Time Value

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Chapter 3

• Time Value
• of Money

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Time Value of Money

• 100 in hand today has a higher value than 100
  to be received 1 year from today. Why?
• The 100 in hand today can be invested and grow
  to more than 100 in 1 year.

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Earning Interest

• Simple Interest Interest is earned on principal
  only.
• Compound Interest Interest is earned on both
  principal and previously accumulated interest.
• All problems in Ch 3 involve compound interest.

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Compound Interest

• Adding compound interest to a cash flow over a
  period of time to find the sum of the principal
  and interest is finding FUTURE VALUE
• Backing compound interest out of a cash flow over
  a period of time to find the amount of principal
  only is finding PRESENT VALUE

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Types of Cash Flows

• Single cash flow a one time cash flow that
  occurs now or some time in the future
• Series of unequal cash flows will probably be
  treated individually as single cash flows and
  then summed
• Series of equal cash flows annuities

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Annuities

• Equal, repeating cash flow that occurs at regular
  time intervals e.g., annually, monthly , weekly,
  etc.
• Annuity due Cash flows occur at beginning of
  period.
• Ordinary annuity Cash flows occur at end of
  period.

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Financial Calculator BA II Plus

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Keys for Time Value of Money

• N of periods over which compound interest is
  earned. For annuity problems, N Total number of
  repeating cash flows.
• I or I/Y Annual interest rate, entered as a
  whole number.
• PV Present value principal value with no
  compound interest included.
• PMT Equal, repeating cash flow.
• FV Future value sum of principal plus compound
  interest earned.

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Settings for Annuities

• Ordinary Annuity
• Hit 2nd BGN. END should show up on your screen.
• Annuity Due
• To change from ordinary annuity to annuity due,
  hit 2nd BGN and 2nd SET. BGN should now show on
  your screen.
• 2nd BGN, 2nd SET will change back and forth
  from END to BGN.

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P/Y and C/Y

• P/Y Payments per year (how many cash flows
  occur in 1 year)
• C/Y Compounding per year (how many times in 1
  year compound interest is added in or backed out)
• Start with these at 1. They will need to be
  changed as problems get more complicated.

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Changing P/Y C/Y

• P/Y is 2nd function of I/Y. Comes preset from
  factory at 12. To change it to 1, pull up P/Y,
  hit 1, then hit ENTER (top row). Should now say
  P/Y 1.
• With P/Y still on screen, hit ? or ? to access
  C/Y. When you change P/Y, C/Y is automatically
  changed to the same setting. Should say C/Y 1.

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• Sometimes, you want P/Y and C/Y at different
  settings. Change P/Y to its desired value then
  access C/Y and change it using the ENTER key (top
  row). Changing C/Y does not affect the setting
  for P/Y (even though the reverse is true).

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Solving Time Value of Money Problems

• Look at given variables
• Will be given 4 out of 5 variables (N, I, PV,
  PMT, FV)
• Must solve for the 5th
• Be careful - these calculators keep the last
  value entered stored. If you dont enter a new
  variable, the last one entered will automatically
  be used.

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Decisions to Make

• Does problem involve adding compound interest in
  and looking for sum of principal and interest at
  some point in the future?
• Does problem involve backing compound interest
  out of a future cash flow(s) and looking for
  total principal only?

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• Does cash flow occur just once or is it
  repeating?
• If repeating, is it the same amount each time?
• Only enter a value for PMT when dealing with
  annuity problems otherwise, enter a zero for it.

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Example 1

• If you deposit 500 today into an account that
  earns 5 annual interest, compounded annually,
  how much will be in the account at the end of 10
  years?
• Finding FV of a single cash flow.

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Example 2

• If you deposit 500 today, 750 at the end of the
  year, and 1,000 at the end of 2 years from now,
  how much will be in the account at the end of 10
  years if the account earns 5 annual interest,
  compounded annually?
• Finding FV of a series of unequal cash flows.

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Example 3

• If you deposit 500 at the end of each year for
  the next 10 years into an account earning 5
  annual interest, compounded annually, how much
  will be in the account at the end of 10 years?
• Finding FV of an ordinary annuity.

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Example 4

• You expect to receive 100,000 from a trust fund
  at the end of 5 years. If interest can be earned
  at an annual rate of 6, compounded annually,
  what is the 100,000 worth to you today?
• Finding PV of a single cash flow.

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Example 5

• You estimate that a proposed project will provide
  cash inflows of 20,000, 30,000, 40,000 and
  50,000, respectively, at the end of each of the
  next 4 years. If annual interest rates are 6,
  compounded annually, what is the present value of
  this cash flow stream?
• Finding PV of unequal series of CF.

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Example 6

• You estimate that a proposed project will provide
  cash inflows of 35,000 per year at the end of
  each year for the next 4 years. If annual
  interest rates are 6, compounded annually, what
  is the present value of this cash flow stream?
• Finding PV of an ordinary annuity.

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Frequently Compounded Interest

• So far, we have assumed that interest is
  compounded once a year (annually)
• However, interest may be compounded more
  frequently than once a year
• Annual - 1 time per year
• Semiannual - 2 times per year
• Quarterly - 4 times per year
• Monthly - 12 times per year
• Weekly - 52 times per year
• Daily - 365 times per year

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C/Y Function on Calculator

• ?Changing the C/Y function will allow you to
  adjust for different frequencies of compounding
• Make sure P/Y reflects the number of payments per
  year (which may be different from the compounding
  frequency)

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Example 7

• You have deposited 1,000 into an account that
  will pay 7 annual interest, compounded
  semiannually. How much will be in the account
  after 6 years?

