chapter # 2. a dollar received today is worth more than a dollar received tomorrow › this is...
TRANSCRIPT
Chapter # 2
A dollar received today is worth more than a dollar received tomorrow› This is because a dollar received today can be
invested to earn interest› The amount of interest earned depends on the
rate of return that can be earned on the investment
Obviously, $10,000 today$10,000 today.
You already recognize that there is TIME VALUE TO MONEYTIME VALUE TO MONEY!!
Which would you prefer -- $10,000 $10,000 today today or $10,000 in 5 years$10,000 in 5 years?
TIMETIME allows you the opportunity to postpone consumption and earn
INTERESTINTEREST.
Why is TIMETIME such an important element in your decision?
Compound InterestCompound InterestInterest paid (earned) on any previous
interest earned, as well as on the principal borrowed (lent).
Simple InterestSimple Interest
Interest paid (earned) on only the original amount, or principal borrowed (lent).
FormulaFormula SI = P(i)(n)
SI: Simple InterestP: Deposit today i: Interest Rate per Periodn: Number of Time Periods
Simple Interest(SI) = P(i)(n)=
$1,000(.07)(2)= $140$140
Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year?
FVFV = P + SI = $1,000 + $140= $1,140$1,140
Future ValueFuture Value is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate.
What is the Future Value Future Value (FVFV) of the deposit?
The Present Value is simply the $1,000 you originally deposited. That is the value today!
Present ValuePresent Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate.
What is the Present Value Present Value (PVPV) of the previous problem?
Assume that you deposit $1,000$1,000 at a compound interest rate of
7% for 2 years2 years.
0 1 22
$1,000$1,000
FVFV22
7%
FVFV11 = PP (1+i)1 = $1,000$1,000 (1.07) = $1,070$1,070
Compound InterestYou earned $70 interest on your
$1,000 deposit over the first year.This is the same amount of interest
you would earn under simple interest.
FVFV11 = P P (1+i)1 = $1,000$1,000 (1.07) = $1,070$1,070
FVFV22 = FV1 (1+i)1 = PP (1+i)2 =
$1,000$1,000(1.07)2
= $1,144.90$1,144.90You earned an EXTRA $4.90$4.90 in Year 2 with
compound over simple interest.
Future ValueFuture ValueSingle Deposit (Formula)Single Deposit (Formula)Future ValueFuture ValueSingle Deposit (Formula)Single Deposit (Formula)
Julie Miller wants to know how large her deposit of $10,000$10,000 today will become at a compound annual interest rate of 10% for 5 5 yearsyears.
0 1 2 3 4 55
$10,000$10,000
FVFV55
10%
Calculation based on general formula: FVFVnn = P (1+i)
FVFV55 = $10,000 (1+ 0.10)= $16,105.10$16,105.10
n5
Assume that you need $1,000$1,000 in 2 years.2 years. Let’s examine the process to determine how much you need to deposit today at a discount rate of 7% compounded annually.
0 1 22
$1,000$1,000
7%
PV1PVPV00
PVPV = FVFV22 / (1+i)2 = $1,000$1,000 / (1.07)2 = FVFV22 / (1+i)2 = $873.44$873.44
0 1 22
$1,000$1,000
7%
PVPV00
PVPV = FVFV11 / (1+i)1
PVPV = FVFV22 / (1+i)2
etc.
Julie Miller wants to know how large of a deposit to make so that the money will grow to $10,000$10,000 in 5 years5 years at a Interest rate of 10%.
0 1 2 3 4 55
$10,000$10,000PVPV
10%
Calculation based on general formula: PVPV = FVFVnn / (1+i)n
PVPV = $10,000$10,000 / (1+ 0.10)5 = $6,209.21$6,209.21
The result indicates that a $10,000 future value that will earn 10% annually for 5 years requires a $6,209.21 deposit
today (present value).
Ordinary AnnuityOrdinary Annuity: Payments or receipts occur at the end of each period.
