Mathematics of Finance Solutions to the examples in

Mathematics of Finance Solutions to the examples in this presentation are based on using a Texas Instruments BAII Plus Financial calculator.

Time Value of Money(TVM) $x today or $x in future? A matter of Preference or Risk? $x today? 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.

TVM capture the Opportunity Cost Through: ØCompounding or determining the Future Values based on present $s, and ØDiscounting or determining the Present values based on future $s

TVM capture the Opportunity Cost Ø Compounding ü Future Value of a single amount ü Future Value of an annuity ü Future Value of uneven cash flows Ø Discounting ü Present Value of a single amount ü Present Value of an annuity ü Present Value of uneven cash flows

Compound Interest Compounding is paying interest both on principle and interest. For a 2 -year savings commitment, the FV 1 = PV + (PV x r) = PV (1 + r) FV 2 = PV (1 + r) + PV (1 + i) x r = PV (1 + r)2 FV 1 = 100 + (100 x. 05) = 100 (1 +. 05) = 105 FV 2 = 100 (1 +. 05) + 100 (1 +. 05) x. 05 = 100 (1 +. 05)2 = 110. 25 Note: Present Value = Principal

Future Value on a Timeline An investment of $100 today in a savings account that pays 5% interest, with interest 5 compounded annually, will result in $110. 25 at the end of year 2. 0 $100 PV 5% 1 $105 2 $110. 25 FV

Future Value, General Formula FVn = PV (1+r)n FV 2 = $100 (1. 05)2 = $110. 25 Lets Put The Calculator to Work!

Future Value on TI BAII Plus Turn the calculator on and change the default setting by: Press 2 nd I/Y Enter Press 1 ENTER These keystrokes will change the frequency of compounding to once per year

Future Value on TI BAII Plus Always Press 2 nd, then FV Enter Press 2 N 5 I/Y 100 PV CPT, FV $110. 25

Future Value Example How much money would be in your savings account after 6 years if you deposit $5, 000 today and the bank pay an annual compound interest rate of 7%? 0 7% $5, 000 1 2 3 4 5 6 FV 6

Future Value Solution ü Calculation based on the formula: FVn = PV (1+r)n FV 5 = $5, 000 (1+ 0. 07)6 = $7, 503. 65 ü Calculator keystrokes: 1. 07 yx 6 5000 =

Future Value on TI BAII Plus Always Press 2 nd, then FV Enter Press 6 N 7 I/Yr 5000 PV CPT, FV 7, 503. 65

Present Value Ø Having FV = PV(1 + r)n then: Ø This represents the Discounting process or the process of determining the present value of a single future cash flow.

Present Value (Graphic) If you need to have a $10, 000 down payment on a house 12 years from now, how much must you save today in an account that pays 7% interest, compounded annually? 0 7% 3 6 9 12 $10, 000 PV 0

Present Value on TI BAII Plus Always Press 2 nd, then FV Enter Press 12 N 7 I/Yr 10000 FV CPT, PV 4, 440. 12 Calculator keystrokes: 1. 07 yx 12 = 1/x 10000 =

Computing “n” or “i” knowing PV and FV Ø If John lends Linda $4, 000 today for a return of $6, 154. 50 after 5 years, what rate of annual compound interest does he earn?

Present Value on TI BAII Plus Always Press 2 nd, then FV Enter Press 5 4000 N +/-, PV 6154. 50 FV CPT, I/Y 9. 00%

Frequency of Compounding General Formula: FVn, m = PV 0[1 + (r/m)] mn n: m: r: FVn, m: PV 0: Number of Years Number of Compounding per Year Annual Interest Rate Future Value at Year n Present value of amounts

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, 000 r = 9%/12 = 0. 75% per month n = 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 ) Always Press 2 nd, then FV Enter 72 Press N 3000 PV 0. 75 I/Y CPT, FV $5, 137. 66

