SPRING 2014

““The Time Value of Money”The Time Value of Money”

SUFE/Webster University SUFE/Webster University FIN C 5000 week 2FIN C 5000 week 2

Recession…

Agenda

• Teams and Companies…

• Getting to know your S&P500 Company…

• Financial Ratio Analysis

• Present Values revisited

• Annuities-Perpetuities

• Capital Budgeting

Fall 2011

Fall 2011

Please use Excel…• This chapter is about straight forward This chapter is about straight forward

calculations of Future Values of cash calculations of Future Values of cash and Present Values of cashand Present Values of cash

• It’s much easier to use Excel if you It’s much easier to use Excel if you know how to work with it…know how to work with it…

• It will save you many hours of It will save you many hours of calculationscalculations

• Please avoid the old fashioned Please avoid the old fashioned calculator…and skip these in the text of calculator…and skip these in the text of chapter 8…chapter 8…

• Nowadays everybody uses Nowadays everybody uses spreadsheets…so you better learn how spreadsheets…so you better learn how to use them if you don’t know…to use them if you don’t know…

Old Fashioned !

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What do you prefer…• $ 1000 now or$ 1000 now or• $ 1000 next year?$ 1000 next year?

• If you are like anybody else If you are like anybody else you know the answer very you know the answer very well…well…

• That’s all of us know there is That’s all of us know there is Time Value to money and Time Value to money and that is what this chapter is that is what this chapter is about…about…

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Future Value

• You put money in the bank at You put money in the bank at % interest: how much money % interest: how much money do you have after…yearsdo you have after…years

• You compound the interest You compound the interest since every year you will since every year you will calculate the interest over the calculate the interest over the amount you put in the bank amount you put in the bank initially and over the interest initially and over the interest gathered in prior years gathered in prior years

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Compounding• Per year; for every year you keep your money in the Per year; for every year you keep your money in the

bank calculate amount times (1+i) where i= the interest bank calculate amount times (1+i) where i= the interest raterate

• Per half year; for every year amount times (1+ i/2) to the Per half year; for every year amount times (1+ i/2) to the power 2 (we will use the symbol j2 indicating that we power 2 (we will use the symbol j2 indicating that we compund twice per year every half year)compund twice per year every half year)

• Per quarter (3 months that is 4 times per year); for every Per quarter (3 months that is 4 times per year); for every year amount times (1+i/4) to the power 4 (j4)year amount times (1+i/4) to the power 4 (j4)

• Per month; amount times (1+i/12) to the power 12 (j12)Per month; amount times (1+i/12) to the power 12 (j12)

• Per day; amount times (1+i/365) to the power 365 (j365)Per day; amount times (1+i/365) to the power 365 (j365)

• Per hour; amount times (…etc.Per hour; amount times (…etc.

• I guess by now you have got the message right? I guess by now you have got the message right? Compounding…

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Unfortunately…

• No bank in this world compounds per No bank in this world compounds per month or per day, leave alone per hour or month or per day, leave alone per hour or per second…. per second….

Future Value…

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But if they would….

• Future Value= Amount* e^ Future Value= Amount* e^ (i*t)(i*t)

• Present Value= Amount* e^ Present Value= Amount* e^ -(i*t)-(i*t)

• This is called continuous interest…This is called continuous interest…

Compounding effects (FV)

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after 1 year

FV $100 5% Annual $5.00

2 half year 2.50000000% $5.06

4 quarter 1.25000000% $5.09

12 month 0.41666667% $5.12

52 week 0.09615385% $5.12

365 day 0.01369863% $5.13

8760 hour 0.00057078% $5.13

525600 minute 0.00000951% $5.13

31536000 second 0.00000016% $5.13

∞ continuous ≈0 $5.13

Compounding effects (PV)

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Today is:

PV $100 5% Annual $95.24

2 half year 2.50000000% $95.18

4 quarter 1.25000000% $95.15

12 month 0.41666667% $95.13

52 week 0.09615385% $95.13

365 day 0.01369863% $95.12

8760 hour 0.00057078% $95.12

525600 minute 0.00000951% $95.12

31536000 second 0.00000016% $95.12

∞ continuous ≈0 $95.12

Using Excel for NPV…

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year cash flows

0 -1000

1 500

2 400

3 300

4 200

5 100

Assume i= 10%

NPV $190.19

IRR% 20.3%

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Make a timeline; it’s helpful

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Let’s warm up with Future Value• If you are familiar with If you are familiar with

calculating calculating compounded interest compounded interest move on to Present move on to Present ValueValue

• If not; try to do the If not; try to do the following quickies and following quickies and check your answers…check your answers…

• If you need more If you need more exercises you will find exercises you will find them after chapter 8 in them after chapter 8 in the textbookthe textbook

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Compound InterestCompound Interest

• Find the accumulated value of $ 100 at 10% pa compounding annually after:

• 10 years

• 20 years

• 30 years

Test your self….

