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14%

no. ofperiods

1 0.990 0.980 0.971 0.962 0.952 0.943 0.935 0.926 0.917 0.909 0.901 0.893 0.885 0.8772 1.970 1.942 1.913 1.886 1.859 1.833 1.808 1.783 1.759 1.736 1.713 1.690 1.668 1.6473 2.941 2.884 2.829 2.775 2.723 2.673 2.624 2.577 2.531 2.487 2.444 2.402 2.361 2.3224 3.902 3.808 3.717 3.630 3.546 3.465 3.387 3.312 3.240 3.170 3.102 3.037 2.974 2.9145 4.853 4.713 4.580 4.452 4.329 4.212 4.100 3.993 3.890 3.791 3.696 3.605 3.517 3.4336 5.795 5.601 5.417 5.242 5.076 4.917 4.767 4.623 4.486 4.355 4.231 4.111 3.998 3.8897 6.728 6.472 6.230 6.002 5.786 5.582 5.389 5.206 5.033 4.868 4.712 4.564 4.423 4.2888 7.652 7.325 7.020 6.733 6.463 6.210 5.971 5.747 5.535 5.335 5.146 4.968 4.799 4.6399 8.566 8.162 7.786 7.435 7.108 6.802 6.515 6.247 5.995 5.759 5.537 5.328 5.132 4.946

10 9.471 8.983 8.530 8.111 7.722 7.360 7.024 6.710 6.418 6.145 5.889 5.650 5.426 5.21611 10.368 9.787 9.253 8.760 8.306 7.887 7.499 7.139 6.805 6.495 6.207 5.938 5.687 5.45312 11.255 10.575 9.954 9.385 8.863 8.384 7.943 7.536 7.161 6.814 6.492 6.194 5.918 5.66013 12.134 11.348 10.635 9.986 9.394 8.853 8.358 7.904 7.487 7.103 6.750 6.424 6.122 5.84214 13.004 12.106 11.296 10.563 9.899 9.295 8.745 8.244 7.786 7.367 6.982 6.628 6.302 6.00215 13.865 12.849 11.938 11.118 10.380 9.712 9.108 8.559 8.061 7.606 7.191 6.811 6.462 6.14216 14.718 13.578 12.561 11.652 10.838 10.106 9.447 8.851 8.313 7.824 7.379 6.974 6.604 6.26517 15.562 14.292 13.166 12.166 11.274 10.477 9.763 9.122 8.544 8.022 7.549 7.120 6.729 6.37318 16.398 14.992 13.754 12.659 11.690 10.828 10.059 9.372 8.756 8.201 7.702 7.250 6.840 6.46719 17.226 15.678 14.324 13.134 12.085 11.158 10.336 9.604 8.950 8.365 7.839 7.366 6.938 6.55020 18.046 16.351 14.877 13.590 12.462 11.470 10.594 9.818 9.129 8.514 7.963 7.469 7.025 6.62321 18.857 17.011 15.415 14.029 12.821 11.764 10.836 10.017 9.292 8.649 8.075 7.562 7.102 6.68722 19.660 17.658 15.937 14.451 13.163 12.042 11.061 10.201 9.442 8.772 8.176 7.645 7.170 6.74323 20.456 18.292 16.444 14.857 13.489 12.303 11.272 10.371 9.580 8.883 8.266 7.718 7.230 6.79224 21.243 18.914 16.936 15.247 13.799 12.550 11.469 10.529 9.707 8.985 8.348 7.784 7.283 6.83525 22.023 19.523 17.413 15.622 14.094 12.783 11.654 10.675 9.823 9.077 8.422 7.843 7.330 6.87326 22.795 20.121 17.877 15.983 14.375 13.003 11.826 10.810 9.929 9.161 8.488 7.896 7.372 6.90627 23.560 20.707 18.327 16.330 14.643 13.211 11.987 10.935 10.027 9.237 8.548 7.943 7.409 6.93528 24.316 21.281 18.764 16.663 14.898 13.406 12.137 11.051 10.116 9.307 8.602 7.984 7.441 6.96129 25.066 21.844 19.188 16.984 15.141 13.591 12.278 11.158 10.198 9.370 8.650 8.022 7.470 6.98330 25.808 22.396 19.600 17.292 15.372 13.765 12.409 11.258 10.274 9.427 8.694 8.055 7.496 7.003

CALCULATING PV WHEN I KNOW C, r, N OR HOW MUCH AM I PAYING FOR MY CAR?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 1PV = €4,000 x - = €4,000 x 2.487 = €9,947.41 .10 .10(1.10)3

ANNUITY TABLE

NUMBER INTEREST RATE OF YEARS 5% 8% 10% 1 .952 .926 .909 2 1.859 1.783 1.736 3 2.723 2.577 2.487 5 4.329 3.993 3.791 10 7.722 6.710 6.145

LOANSLOANS

IN SOME LOANS, PRINCIPAL IS REPAID OR

AMORTIZED OVER LIFE OF LOAN

BORROWER MAKES FIXED PAYMENT EACH PERIOD

– EXAMPLES: MORTGAGES, CONSUMER LOANS

.

LOANSLOANS

EXAMPLE: AMORTIZATION SCHEDULE FOR 5-YEAR , €5,000

LOAN, 9% INTEREST RATE, ANNUAL PAYMENTS IN

ARREARS.

SOLVE FOR PMT AS ORDINARY ANNUITY PMT=€1,285.46

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

1 5,000 1,285.46 450.00 835.46 4,164.542 4,165 1,285.46 374.81 910.65 3,253.883 3,254 1,285.46 292.85 992.61 2,261.274 2,261 1,285.46 203.51 1,081.95 1,179.325 1,179 1,285.46 106.14 1,179.32 0

INTEREST DECLINES EACH PERIODAMORTIZATION OF PRINCIPAL INCREASES OVER

TIME

0

2000

4000

6000

8000

10000

12000

14000

1 6 11 16 21 26

Year

$

AMORTIZING LOAN

AMORTIZING LOAN

Year

$

AMORTIZATION

INTEREST

30

NOMINAL AND REAL RATES OF INTEREST

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 AND REAL RATES OF INTEREST

NOMINAL CASH FLOW FROM BANK Deposit IS €1,100IF INFLATION IS 6% OVER THE YEAR, REAL CASH

FLOW IS

REAL CASH FLOW =

€1,037.741.06

€1,100

NOMINAL CASH FLOW(1 AVERAGE INFLATION RATE )t

MEASURED IN CONSTANT POUNDS (EUROS OF CONSTANT PURCHASING POWER)

NOMINAL RETURNSNOT ADJUSTED FOR INFLATION

NOMINAL RETURNSNOT 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

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 = €1,000x1.120

= €6,727.50FUTURE REAL CASH FLOW

€2,097.671.06

€6,727.5020

NOMINAL RATE OF RETURN 10%

NOMINAL RATE OF RETURN 10%

REAL RATE OF RETURN

1.11.06

1 3.774%

FISHER EQUATION

(1+ rnominal) = (1+ rreal)(1+EXPECTED INFLATION RATE)

= 1 + rreal + EXPECTED INFLATION RATE + rreal (EXPECTED INFLATION RATE)

APPROXIMATELY, rnominal = rreal +EXPECTED INFLATION RATE

When I earn a real return, rreal,

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 rreal times the expected

inflation rate, to compensate me for the

effect of inflation on the interest.