Annuity DueAnnuity Due: Payments or receipts occur at the beginning of each period.
An AnnuityAn Annuity represents a series of equal payments (or receipts) occurring at regular intervals.
Student Loan Payments Car Loan Payments Insurance Premiums Mortgage Payments Retirement Savings
0 1 2 3
$100 $100 $100
(Ordinary Annuity)EndEnd of
Period 1EndEnd of
Period 2
Today EqualEqual Cash Flows Each 1 Period Apart
EndEnd ofPeriod 3
0 1 2 3
$100 $100 $100
(Annuity Due)BeginningBeginning of
Period 1BeginningBeginning of
Period 2
Today EqualEqual Cash Flows Each 1 Period Apart
BeginningBeginning ofPeriod 3
FVAFVAnn = R(1+i)n-1 + R(1+i)n-2 + ... + R(1+i)1 + R(1+i)0
R R R
0 1 2 n n n+1
FVAFVAnn
R = Periodic Cash Flow
Cash flows occur at the end of the period
i% . . .
FVAFVA33 = $1,000(1.07)2 + $1,000(1.07)1 +
$1,000(1.07)0
= $1,145 + $1,070 + $1,000 = $3,215$3,215
$1,000 $1,000 $1,000
0 1 2 3 3 4
$3,215 = FVA$3,215 = FVA33
7%
$1,070
$1,145
Cash flows occur at the end of the period
The future value of an ordinary annuity can be viewed as occurring at the endend of the last cash flow period, whereas the future value of an annuity due can be viewed as occurring at the beginningbeginning of the last cash flow period.
FVAFVAnn = R (FVIFAi%,n) FVAFVA33 = $1,000 (FVIFA7%,3)
= $1,000 (3.215) = $3,215$3,215Period 6% 7% 8%
1 1.000 1.000 1.0002 2.060 2.070 2.0803 3.184 3.215 3.2464 4.375 4.440 4.5065 5.637 5.751 5.867
FVADFVADnn = R(1+i)n + R(1+i)n-1 + ... + R(1+i)2 +
R(1+i)1 = FVAFVAn n (1+i)
R R R R R
0 1 2 3 n-1n-1 n
FVADFVADnn
i% . . .
Cash flows occur at the beginning of the period
FVADFVAD33 = $1,000(1.07)3 + $1,000(1.07)2 +
$1,000(1.07)1
= $1,225 + $1,145 + $1,070 = $3,440$3,440
$1,000 $1,000 $1,000 $1,070
0 1 2 3 3 4
$3,440 = FVAD$3,440 = FVAD33
7%
$1,225
$1,145
Cash flows occur at the beginning of the period
FVADFVADnn = R (FVIFAi%,n)(1+i)FVIFA i ni n = (1+i)n -1/ i
FVADFVAD33 = $1,000 (FVIFA7%,3)(1.07)= $1,000 (3.215)(1.07) =
$3,440$3,440Period 6% 7% 8%1 1.000 1.000 1.0002 2.060 2.070 2.0803 3.184 3.215 3.2464 4.375 4.440 4.5065 5.637 5.751 5.867
PVAPVAnn = R/(1+i)1 + R/(1+i)2
+ ... + R/(1+i)n
R R R
0 1 2 n n n+1
PVAPVAnn
R = Periodic Cash Flow
i% . . .
Cash flows occur at the end of the period
PVAPVA33 = $1,000/(1.07)1 + $1,000/(1.07)2 +
$1,000/(1.07)3
= $934.58 + $873.44 + $816.30 = $2,624.32$2,624.32
$1,000 $1,000 $1,000
0 1 2 3 3 4
$2,624.32 = PVA$2,624.32 = PVA33
7%
$ 934.58$ 873.44 $ 816.30
Cash flows occur at the end of the period
The present value of an ordinary annuity can be
viewed as occurring at the beginningbeginning of the first cash flow
period, whereas the present value of an annuity due can be viewed as occurring at the endend
of the first cash flow period.