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Your answer

• $100* 1.10^10= $ 259.37

• $100*1.10^20= $ 672.75

• $100*1.10^30=$ 1,744.94

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CompoundedCompounded

• Chin Li puts $ 250 in a savings account at 7,3% pa How much money will be in her account after 180 days if interest is compounded on a daily basis (j365=7.3%)?

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Your answer

• 7.3% per year is equivalent 7.3%/365 days or 0.02% per day so in 180 days the account has grown to 250* (1+0,02%)^180= $ 259.16

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Find accumulated values ofFind accumulated values of

• $ 750 after 6 years at 10% pa compounded every half year

• $ 750 after 6 years convertible quarterly

• $ 1500 at j4=12% after 8.25 years

• $ 1500 after 7 years and 8 months at 12% compounded monthly

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Your answers

• $ 750*(1.05)^12= $ 1346.89

• $ 750* (1.03)^24= $ 1524.60

• $ 1500* (1.03)^33= $ 3978.50

• $ 1500* (1.01)^92= $ 3746.78

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Present Value• Now we calculate the equivalent value Now we calculate the equivalent value

per today (the present) of a future per today (the present) of a future cash flowcash flow

• For instance how much money do I For instance how much money do I need to put in the bank today if I want need to put in the bank today if I want to have $100,000 in 5 years from now to have $100,000 in 5 years from now and my bank offers me a semi-anual and my bank offers me a semi-anual interest rate of 4% per annum interest rate of 4% per annum (annum=year) (annum=year)

• So instead of multiplying an amount So instead of multiplying an amount with (1+i)m per year, we now divide with (1+i)m per year, we now divide by (1+i) per year by (1+i) per year

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Don’t forget to draw the timeline

Backward Time

S=$2000

P=?

P= Present Value = S/(1+i)N

= $ 2000/(1+7%)^3= $ 1,632.60

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Present value• Can also be compounded per half Can also be compounded per half

year, quarter, month, week, hour etc.year, quarter, month, week, hour etc.

• The formulas are similar to Future The formulas are similar to Future Value but remember in case of Value but remember in case of Present Value we divide by (1+i) or Present Value we divide by (1+i) or multiply by 1/(1+i) per annummultiply by 1/(1+i) per annum

• So for half yearly compounding over So for half yearly compounding over 3 years we multiply the future 3 years we multiply the future amount with:amount with:

• 1/(1+i/2)^6 ( we have 6 half years)1/(1+i/2)^6 ( we have 6 half years)

• And for monthly compounding 5 And for monthly compounding 5 years:years:

• 1/(1+i/12)^60 (we have 60 months) 1/(1+i/12)^60 (we have 60 months)

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Compounding againCompounding again

• How much money should Ming Wei invest on May 1st 2003 at j12=9% in order to have $ 1125 on 1st April 2005 ?

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Your answer

• Now we have to calculate the PV (present value) of $ 1125 compounding per month at 9%/12= 0.75% per month and over n= 23 periods (check it!) so:

• $1125/(1.0075)^23= $ 947.36

This is 0.75% or 0.0075

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Sometimes

• You will have to calculate the You will have to calculate the i(nterest)i(nterest)

• Or the number of periods that Or the number of periods that the money is outstanding…the money is outstanding…

• Don’t worry Excel has got Don’t worry Excel has got perfect functions to help you do perfect functions to help you do itit

• You can find the interest with You can find the interest with the IRR% function (Internal the IRR% function (Internal Rate of Return)Rate of Return)

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Find the nominal rate of interestFind the nominal rate of interest

• Find the nominal interest rate pa compounded per half year (semi annually) at which $2500 grows to $4000 in 5 years

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Your answer

• $2500*(1+i/2)^10=$4000 what is i?

