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Unformatted text preview: The Time Value of Money
A. Terminology
1. Compounding
a. Compound interest b. Simple interest 2. Discounting 3. Annuity
A series of fixedsum payments for a specific number of payments.
a. Ordinary annuity b. Annuity due 4. Components of TVM Problems
a. Present Value (PV)
• A single cash flow (lump sum) at time period 0.
b. Future Amounts
• Future Value (FV)  A single cash flow at any future time point
• Annuity Payment (PMT)  series of fixedsum payments
• Cash Flows (CFi) – uneven series of payments
c. rate of return ( i )
• interest rate, discount rate, yield or required rate
d. length of time
• years (n)
• periods per year (m) 4 The Time Value of Money
B. Future Value of Cash Flows
1. Future Value of a Lump Sum [ FV = PV ⎡ FVF i ,nm ⎤ = PV (1 +
⎢
⎥
m
⎣
⎦ i m )nm ] Q1: What is the value of one hundred dollars in one year if it is invested at an annual
yield of 5%? Q2: What is the value of one hundred dollars in three years if it is invested at an annual
yield of 5%? Q3: What is the value of one hundred dollars in three years if it is invested at an annual
yield of 5% compounded monthly? 5 The Time Value of Money
B. Future Value of Cash Flows
2. Future Value of an Ordinary Annuity FVA = ANN (1 + i ) N −1 + ANN (1 + i ) N −2 + K + ANN (1 + i ) + ANN (1 + i ) + ANN
2 1 ⎡ (1 + i )N − 1⎤
⎡N
N −t ⎤
FVA = ANN ⎢∑ (1 + i ) ⎥ = ANN ⎢
⎥
i
⎣ t =1
⎦
⎣
⎦
N
⎡ (1 + i m ) − 1⎤
FVA = ANN ⎢
⎥ = ANN FVAF i m,nm
i
m
⎣
⎦ [ What is the future value of a $100 ordinary annuity which will be received for 3 years at
12%? 3. Future Value of an Annuity Due FVAD = ANN ⎡ FVAF i ,nm ⎤ (1 +
⎢
⎥
m
⎣
⎦ i m ) What is the future value of a $100 annuity due which will be received for 3 years at
12%? 6 The Time Value of Money
C. Present Value of Cash Flows
1. Present Value of a Lump Sum ⎡
⎤
1
PV = FV ⎡ PVF i ,nm ⎤ = FV ⎢
nm ⎥
⎢
⎥
m
⎣
⎦
⎣ (1 + i m ) ⎦
Q1: What is the value today of one hundred and five dollars that will be received at the
end of one year? Assume a 5% interest rate compounded annually. Q2: If you are going to receive $100 three years from today, what is it worth to you
today? Assume a 5% interest rate compounded annually. 7 The Time Value of Money
C. Present Value of Cash Flows
2. Present Value of an Ordinary Annuity PVA = ANN 1
1
1
1
+ ANN
+ ANN
+ K + ANN
1
2
3
(1 + i )
(1 + i )
(1 + i )
(1 + i )N ⎡ 1
1
1
1 ⎤
+
+
+K+
PVA = ANN ⎢
1
2
3
(1 + i )N ⎥
⎣ (1 + i ) (1 + i ) (1 + i )
⎦
1
⎡1 − (1+i )t ⎤
⎡N
1 ⎤
= ANN ⎢
PVA = ANN ⎢∑
⎥
t ⎥
⎢ i ⎥
⎣ t =1 (1 + i ) ⎦
⎣
⎦
1
⎡
⎤
⎢1 − (1 + i )nm ⎥
m
⎥ = ANN PVAF i ,nm
PVA = ANN ⎢
m
i
⎢
⎥
m
⎢
⎥
⎣
⎦ [ Question: What is the present value of a $100 ordinary annuity received each year for 3
years, with annual discounting at 10.0%? 3. Present Value of an Annuity Due ( PVAD = ANN ⎡ PVAF i ,mn ⎤ 1 + i
m
⎢
⎥
m
⎣
⎦ ) Question: What is the present value of $100 annuity due received at the beginning of