PVAPVAnn = R (PVIFAi%,n)
PVAPVA33 = $1,000 (PVIFA7%,3)= $1,000 (2.624) =
$2,624$2,624Period 6% 7% 8%1 0.943 0.935 0.9262 1.833 1.808 1.7833 2.673 2.624 2.5774 3.465 3.387 3.3125 4.212 4.100 3.993
PVADPVADnn = R/(1+i)0 + R/(1+i)1 + ... + R/(1+i)n-1 = PVAPVAn n (1+i)
R R R R
0 1 2 n-1n-1 n
PVADPVADnn
R: Periodic Cash Flow
i% . . .
Cash flows occur at the beginning of the period
PVADPVADnn = $1,000/(1.07)0 + $1,000/(1.07)1 + $1,000/(1.07)2 = $2,808.02$2,808.02
$1,000.00 $1,000 $1,000
0 1 2 33 4
$2,808.02 $2,808.02 = PVADPVADnn
7%
$ 934.58$ 873.44
Cash flows occur at the beginning of the period
PVADPVADnn = R (PVIFAi%,n)(1+i)
PVADPVAD33 = $1,000 (PVIFA7%,3)(1.07) = $1,000 (2.624)(1.07) =
$2,808$2,808Period 6% 7% 8%1 0.943 0.935 0.9262 1.833 1.808 1.7833 2.673 2.624 2.5774 3.465 3.387 3.3125 4.212 4.100 3.993
Julie Miller will receive the set of cash flows below. What is the Present Value Present Value at a discount rate of 10%10%? 0 1 2 3 4 55
$600 $600 $400 $400 $100$600 $600 $400 $400 $100
PVPV00
10%10%
0 1 2 3 4 55
$600 $600 $400 $400 $100$600 $600 $400 $400 $10010%
$545.45$545.45$495.87$495.87$300.53$300.53$273.21$273.21$ 62.09$ 62.09
$1677.15 $1677.15 = = PVPV00 of the Mixed Flowof the Mixed Flow
General Formula:FVn = PVPV00(1 + [i/m])mn
n: Number of Yearsm: Compounding Periods per Year
i: Annual Interest RateFVn,m: FV at the end of Year n
PVPV00: PV of the Cash Flow today
Julie Miller has $1,000$1,000 to invest for 2 years at an annual interest rate of 12%.
Annual FV2 = 1,0001,000(1+ [.12/1])(1)(2)
= 1,254.401,254.40Semi FV2 = 1,0001,000(1+ [.12/2])(2)(2)
= 1,262.481,262.48
Qrtly FV2 = 1,0001,000(1+ [.12/4])(4)(2)
= 1,266.771,266.77Monthly FV2 = 1,0001,000(1+ [.12/12])(12)(2)
= 1,269.731,269.73
Daily FV2 = 1,0001,000(1+[.12/365])(365)(2) = 1,271.201,271.20
The term loan amortization refers to the determination of equal periodic loan payments.
Julie Miller is borrowing $10,000 $10,000 at a compound annual interest rate of 12%. Amortize the loan if annual payments
are made for 5 years.Step 1: Payment
PVPV00 = R (PVIFA i%,n)
$10,000 $10,000 = R (PVIFA 12%,5)
$10,000$10,000 = R (3.605)
RR = $10,000$10,000 / 3.605 = $2,774$2,774
End of Year
Payment Interest Principal Ending Balance
0 --- --- --- $10,000
1 $2,774 $1,200 $1,574 8,426
2 2,774 1,011 1,763 6,663
3 2,774 800 1,974 4,689
4 2,774 563 2,211 2,478
5 2,775 297 2,478 0
$13,871 $3,871 $10,000
[Last Payment Slightly Higher Due to Rounding]
2. Calculate Debt Outstanding:2. Calculate Debt Outstanding: The quantity of outstanding debt may be used in financing the day-to-day activities of the firm.
1.1. Determine Interest Expense: Determine Interest Expense: Interest expenses may reduce taxable income of the firm.