Annuities u An Annuity represents a series of equal payments (or receipts) over EQUAL intervals. Ø Annuities Can Be: ü Ordinary (starting at the end of each period) or ü Due (starting at the beginning of each period) Ø Example of Annuities Are: ü Any kind of installment payment for retiring a loan ü Insurance Premiums ü Savings for Retirement

Future Value of an Ordinary Annuity -- FVA 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? FVA 3 = $4, 000(1. 07)2 + $4, 000(1. 07)1 + $4, 000(1. 07)0 = $12, 610 0 7% 1 End of Year $4, 000 2 $4, 000 3 $4, 000 $4, 280 $4, 579. 60 $12, 859. 60 = FVA 3

Future Value (TI BAII Plus) Always Press 2 nd, then FV Enter 3 4000 7 Press N PMT I/Y CPT, FV $12, 859. 60

Present Value of an Ordinary Annuity -- PVA 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? PVA 3 = $2, 500/(1. 06)1 + $2, 500/(1. 06)2 + $2, 500/(1. 06)3 + $2, 500/(1. 06)4 = $8, 662. 76 Yearend 0 1 2 3 4 6% $2, 500 $2, 358. 49 $2, 224. 99 $2, 099. 05 $1, 980. 23 $8, 662. 76 = PVA 3 $2, 500

Present Value on TI BAII Plus Always Press 2 nd, then FV Enter 4 2500 6 Press N PMT I/Y CPT, PV $8, 662. 76

PV of Unequal Cash Flows 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? 0 2 3 $3000 PV 0 1 $5000 7, 000 12%

Unequal Cash Flow Solution 0 2 $3, 000 12% 1 $5, 000 $2, 678. 57 $3, 985. 97 $4, 982. 46 $11, 647. 00 = PV 0 3 $7, 000

Unequal Cash Flow Solution (TI BAII Plus) Press CF 2 nd, then CE/C Enter Press 0 ENTER 3000 ENTER 1 ENTER 5000 ENTER 1 ENTER 7000 ENTER 1 ENTER NPV 12 ENTER $11, 647. 00 CPT 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 Enter 932. 50 Press +/-, PV (. 11 1000) 2= 1000 PMT Always Press 2 nd, then FV FV 20 2 = N CPT, I/Y 5. 945% for 6 months or 11. 89% annually 0 1 2 55 55 ………. … 40 55 1000

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 ü $32, 716. 13 ü $32, 375. 73 ü $556, 280. 63 HELP!

TI BAII Plus Solution to #1 Always Press 2 nd, then FV Enter Press 3 12 = N 9 12 = I/Y 25, 000 PV CPT, FV 32, 716. 13 FV = 25000 (1 +. 0075)36 Click for Next Problem

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 ü $40, 020. 76 ü $45, 637. 57 ü $100, 890. 11 HELP!

TI BAII Plus Solution to #2 Enter Press 10000 Always Press 2 nd, then FV PMT 7 N 12 I/Y CPT PV $45, 637. 57 PVA=10000/(1. 12)1 + 10000/(1. 12)2 +…+ 10000/(1. 12)7 Click for Next Problem 0 1 10000 2 10000 3 10000 4 … 10000. . . 7 10000

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 ü $843. 78 ü $785. 18 ü $388. 33 HELP!

TI BAII Plus Solution to #3 Enter Press 30 PMT 1000 FV 4 I/Y 50 Always Press 2 nd, then FV N CPT, PV Click for Next Problem 0 PVb 1 2 30 30 ………. … 50 30 1000

Excellent! A job well done! Click for Next Problem

Calculating the Future Value ü When the frequency of compounding is more than once per year you should adjust both the discount rate, and the time. ü Determine the future value of single amount. Click to return

The Worth of a Contract ü The worth of any asset is the present value of its future cash flows. ü Terms such as “per year”, “annually”, “every year” are indications that the cash flows are annuities. Click to return

Valuing a Bond ü Consider the coupon payments as annuity and the face value of the bond as a single cash flow at maturity. ü Remember that you should adjust the time, the discount rate, and the interest payments to reflect the semi-annual compounding. Click to return