• (1+i/2)^10=$4000/$2500• 1+i/2=(4000/2500)^0.10• i= (((4000/2500)^0.10)-1)*2

• i= 9.6245%

2500

4000

2500(1+i/2)^10=4000

1+i/2=(4000/2500)^0.10

i/2=(4000/2500)^0.10-1

i=2*((4000/2500)^0.10-1) i= 9.62%9.62%

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Future and Present Value of Annuities

• Instead of compounding Instead of compounding interest over one amount interest over one amount either forward (future value) either forward (future value) or backward (present valueor backward (present value

• Often we have to calculate Often we have to calculate the future or present value the future or present value in case the same amount in case the same amount enters the bank account enters the bank account every yearevery year

• This is called an annuity This is called an annuity

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Ordinary Annuities• All payments are done at the end of the periodAll payments are done at the end of the period

• So an ordinary annuity of $100 3 years at 5% per So an ordinary annuity of $100 3 years at 5% per annum (pa) returns:annum (pa) returns:

• $100 stays 2 years in the account since it entered $100 stays 2 years in the account since it entered the account at the end of period 1 it adds up to ; the account at the end of period 1 it adds up to ; $100*(1+5%)^2 at the end of period 3$100*(1+5%)^2 at the end of period 3

• $100 stays 1 year in the account and adds up to $100 stays 1 year in the account and adds up to $100*(1+5%) at the end of period 3$100*(1+5%) at the end of period 3

• And $100 enters the account at the end of period 3 And $100 enters the account at the end of period 3 but it has no time to gather any interestbut it has no time to gather any interest

• Adding all up: Adding all up: $ 315,25$ 315,25

Annuities in General

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Annuity due

• Now all amounts are paid at the Now all amounts are paid at the beginning of the periodbeginning of the period

• So the former example in case So the former example in case of an annuity due will result in:of an annuity due will result in:

• Now the first $100 is 3 years in Now the first $100 is 3 years in the account; $100*(1+5%)^3the account; $100*(1+5%)^3

• The next one 2 years; The next one 2 years; $100*(1+5%)^2$100*(1+5%)^2

• The last one 1 years; The last one 1 years; $100*(1+5%)$100*(1+5%)

• Add it all up: Add it all up: $ 331,01$ 331,01

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Annuities

• Let us flex our financial skill muscles again:

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Ordinary annuitiesOrdinary annuities

• Henry saves $600 each half year and invests it at 13% (convertible semi annually) How much money has Henry got after 10 years?

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Your answer• After a half year Henry puts in his first $600 and

so fort every half year• Starting today the first $600 will stay in the

account 9.5 years• The second amount of $600 9 years• The third amount 8.5 years etc.• The last amount of $600 will have no time to

gather any interest • Adding up all these future values for this annuity

will give: $ 22,695.19+$600=$ 23,295.19

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How many paymentsHow many payments

• How many payments will be paid per half year if the first payment is on 1st july 1997 and the last on 1st january 2006?

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Your answer

• Draw a timeline!

• 1st july 1997=1, 1st january 1998=2, etc.1st january 1999=4, 2000=6, 2001=8, 2002=10, 2003=12, 2004=14, 2005=16, 1st january 2006= 18!

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Calculate Present ValueCalculate Present Value

• For each of the ordinary annuities:• $500 payment, 6 months payment

period, 12 years long, j2=9.5%• $1000 payment, yearly payment

period, 8 years long, j=10.8%• $250 payment, monthly payment

period, 10 years long, j12=10.8%• $500 payment, 3 monthly

payment period, 8 years long, j4=9.8%

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Your answers• There are 24 times $500 payments after every There are 24 times $500 payments after every

half year the last $500 is discounted at half year the last $500 is discounted at 1.0475^24, the fore last at 1.0475^23 etc. (the 1.0475^24, the fore last at 1.0475^23 etc. (the first one at 1,0475) add the all up and you get: $ first one at 1,0475) add the all up and you get: $ 7070.277070.27

• There are 8 yearly payments of $1000 discounted There are 8 yearly payments of $1000 discounted at 10.8% add it up : $ 5183.03at 10.8% add it up : $ 5183.03

• There are 120 payments of $ 250 the first amount There are 120 payments of $ 250 the first amount will be discounted at (1+10.8%/12) the second at will be discounted at (1+10.8%/12) the second at (1+10.8%/12)^2 etc. Excel will do the job for you; (1+10.8%/12)^2 etc. Excel will do the job for you; add it all up and $ 18298.89 is your answeradd it all up and $ 18298.89 is your answer

