Interest and Equivalence

1
Interest and Equivalence
2
Time Value of Money

• Question Would you prefer 100 today or 100
  after 1 year?
• There is a time value of money. Money is a
  valuable asset, and people would pay to have
  money available for use. The charge for its use
  is called interest rate.
• Question Why is the interest rate positive?
• Argument 1 Money is a valuable recourse, which
  can be rented, similar to an apartment.
  Interest is a compensation for using money. 
• Argument 2 Interest is compensation for
  uncertainties related to the future value of the
  money.

3
Simple Interest

• Simple interest is interest that is computed on
  the original sum. If you loan an amount P for n
  years at a rate of i a year, then after n years
  you will have 
• P n ? (i P) P n ? i ? P P (1 i
  n). (derive it)
• Note. Interest is usually compound interest, not
  simple interest.
• Example 3-3 You loan your friend 5000 for five
  years at a simple interest rate of 8 per year.  
• At the end of each year your friend pays you 0.08
  ? 5000 400 in interest.
• At the end of five years your friend also repays
  the 5000.
• After five years your friend has paid you.  
• 5000 5 ? 400 5000 5 ? 0.08 ? 5000.
• Note The borrower has used the 400 for 4 years
  without paying interest on it.

4
Compound Interest

• Compounded interest is interest that is charged
  on the original sum and un-paid interest.
• You put 500 in a bank for 3 years at 6 compound
  interest per year.
• At the end of year 1 you have (1.06) ? 500
  530. 
• At the end of year 2 you have (1.06) ? 530
  561.80. 
• At the end of year 3 you have (1.06) ? 561.80
  595.51. 
• Note  
• 595.51 (1.06) ? 561.80
• (1.06) (1.06) 530
• (1.06) (1.06) (1.06) 500 500
  (1.06)3

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Single Payment Compound Amount Formula

• If you put P in the bank now at an interest rate
  of i for n years, the future amount you will have
  after n years is given by
• F P (1i)n 
• The term (1i)n is called the single payment
  compound amount factor.
• Factor used to compute F, given P, and given i
  and n.
• Handy Notation.
• (F/P,i,n) (1i)n
• F P (1i)n P (F/P,i,n). 

Factor used to compute F, given P, and given i
and n.
6
Present Value

• Example 3-5.
• If you want to have 800 in savings at the end of
  four years, and 5 interest is paid annually, how
  much do you need to put into the savings account
  today?
• We solve P (1i)n F for P with i 0.05, n
  4, F 800. 
• P F/(1i)n F(1i)-n ( P F (P/F,i,n) )
• 800/(1.05)4 800 (1.05)-4 800 (0.8227)
  658.16.
• Single Payment Present Worth Formula
• P F/(1i)n F(1i)-n

7
Present Value

• Example You borrowed 5,000 from a bank and you
  have to pay it back in 5 years. There are many
  ways the debt can be repaid.
• Plan 1 At end of each year pay 1,000 principal
  plus interest due
• (a) (b) (c) (d)
  (e) (f)
• Yr. Amnt. Owed Int. Owed Total Owed
  Princip. Total Payment Begin. of Yr
  0.08 b b c Payment
  
• 1 5,000 400 5,400
  1,000 1,400
• 2 4,000 320 4,320
  1,000 1,320
• 3 3,000 240 3,240
  1,000 1,240
• 4 2,000 160 2,160
  1,000 1,160
• 5 1,000 80
  1,080 1,000 1,080
• 1,200 5,000 6,200

8
Present Value

• Plan 2 Pay interest due at end of each year and
  principal at end of five years.
• (a) (b) (c)
  (d) (e)
  (f)
• Yr. Amnt. Owed Int. Owed Total Owed
  Princip. Total Payment Begin. of Yr
  0.08 b b c Payment
  
• 1 5,000 400
  5,400 0 400
• 2 5,000 400 5,400
  0 400
• 3 5,000 400 5,400
  0 400
• 4 5,000 400 5,400
  0 400
• 5 5,000 400 5,400
  5,000 5,400
• 2,000 5,000 7,000

9
Present Value

• Plan 3 Pay in five end-of-year payments
• (a) (b) (c)
  (d) (e)
  (f)
• Yr. Amnt. Owed Int. Owed Total Owed
  Princip. Total Payment Begin. of Yr
  0.08 b b c Payment
• 1 5,000 400 5,400
  852 1,252
• 2 4,148 332 4,480
  921 1,252
• 3 3,227 258 3,485
  994 1,252
• 4 2,233 179 2,412
  1,074 1,252
• 5 1,159 93 1,252
  1,159 1,252
  1,261 5,000 6,261

10
Present Value

• Plan 4 Pay principal and interest in one payment
  at end of five years.
• (a) (b) (c)
  (d) (e)
  (f)
• Yr. Amnt. Owed Int. Owed Total Owed
  Princip. Total Payment Begin. of Yr
  0.08 b b c Payment
• 1 5,000 400 5,400
  0 0
• 2 5,400 432 5,832
  0 0
• 3 5,832 467 6,299
  0 0
• 4 6,299 504 6,802
  0 0
• 5 6,802 544 7,347
  5,000 7,347 2,347
  5,000 7,347

Compound Interest interest is charged on the
unpaid interest.
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Present Value
12
Present Value
13
Present Value
14
Present Value
15
Present Value
16
Quarterly Compounded Interest Rates

• Example 3-6. You put 500 in a bank for 3 years
  at 6 compound interest per year. Interest is
  compounded quarterly. 
• The bank pays you i 0.06/4 0.015 every 3
  months 1.5 for 12 periods (4 periods per year ?
  3 years). 
• At the end of three years you have F P (1i)n
  500 (1.015)12 500 (1.19562) ? 597.81
•   (598 in text due to rounding)
• Note. Usually the stated interest is for a
  1-year period. If it is compounded quarterly
  then an interest period is 3 months long. If the
  interest is i per year, each quarter the interest
  paid is i/4 since there are four 3-month periods
  a year.

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• Example 3-7. In 3 years, you need 400 to pay a
  debt. In two more years, you need 600 more to
  pay a second debt. How much should you put in
  the bank today to meet these two needs if the
  bank pays 12 per year?  
  
  
• bank cash flows (bank point of
  view) 
• Solution
• P 400(P/F,12,3) 600(P/F,12,5)
• 400 (0.7118) 600 (0.5674)
• 284.72 340.44 625.16.

P
2
3
4
5
1
600
400
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• Borrower point of viewYou borrow money from the
  bank to start a business.
• Investors point of viewYou invest your money in
  a bank and buy a bond.

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• Example 3-8. In 3 years, you need 400 to pay a
  debt. In two more years, you need 600 more to
  pay a second debt. How much should you put in
  the bank today to meet these two needs if the
  bank pays 15 instead of 12 per year?
• Will we need more or less than the amount,
  625.16, we computed when i is 12? (less)
•  Solution
• P 400(P/F,15,3) 600(P/F,15,5)
• 400 (0.6575) 600 (0.4972) 561.32.

P
400
600
20

• Remark. The Yellow Pages in the text book
  tabulate
• Compound Amount Factor
• (F/P,i,n) (1i)n  
• Present Worth Factor and
• (P/F,i,n) (1i)-n
•  
• These terms are in columns 2 and 3, identified as
   
• Compound Amount Factor Find F Given P F/P 
• Present Worth Factor Find P Given F P/F