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Unformatted text preview: L ecture 1 Introduction FINANCE IS THE INTERACTION OF : TIME MONEY and UNCERTAINTY All ﬁnancial decisions can be mapped onto % X1
0 1 % X2
2 3 ... X0 % X3 1 Our focus in this course is on:
corporate ﬁnancial decisions the central role of the ﬁnancial manager in creating value via these decisions Business Organizations: Sole Proprietorships Partnerships Limited liability companies Corporations Sole Proprietorships 2 Partnerships and L LCs 3 Corporate Revenues C orporation legal entity unlimited life limited liability of owners separation of ownership and management ʻdouble taxationʼ 4 We assume that the environment for ﬁnancial decision m aking is c haracterized as: Perfect Markets: I n perfect capital markets investors can trade existing
ﬁnancial claims freely, without transaction costs or other restrictions. No single trader can affect market prices b y his/her activities. E fﬁcient Markets: A market is efﬁcient if at each point in time prices fully
reﬂect all available information. Complete Markets: I n a complete market investors can o btain any
pattern of payoffs o ver time. The Time Value of Money Compound Interest Formulas Cn: Future value at end of n periods of C0 dollars today C1 = C0 + rC0 (deposit C0 for 1 period) C1 = C0(1 + r) C2 = C1 + rC1 (deposit C1 for 1 period) C2 = C1(1 + r) = C0(1 + r)2 Recursively Cn = C0(1 + r)n FVn = PV0(1 + r)n PV0 = FVn/(1 + r)n 5 Example: $ 100 at 5% for 5 years Year 1 2 3 4 5 Beginning A mount 1 00 1 05 1 10.25 1 15.76 1 21.55 Interest 5 5 .25 5 .51 5 .79 6 .08 Ending A mount 1 05 1 10.25 1 15.76 1 21.55 1 27.63 C5 = 100(1.05)5 = 100 ! 1.2763 = 127.63 D oubling your money Let Cn = 2C0 n* = ln2/ln(1 + r) = Rule of 72: n* =
.69315 ln(1 + r ) 72 100 r Recommendation: Use your calculator! R ules of Time Travel w ith Money Only cash flows that occur at the same point in time can be compared To move a cash flow forward  c ompound 6 Another example: Present Values/Discounting FVn = PV0(1 + r)n PV0 = FVn/(1 + r)n Present values are ADDITIVE PV(C1,C2,...,Cn) = PV1(C1) + PV2(C2) + ... + PVn(Cn) =
! PVt(Ct)
n t =1 To move a cash ﬂow backward  discount 7 Example : "Claim X" : $100 a year at the end of each year for 10 years. "Claim Y" : $900 at the end of year 3. Which one would you choose? Your choice depends on the interest rate Interest rate (%) 0 2 5 8 10 12 PV(X) ($) 1 000 8 98 7 72 6 71 6 14 5 65 PV(Y) ($) 9 00 8 48 7 77 7 14 6 76 6 41 The effect of compounding frequency : Put $1.00 in the bank today. At what annual percentage rate of return will you get an effective return of $1.08 i n one year? Compounding Annual Effective Annual Rate Percentage Rate A nnual Semiannual Quarterly M onthly Daily Continuous 8 .0000% 7 .8461% 7 .7706% 7 .7208% 7 .6968% 7 .6961% (1 + .080000/1)1 = 1.08 (1 + .078461/2)2 = 1.08 (1 + .077706/4)4 = 1.08 (1 + .077208/12)12 = 1.08 (1 + .076968/365)365 = 1.08 e.076961 = 1 .08 8 Continuous Compounding :
If the Annual Percentage Rate ( APR) is r% and funds are compounded m times a year, the effective annual rate of interest, (EAR), im is 1 + im = (1 + r/n)m As the number of periods m ! " we have continuous compounding. 1
n !" + i" = er
m note Lim (1 + r/m) = er e = 2 .7183 . .. APRs and E ARs r(%)(APR) 5 10 15 20 i! (%)(EAR) 5.13 10.52 16.18 22.14 A dramatic contrast : Future value of $100 @ 20% per year after 50 years is $910,043.82 Future value of $100 @ 20% per year compounded continuously after 50 years is $2,202,646.58 ΔFuture value =1,292,602.76! 9 Example : ten years ago, a company started with an investment of $100,000. Over time, the company has invested in various projects, with the following results To what amount has the investment grown? Annuities : a ﬁxed number of level,regular cash ﬂows . Let C be the cashflow received at time t = 1,2,...,n The present value of this payment stream can be written as:
C C C PV(An) = (1 + r) + (1 + r) 2 + ... + (1 + r) n
N ="
!