• There are 32 payments (8 years and 4 times per There are 32 payments (8 years and 4 times per year) of $500; the first payment of $500 will be year) of $500; the first payment of $500 will be discounted at (1+9.8%/4) and the second by discounted at (1+9.8%/4) and the second by (1+9.8%/4) ^2 etc. add it all up and $ 11001,81 is (1+9.8%/4) ^2 etc. add it all up and $ 11001,81 is your answeryour answer

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Used CarUsed Car

• A used car sells for $ 10,000 in cash OR:

• $2000 deposit plus 6 instalments of $ 1400 per month for 6 months

• What is the implied interest in the instalment plan? (j 12)

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Your answer• 10.000=2000+1400/(1+i/12)+1400/(1+i/

12)^2+1400/(1+i/12)^3+…+1400/(1+i/12)^6• So 8000=1400*(1/(1+i/12)+…+1/(1+i/12)^6)• 8000/1400= (1/(1+i/12)+…+1/(1+i/12)^6)• 5.7143=(1/(1+i/12)+…+1/(1+i/12)^6)• Excel will solve it; i=16.945% with IRR%• Note; assume 0= -5.7143+1/(1+i/12)+….

year 1 year 2 year 3 year 4 year 5 year 6

-5.7143 1 1 1 1 1 1

1.41%1.41% per monthper month

16.94%16.94% papa

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Changing interest ratesChanging interest rates• On may 1st 1985 Minnie

deposited $100 in a savings account which paid 8% pa convertible half yearly and continued to make the same deposits every 6 months.

• After may 1st 1997 the bank paid 9% pa compounded half yearly.

• How much will be in the account just after the deposit on 1st november 2005? Nobody likes them; everybody Nobody likes them; everybody

needs them…needs them…

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Your answer• 11stst may 1985 the first $100 enters the account and will be there until 1 may 1985 the first $100 enters the account and will be there until 1stst may 1997 at may 1997 at

(1+8%/2) or 1.04 growth rate per half year(1+8%/2) or 1.04 growth rate per half year• Its 12 years or 24 half years compounded at 4% per half yearIts 12 years or 24 half years compounded at 4% per half year• 11stst november 1985 the next $ 100 enters the account and will be compounded at 4% for november 1985 the next $ 100 enters the account and will be compounded at 4% for

23 periods etc23 periods etc• 11stst november 1996 the last amount of $100 enters the account that will benefit the 8% november 1996 the last amount of $100 enters the account that will benefit the 8%

raterate• This amount will be up rated at 1.04 (only 1 period)This amount will be up rated at 1.04 (only 1 period)• So on 1So on 1stst may 1997 just after the deposit on that date you can calculate that there is $ may 1997 just after the deposit on that date you can calculate that there is $

4164.59 in the account4164.59 in the account• From that moment the rate will be 9% compounded half yearly up to 1From that moment the rate will be 9% compounded half yearly up to 1stst november 2005 november 2005

(for 8 and a half years)(for 8 and a half years)• So the $ 4164.59 will grow to 4164.59*(1.045)^17 or So the $ 4164.59 will grow to 4164.59*(1.045)^17 or $ 8801.35$ 8801.35• But there will also be new deposits from 1But there will also be new deposits from 1stst november 1997 etc, every half year; the first november 1997 etc, every half year; the first

deposit on 1deposit on 1stst november 1997 will enjoy 1.045 up rating every half year for 16 periods (8 november 1997 will enjoy 1.045 up rating every half year for 16 periods (8 years) etc.years) etc.

• So add it all up and the new deposits from 1So add it all up and the new deposits from 1stst nov 1997 including 1 nov 1997 including 1stst nov 2005 will be a nov 2005 will be a total amount of: total amount of: $ 2474.17$ 2474.17

• Add the bold figures and you have your answer: $ 8801.35+$ 2474.17=Add the bold figures and you have your answer: $ 8801.35+$ 2474.17=$ 11275.52$ 11275.52

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Compound interest and loansCompound interest and loans

• Colin buys a new car worth $ 25,000.-

• He pays $ 2,500 deposit• The balance he will pay with

level payments at the end of each quarter for 2 years

• The interest is (j4) 10% pa• Find the quarterly instalment• Construct the loan repayment

schedule for the first 4 quarters

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Your answer• The balance is $ 22,500.- This amount has to The balance is $ 22,500.- This amount has to

be paid back in 8 (2 years is 8 quarters) equal be paid back in 8 (2 years is 8 quarters) equal instalments including the interestinstalments including the interest