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Example 8

• You will receive 20,000 in 8 years. If you can
  earn interest at an annual rate of 9, compounded
  daily, what is the 20,000 worth to you today?

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Example 9

• You are going to make quarterly deposits of 500
  into an account that will pay an annual rate of
  6, compounded quarterly. How much will be in the
  account after 30 years?

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Example 10

• You plan to deposit 500 per year into an account
  that will pay 6 interest, compounded quarterly.
  How much will be in the account after 15 years

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Effective Annual Rate of Interest

• Simple formula converts any frequently compounded
  rate of interest into an equivalent annually
  compounded rate of interest
• EAR (1 i/m)m - 1
• m number of times interest is compounded in 1
  year
• i stated annual rate of interest

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Example 11

• Which deposit has the better rate?
• Bank A 8 annual rate, compounded semiannually
• Bank B 7.9 annual rate, compounded weekly
• Bank C 7.8 annual rate, compounded daily

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Finding Rates of Return

• For simple interest, can use formula introduced
  in Chapter 1 ROR (Ending value - Starting
  value)/ Starting Value
• For compound interest, MUST use Time Value of
  Money (All problems in Ch 3 involve compound
  interest)
• Enter N, PV, PMT, FV and solve for I

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• WARNING!
• Calculator cant find an interest rate unless it
  can identify outflows vs. inflows
• Outflows negative inflows positive
• Enter the PV as the negative number

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Example 12

• You deposited 1,000 into an account. After 5
  years, the balance is 1,762.34. Find average
  annual rate of interest earned.

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Example 13

• You purchased 1000 shares of XYZ stock in 1999
  for 35 per share. The stock has split twice over
  the past 10 years (2/1 splits). The stock now
  sells for 70 per share. Find the average annual
  rate of return over the past 10 years.

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Example 14

• If you invest 50,000 today, you will receive 10
  year-end payments of 7,500. What average annual
  rate of interest are you earning?

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Example 15

• What average annual rate of interest would you
  need to earn to make 40 year-end deposits of
  2,000 grow to 1,000,000 by the end of the 40th
  year?

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LOANS

• Funds received from a loan PRINCIPAL (amount of
  loan)
• Principal will be repaid by a series of equal
  payments (PMT)
• PMT includes repayment of principal and interest
  charges
• Principal PV (sum of PMTS with interest backed
  out)

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Monthly Repayments

• Most loans involve monthly payments
• P/Y needs to be set on 12 for monthly payments
• N total number of monthly payments
• Lenders charge interest monthly
• C/Y needs to be set on 12
• I stated annual rate (whole number)

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Mortgages

• Loan to buy a property
• Lender holds title to property until loan is paid
  off
• Borrower normally has to pay part of purchase
  price in cash - Down Payment
• Down payment is of loan (e.g.20)

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Example 16

• Purchase price 450,000
• Down payment requirement 20
• Length (term) of loan 30 years
• Interest rate annual rate of 8, compounded
  monthly
• Find monthly payment

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Escrow

• Monthly payment P I
• Some lenders require payments for an escrow
  account in addition to P I
• Property taxes and homeowners insurance are paid
  from the escrow account
• Total pmt P, I, Escrow

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• If you have a fixed interest rate loan, monthly
  payment for P I stays constant
• Escrow is adjusted as taxes and insurance rates
  change
• Total monthly payment usually goes up each year

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Qualifying for a Mortgage

• As a rule of thumb, lenders say your total
  monthly debt should not exceed 1/3 of your gross
  monthly income
• Total monthly debt includes credit cards, car
  loans, student loans, etc. as well as the
  mortgage

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Total Interest Paid over Loan

• Amount of monthly payment x total number of
  payments total paid on loan
• Total paid - principal borrowed total interest
  paid over life of loan
• Often pay more in interest than you borrowed in
  principal

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Splitting PMT into P I

• How does lender know how much of your monthly
  payment to use to repay principal and how much
  its getting for interest charges?
• How monthly payment split between P I
  Amortization Schedule

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Format of Amortization Schedule

• Month
• Beginning Balance (Principal)
• PMT
• Interest (Monthly rate x BB)
• Principal Repaid (PMT - Interest)
• Ending Balance (BB - Princ. Repaid)

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Payoff of a Loan

• Payoff principal still owed on a loan
• Look at remaining payments
• They include principal still owed plus future
  interest charges.
• You dont owe the interest charges if you pay off
  the loan early.

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• To find the principal still owed, back the
  interest out of the remaining payments
• Find the PVA of the remaining monthly payments
• Back interest out at the same rate it was added
  in originally
• PVA Payoff Principal still owed

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Example 16 Continued

• Suppose we want to sell the house after 12 years.
  What will be the payoff?
• Back the interest out of the 18 years of monthly
  payments that have yet to be made.

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Amortization Schedule on BA II Plus Calculator

• Hit 2nd AMORT, 2nd CLR WORK
• P1 First cash flow to evaluate (enter in months
  if loan requires monthly pmts)
• P2 Last cash flow to evaluate
• BAL Principal still owed (payoff)
• PRN Principal repaid
• INT Interest paid