each of the next 3 years (starting today) with an annual discount rate of 10.0%? 8 The Time Value of Money
C. Present Value of Cash Flows
4. Present Value Practice Problem
Congratulations, your diligence has paid off and you have just won the Publisher
Clearing House grand prize ONE MILLION DOLLARS (no this is not the thousandth
letter that tells you that you won, you actually won). You will receive the prize money in
ten annual payments of $100,000. If current interest rates are 7%, what is the grand
prize worth to you today? 5. Present Value Relationship
a. The greater the interest rate, the _____ the present value of a cash flow. b. The further into the future the money is received, the ______ the present value. 9 The Time Value of Money
C. Present Value of Cash Flows
6. Present Value of an Uneven Series 1
1
1
1
+ CF2
+ CF3
+ K + CFN
1
2
3
(1 + i )
(1 + i )
(1 + i )
(1 + i )N
PV = CF1 [PVFi ,1 ] + CF2 [PVFi , 2 ] + CF3 [PVFi ,3 ] + K + CFN [PVFi , N ]
PV = CF1 Example: What is the present value of the following cash flows if received at the end of
each year and the current rate of return is equal to 8%?
Year(N)
Cash Flow
1
$1000
2
1000
3
1200
4
2000
5
2000
PV = 1000[PVF0.08,1] + 1000[PVF0.08,2] + 1200[PVF0.08,3] + 2000[PVF0.8,4] + 2000[PVF0.08,5] 10 The Time Value of Money
D. Calculation of Annuity Payments
1. The Mortgage Constant
1
Derivation : PV = ANN ⎡ PVAF i ,nm ⎤ ⇒ PV
= ANN ⇒ PV ⎡ MC i ,nm ⎤ = ANN
⎥
⎢
⎥
⎢
m
m
⎦
⎣
⎦
⎣
⎤
⎡ PVAF
i , nm
⎥
⎢
m
⎦
⎣ ⎡
⎤
⎤
⎢ i m ⎥ = PV ⎡ MC
ANN = PV
i , nm ⎥
⎢
⎢ 1− 1 nm ⎥
m
⎣
⎦
⎢ (1+ i m ) ⎥
⎣
⎦
EXAMPLE: Find the monthly mortgage payments required for a 30 year fixed rate
mortgage with principal of $150,000 and an annual contract rate of 9%. 2. The Sinking Fund Factor
1
= ANN ⇒ FV ⎡ SFF i ,nm ⎤ = ANN
Derivation : FV = ANN ⎡ FVAF i ,nm ⎤ ⇒ FV
⎢
⎥
⎢
⎥
m
m
⎣
⎦
⎣
⎦
⎡ FVAF
⎤
i , nm
⎢
⎥
m
⎣
⎦ ⎡ i
⎤
ANN = FV ⎢ i mnm ⎥ = FV ⎡ SFF i ,nm ⎤
⎢
⎥
(1+ m ) −1 ⎦
m
⎣
⎦
⎣
EXAMPLE: Assume you currently have two years of undergraduate education left and
you plan to attend grad school for another 4 years. After you graduate with your
doctorate, you would like to purchase a home and feel you need a down payment of
$20,000. You believe the average interest rate for this time period to be 8%. Compute
the monthly payment required to accumulate $20,000 by the time you leave grad school. 11 Mortgage Analysis
A. Terminology
1. Mortgage Loan 2. Amortization • The periodic repayment of debt in installments of principal and interest.
3. Typical Fixed Rate Mortgage • Fixed Rate  Constant Payment
• Fully amortizing
• Payments include principal and interest
4. Amortization Table • A table in which the payments are divided into their principal and interest components 12 Mortgage Analysis
B. Mortgage Mechanics
1. What is the amount of the payment necessary to amortize a mortgage loan? 2. What is the amount outstanding (or the proportion outstanding) on a mortgage loan at
any point in time during the mortgage maturity?