• The installment R can be calculated from: $ The installment R can be calculated from: $ 22,500=R( 22,500=R( ΣΣ1/(1+0.025)^t) there are in total 8 1/(1+0.025)^t) there are in total 8 instalments; The PV of all future instalments instalments; The PV of all future instalments should add up to $ 22,500should add up to $ 22,500

• 7.170137*R=$ 22500 7.170137*R=$ 22500 so R= $ 3138.02so R= $ 3138.02• See the scheme on next slide: See the scheme on next slide:

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Payment Scheme:

Quarter Quarter DebtDebt InstalmentInstalment InterestInterest Down PaymentDown Payment

11 2250022500 3138.0153138.015 562.50562.50 2575.5152575.515

22 19924.4919924.49 3138.0153138.015 498.1121498.1121 2639.9032639.903

33 17284.5817284.58 3138.0153138.015 432.1146432.1146 2705.902705.90

44 14578.6814578.68 3138.0153138.015 364.467364.467 2773.5482773.548

55 11805.1311805.13 3138.0153138.015 295.1283295.1283 2842.8872842.887

66 8962.2478962.247 3138.0153138.015 224.0562224.0562 2913.9592913.959

77 6048.2886048.288 3138.0153138.015 151.2072151.2072 2986.8082986.808

88 3061.483061.48 3138.0153138.015 76.5370176.53701 3061.4783061.478

00

A total of $ 25,104.12 will be paid back in total for a “loan” of $ 22,500A total of $ 25,104.12 will be paid back in total for a “loan” of $ 22,500

(growing) Perpetuity

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Perpetuity P= $C/(1+i)+$C/(1+i)^2+…….+$C/(1+i)^n with n=∞

Proof that this is the same as P= $C/i….

Growing Perpetuity:

P=$C(1+g)/(1+i)+……………………$C(1+g)^n/(1+i)^n with n=∞

Proof that this is the same as P= $C(1+g)/(i-g) if i>g

We will use above properties….

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Week 2: Homework Assignment

• Chose a company from the valueline documents (this is a Dow Chose a company from the valueline documents (this is a Dow Jones 30 company)Jones 30 company)

• Go to: www.valueline.comGo to: www.valueline.com• Read the valueline document and take a look at the financialsRead the valueline document and take a look at the financials• Now calculate Free Cash Flow (FCF)=(NOPAT+depreciation+/- Now calculate Free Cash Flow (FCF)=(NOPAT+depreciation+/-

change in Working Capital- Capital Spendingchange in Working Capital- Capital Spending• Do this for all the years available on the valueline document Do this for all the years available on the valueline document

(including FY 2012, 2015)(including FY 2012, 2015)• Assume that the Cost of Capital (WACC%)=10% if your company is Assume that the Cost of Capital (WACC%)=10% if your company is

only Equity funded and 7% if your company has Debt on the only Equity funded and 7% if your company has Debt on the Balance sheet…Balance sheet…

• Forecast the FCF for FY 2011 etc. years (assume an endless stream Forecast the FCF for FY 2011 etc. years (assume an endless stream of FCF’s)of FCF’s)

• Now calculate the Present Value and use the WACC% as discount Now calculate the Present Value and use the WACC% as discount factor (i)factor (i)

• The present value you have calculated is an estimate for the value The present value you have calculated is an estimate for the value of the company you have chosenof the company you have chosen

• Do it in Excel…and save time…Do it in Excel…and save time…• Have fun! Have fun!

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Ad homework

• You can chose from the following documents

• The full docs you will find on line: www.valueline.com

• The attached summaries are for your convenience

• In case you need more information• Go to the website of your company• Go to search• Search for “investor relations”• Download financial information from

there…

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WC(2003)=14% of SalesWC(2003)=14% of Sales

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WC(2003)=3,6% of SalesWC(2003)=3,6% of Sales

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These calculation techniques you should master…

• We will use them in the near We will use them in the near future to calculate the present future to calculate the present values of future cash flows of values of future cash flows of companiescompanies

• Next week we will discuss Next week we will discuss

Risk and Return Risk and Return Getting the noses in the same direction…