n =1 Cn (1 + r) n ! Useful shortcuts : Using the standard technique to sum a finite geometric series, the present value of an annuity is
n C# # 1 & & %1 " % (( P V ( An) = r % $ 1 + r ' ( $ ' The Future (compounded) value of the annuity at ! time t = n is: FV(An)= PV(An)(1 + r)n
n FV(An)= r ((1 + r) " 1) C ! 10 Perpetuities: a level series of cash ﬂows that last forever PRESENT VALUE OF A PERPETUITY This is a special case when n ! " C C C + + ... + + ... PV(A")= (1 + r) (1 + r) 2 (1 + r) n
C P V ( A") = r # # Valuing a perpetuity when cash flows grow at constant rate
1 C C(1 + g) C (1 + g) n #1 PV(GA!)= (1 + r) + (1 + r ) 2 + ... + (1 + r) n + ... Simplifying,
" C
P V ( G A!)= r # g Assumption: g < r otherwise the sum $ ! Can you write an "xpression for an annuity when cash e flows grow? i.e., what is PV(GAn)? TVM: Applications 11 Mortgage Payments: GIVEN: Length of the mortgage's life: 25 years Payments made once a year Interest rate: 8% Amount ﬁnanced: $300,000 What will be the required annual payments? Analysis: Let the annual payments be $X. Clearly, the above problem is identical to an annuity of $X a year for 25 years with r = 8% and PV(A25)= 300,000. 25 year 8% mortgage on a $300,000 house : distribution of payments
Year Total Payment Interest Payments Amount % of Total 1 2 5 10 15 20 25 2 8,100 2 8,100 2 8,100 2 8,100 2 8,100 2 8,100 2 8,100 2 4,000 2 3,670 2 2,520 1 9,900 1 6,050 1 0,390 2 ,080 8 5.4 8 4.2 8 0.1 7 0.8 5 7.1 3 7.0 7 .4 4 ,100 4 ,430 5 ,580 8 ,200 1 2,050 1 7,710 2 6,020 Principal Returned Amount % of Total 1 4.6 1 5.8 1 9.9 2 9.2 4 2.9 6 3.0 9 2.6 295,900 291,470 275,890 240,520 188,550 112,200 0 Amt. of Loan Outstanding 12 VALUE OF A TAX DEFERRED SAVINGS PLAN FOR RETIREMENT (an IRA). Starting at age 29, if y ou invest $2000 in an IRA yielding 10%, what cash ﬂow can you expect at retirement ?
FV(A35)= 2000 [(1.1)35  1] = $542,049
.1 IRA investments and interest income are tax free. Taxes are paid on withdrawal. Assume your tax rate at retirement is 30%. What aftertax annuity can you expect for 20 years? 542,049 = X & " 1 $ 20 ) 1! .1 ( # 1.1% + ' * or X = $63,663. Annuity is based on taxfree interest because taxes are paid on withdrawal. The aftertax annuity = 63,663 (1  0.3) = 44,568 Alternatively… Suppose you invested in an ordinary savings account and your tax rate during your working years is 50%. How would the above analysis change? What annuity can you expect? Analysis Aftertax value of investment 2,000 x 0.5 = 1,000 Aftertax interest rate during your working years 10% (1  0.5) = 5% Balance in your account upon retirement: V35 =
1000 (1.05)35 ! 1 = $90,320 .05 [ 13 Analysis Aftertax interest during retirement 1 0% ( 1  0 .3) = 7 % Twenty year annuity during retirement set or 9 0,320 =
X & " 1 $ 20 ) (1 ! + .07 ' # 1.07 % * X = $ 8,526 Note that the annuity is compounded at the aftertax interest rate because taxes are paid on the interest income when it is realized. The advantage o f tax free compounding SUMMARY IRA Balance at Retirement AfterTax A nnuity $ 542,049 $ 44,568 Savings Account $ 90,320 $ 8,526 14 ...
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