Mathematics of Finance
Solutions to the examples in this presentation are based on using a Texas
Instruments BAII Plus Financial calculator.
$x today$x today?
BUT WHY?
Postponement of today’s opportunities for investments or consumption to the future would result in OPPORTUNITY COST. TVM captures and explains such lost opportunities.
$x today$x today or $x in future?$x in future?
A matter of Preference or Risk?
Time Value of Money(TVM)
TVM capture the Opportunity Cost
Through:Compounding or determining the
Future Values based on present $s, and
Discounting or determining the Present values based on future $s
CompoundingFuture Value of a single amountFuture Value of an annuityFuture Value of uneven cash flows
DiscountingPresent Value of a single amountPresent Value of an annuityPresent Value of uneven cash flows
CompoundingFuture Value of a single amountFuture Value of an annuityFuture Value of uneven cash flows
DiscountingPresent Value of a single amountPresent Value of an annuityPresent Value of uneven cash flows
TVM capture the Opportunity Cost
Compound Interest
Compounding is paying interest both on principle and interest. For a 2-year savings commitment, the
FV1 = PV + (PV x r) = PV (1 + r)
FV2 = PV (1 + r) + PV (1 + i) x r = PV (1 + r) (1 + r) = PV (1 + r)2
FV1 = 100 + (100 x .05) = 100 (1 + .05) = 105
FV2 = 100 (1 + .05) + 100 (1 + .05) x .05 = 100 (1 + .05)2
= 110.25Note: Present Value = Principal
An investment of $100 today in a savings $100 today in a savings account that pays 5account that pays 5% interest, with interest compounded annually, will result in $110.25 at the end of year 2.
Future Value on a Timeline
0 1 2
$100$100
FVFV
5%
$105$105 $110.25$110.25
PVPV
FVFVnn = PVPV (1+r)n FVFV22 = $100$100 (1.05)2 = $110.25$110.25
Future Value, General Formula
Lets Put The Calculator to Work!
Future Value on TI BAII Plus
Turn the calculator on and change the default setting by:
2nd
Enter
1
I/Y
Press
ENTER
Press
These keystrokes will change the frequency of compounding to once
per year
2
5
100
N
I/Y
PV
CPT, FV
Always Press 2nd, then FV
PressEnter
$110.25
Future Value on TI BAII Plus
How much money would be in your savings account after 6 years if you deposit $5,000$5,000 today and the bank pay an annual compound interest rate of 7%?
Future Value Example
0 1 2 3 4 5 65 6
$5,000$5,000
FVFV66
7%
Calculator keystrokes: 1.07 yx 6 5000 =
Future Value Solution
Calculation based on the formula:FVFVnn = PV (1+r)n
FVFV55 = $5,000 (1+ 0.07)6
= $7,503.65$7,503.65
7
5000
I/Yr
PV
CPT, FV
N6
Always Press 2nd, then FV
PressEnter
7,503.65
Future Value on TI BAII Plus
Present ValuePresent Value
Having FV = PV(1 + r)n then:
This represents the Discounting process or the process of determining the present value of a single future cash flow.
nrFVPV
)1(
If you need to have a $10,000$10,000 down payment on a house 12 years from now, years from now, how much must you save today in an account that pays 7% interest, compounded annually?
$10,000$10,000
Present Value (Graphic)
0 3 6 6 9 127%
PVPV00
Present Value on TI BAII Plus
7
10000
I/Yr
FV
CPT, PV
N12
Always Press 2nd, then FV
PressEnter
4,440.12
Calculator keystrokes:
1.07 yx 12 = 1/x 10000 =
Computing “n” or “i” knowing PV and FV
If John lends Linda $4,000 today for a If John lends Linda $4,000 today for a return of $6,154.50 after 5 years, what rate return of $6,154.50 after 5 years, what rate of annual compound interest does he earn?of annual compound interest does he earn?
4000
6154.50
+/-, PV
FV
CPT, I/Y
N5
Always Press 2nd, then FV
PressEnter
9.00%
Present Value on TI BAII Plus
General Formula:
FVn,m = PVPV00[1 + (r/m)] mn
n: Number of Years
m: Number of Compounding per Year
r: Annual Interest Rate
FVn,m: Future Value at Year n
PVPV00: Present value of amounts
Frequency of Compounding
Frequency of Compounding
Example:If your deposit of $3,000 in a savings account, paying monthly compounded interest based on a 9% annual rate, is maintained for six years how much will be in the account at that time?
PV = $3,000r = 9%/12 = 0.75% per monthn = 6 x 12 = 72 months
Solution, based on formula:
FV= PV (1 + r)n
= 3,000(1.0075)72
= 5,137.66
Calculator Keystrokes:
1.0075 yx 72 3000 =
Frequency of Compounding on (TI BAII Plus )
3000
0.75
PV
I/Y
CPT, FV
N72
Always Press 2nd, then FV
PressEnter
$5,137.66
Annuities
Annuities Can Be:Ordinary (starting at the end of each period) orDue (starting at the beginning of each period)
Example of Annuities Are: Any kind of installment payment for retiring a loan Insurance Premiums Savings for Retirement
An AnnuityAn Annuity represents a series of equal payments (or receipts) over EQUAL intervals.
A plan to save $4,000 a year at the end of each year for three years would result in how much savings, considering that your savings account pays 7% interest, compounded annually?