a. Outstanding mortgage balance at time t (OMBt) b. Proportion outstanding at the end of period t. c. Proportion paid at the end of period t. 3. Principal paid over a given period of time 4. Interest paid over a given period of time
a. GENERAL METHOD (ALWAYS WORKS) b. SPECIFIC METHOD ONLY WORKS FOR ONE PERIOD OR IN SPECIAL CASES
WHEN THE MORTGAGE BALANCE DOES NOT CHANGE 13 Mortgage Analysis
C. Amortization Tables
Example w/ monthly payments and compounding
Assume that you borrow $80,000 at 8% interest in the form of a Fixed Rate Mortgage
(FRM) with monthly payments and a maturity of 30 years. ● What is the annual debt service? ● What is the outstanding mortgage balance at the end of the first year? ● What is the amount of principal paid over the first year? ● What is the amount of interest paid over the first year?
● How much principal & interest did you pay over the first year? ● How much principal did you pay over the first year? ● How much principal did you pay over the first year? 14 Mortgage Analysis
C. Amortization Tables
Example w/ monthly payments and compounding
Assume that you borrow $80,000 at 8% interest in the form of a Fixed Rate Mortgage
(FRM) with monthly payments and a maturity of 30 years.
● What is the annual debt service in the second year? ● What is the amount of principal paid over the fifth year? ● What is the amount of interest paid over the fifth year? 2. Principal and Interest
● Concept Check: Under which loan will you pay more interest in the first year?
I. PV = 80,000, i = 8%, n = 30, m = 12 (monthly)
II. PV = 80,000, i = 8%, n = 30, m = 1 (annual) 15 Mortgage Analysis
C. Amortization Tables
2. Principal and Interest
● The graphing of payments 3. Monthly vs. Annual Compounding
MTG
PV=80000
i = 8% n = 30
m = 12
n = 30
m=1
n = 15
m = 12
n = 15
m=1 Annual Debt
Service Total Interest &
Principal Total
Principal Total
Interest $211,324.20 $80,000 $131,324.20 $7106.19 $213,185.84 $80,000 $133,185.84 $9174.26 $137,613.90 $80,000 $57,613.90 $140,195.45 $80,000 $60,195.45 $7044.14
(587.01166)(12) (764.52167)(12) $9,346.36 4. Building an amortization table with a financial calculator 16 Practice Problem I:
The Whosits desire to purchase a house, and they open a savings account that pays 5.75
percent interest, with monthly compounding. If they deposit $100 per month beginning 1 month
from now and they must have 20 percent of the price of a house as a down payment, what price
can they pay for a house after saving for 5 years? Practice Problem II:
Mr. and Mrs. Smith are considering the purchase of a house. They can afford to make a house
payment of $500 per month. If the current mortgage interest rate is 12 percent, with monthly
payments for 30 years, and the equity is 20 percent of the purchase price, can they buy a house
costing $60,000? Practice Problem III:
John Brown obtained a 30 year mortgage loan five years ago for $130,000 at 8% interest
compounded monthly (monthly payments). He would like to pay down the mortgage balance by
$5,000, by including an additional check ($5,000) with his 60th payment. Assuming the loan
mortgage maturity is shortened (i.e., payments remain fixed at their previous level), how many
payments will John still need to make after the 60th payment (how many payments remain)? Practice Problem IV:
What is the internal rate of return of the mortgage based on the following information:
Loan:
$100,000
Interest rate:
7 percent
Term:
180 months
Up front costs: 5 percent of loan amount Practice Problem V:
You pay $1,000 per acre for a tract of land, and your opportunity cost (rate) is 7 percent. You
hold the land 8 years and pay $100 in taxes each year. What price per acre must you sell the
land for to break even with your opportunity cost (rate)? Practice Problem VI:
If the present value of the following cash flows equals $10,260 when discounted at an annual
rate of return of 5%, what is the value of the third cash flow? 17 Name:_______________________________(please print)
Student ID: _______________________________
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This note was uploaded on 10/05/2010 for the course FIN 3000 taught by Professor Ackute during the Spring '10 term at Kennesaw.
 Spring '10
 Ackute

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