FVAFVA33 = $4,000(1.07)2 + $4,000(1.07)1 + $4,000(1.07)0
= $12,610$12,610
Future Value of an Ordinary Annuity -- FVA
0 1 2 3
$4,000 $4,000 $4,000
$12,859.60 = FVA$12,859.60 = FVA3
End of Year
7%
$4,280
$4,579.60
Future Value (TI BAII Plus)
4000
7
PMT
I/Y
CPT, FV
N3
Always Press 2nd, then FV
PressEnter
$12,859.60
Jamshid was approved for a business loan, which required $2,500 annual payment at the end of each next 4 years. The loan carried an annual interest rate of 6%. What was the amount of this loan?
PVAPVA33 = $2,500/(1.06)1 + $2,500/(1.06)2 + $2,500/(1.06)3 + $2,500/(1.06)4
= $8,662.76$8,662.76
Present Value of an Ordinary Annuity -- PVA
$2,500 $2,500 $2,500 $2,500
0 1 2 3 3 4Yearend
6%
$8,662.76 = PVA$8,662.76 = PVA33
$2,358.49$2,224.99 $2,099.05$1,980.23
Present Value on TI BAII Plus
2500
6
PMT
I/Y
CPT, PV
N4
Always Press 2nd, then FV
PressEnter
$8,662.76
Your investment advisor recommends a security that provides $3,000, $5,000, and $7,000 respectively at the end of each of the next 3 years. If you require 12% return on this security, how much would you be willing to pay for it?
PV of Unequal Cash Flows
0 1 2 3
$3000 $5000 7,000 $3000 $5000 7,000
PVPV00
12%12%
Unequal Cash Flow Solution
0 1 2 3
$3,000 $5,000 $7,000$3,000 $5,000 $7,00012%
$2,678.57$2,678.57$3,985.97$3,985.97$4,982.46$4,982.46
$11,647.00 $11,647.00 = = PVPV00
Unequal Cash Flow Solution (TI BAII Plus)
Enter
0
3000
5000
Press
7000
Press CF2nd, then CE/C
ENTER
ENTER
ENTER
ENTER
1
1
1
ENTER
ENTER
ENTER
ENTER
NPV
12 CPT
$11,647.00 Frequency of the cash flows
Computing Yield to Maturity
DXL Industries bond is currently selling for $932.50. This bond is having a coupon interest rate of 11%, and will mature in 20 years. Considering that the bond’s face value is $1,000 and pays interest semiannually, what is the yield to maturity (YTM) on this bond?
YTM Solution on TI BAII Plus
932.50 +/-, PV
1000
(.11 1000) 2=
20 2 =
PMT
FV
N
CPT, I/Y
5.945% for 6 months or 11.89% annually
0 1 2 ……….… 40 55 55 55
1000
Always Press
2nd, then FV
Enter Press
Check your command of the Concepts
Click one of the following problems
1
2
3
Problem #1
Morgan deposited $25,000 in a new savings account that is paying 9% annual interest rate compounded monthly. She will not be able to withdraw her deposit within the next 3 years. What will be the size of deposits in her account in 3 years?
Problem 1 - Select one
$32,716.13$32,375.73$556,280.63
HELP!
TI BAII Plus Solution to #1TI BAII Plus Solution to #1
I/Y
N
PV
CPT, FV
32,716.13
25,000
9 12 =
3 12 =
Always Press 2nd, then FV
PressEnter
Click for Next Problem
FV = 25000 (1 + .0075)36
Problem #2
You currently receive $10,000 per year on a contract. You expect it to run another 7 years. Someone wants to buy the contract from you. If you can earn 12% on other investments of this quality, how much would you be willing to sell the contract for?
Possible Answers - Problem 2
$40,020.76$45,637.57$100,890.11
HELP!
TI BAII Plus Solution to #2TI BAII Plus Solution to #2
PMT
PVA=10000/(1.12)1 + 10000/(1.12)2 +…+ 10000/(1.12)7
10000
I/Y
N
PV
7
12
CPT
Always Press 2nd, then FV
PressEnter
0 1 2 3 4 … 7
10000 10000 10000 10000 ... 10000
$45,637.57
Click for Next
Problem
Problem #3
Thompson Corp. has issued a bond with a face value of $1,000. The bond carries a coupon interest rate of 6%, pays interest semi-annually, and will mature in 25 years. How much would you pay for this bond if your required return on similar investments is 8%?
Possible Solutions - Problem 3
$843.78$785.18$388.33
HELP!
TI BAII Plus Solution to #3TI BAII Plus Solution to #3
Enter Press
1000
30
4
50
PMT
FV
I/Y
N
CPT, PV
30 30 30 1000
0 1 2 ……….… 50
Always Press 2nd, then FV
n
t
ntb rr
IPV
)1(
1000
)1(
1
PVb
Click for Next
Problem
Excellent!
A job well done!
Click for Next Problem
Calculating the Future Value
When the frequency of compounding is more than once per year you should adjust both the discount rate, and the time.
Determine the future value of single amount.
Click to return
The Worth of a Contract
The worth of any asset is the present value of its future cash flows.
Terms such as “per year”, “annually”, “every year” are indications that the cash flows are annuities.
Click to return
Valuing a Bond
Consider the coupon payments as annuity and the face value of the bond as a single cash flow at maturity.
Remember that you should adjust the time, the discount rate, and the interest payments to reflect the semi-annual compounding.
Click to return
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