Basic Annuities

A Nice Little Chart for Basic Annuities:

annuities immediate and annuities dueI used the image occlusion feature on my memorization software to make cards like so:

basic annuity sample cardThere is a simple, subtle, and significant difference between annuities immediate and annuities due, and these cards help to illustrate the difference.  Really the “immediate” and “due” terms are not important.  The important thing is to recognize there you are measuring from, in relation to the payments.

Some Easy Time Value Money Exercises for Exam FM

Mathematics texts of a certain level contain lots of difficult problems, yet seldom any easy exercises.  For me, creating simple practice problems is an essential part of learning.  I have plenty of difficult material to learn.  Practicing the easy stuff is a great way of mastering the fundamentals.

I have had this same philosophy in anything I have learned in life.  Each thing that you learn, now becomes something that you can practice.  As you expand your forward knowledge, you gain an entirely different understanding of the things that you learned in the past.

After studying calculus for one semester, the definition of the derivative which you learned in week 3 seems too cumbersome and inefficient to bother with.  Then, a couple of semesters later, you encounter functions for which your quick and easy rules of derivatives no longer apply.  Then, you will be glad that you memorized the definition of derivative back in high school:

f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}

During the 25 years of my life that I was an avid juggler, I had a rule of working forward and backward each day.  When my new work (say juggling five or seven balls), would get too difficult, I would retreat for a while to easier stuff, then move back to the new.

So here are some easy problems to practice.

  1. With compound interest i = 0.05, what is present value of 10,000 dollars in 40 years?
  2. With compound interest i = 0.03, what is present value of 2,000 dollars in 9 years?
  3. With compound interest i = 0.01, what is present value of 10,000 dollars in 5 years?
  4. i = 5%.  v = ?
  5. i = 9%.  v = ?
  6. d = 0.055.  v = ?
  7. d = 0.0025.  v = ?
  8. v = 0.96. d = ?
  9. v = 0.85. d = ?
  10. The present value of $50000 payable in 30 years at an effective annual discount rate of 5%.
  11. The present value of $1000 payable in 15 years at an effective annual discount rate of 4%.
  12. Rate is 2% per quarter. Effective rate i = ?
  13. Rate is 1% per month. Annual Nominal rate = i(12)= ?
  14. Rate is 0.25% per month. Effective rate i = ?
  15. Rate is 0.25% per month. Annual Nominal rate = i(12)
  16. Nominal Annual rate is i(2) = 6%. Effective rate i = ?
  17. Nominal Annual rate is i(12)= 12%. Monthly rate = ?
  18. i = 0.09. v = ?
  19. i = 0.06. d = ?
  20. Effective Yearly Interest Rate =  6.1679%.
    i(12) = ?
  21. Effective Yearly Interest Rate = 2.27543%, Compounded daily.
    Nominal Annual Rate ?
  22. d = 0.06. i = ?
  23. $10000 today yields $100000 in 40 years.
    Interest Rate?
  24. $10 today yields $20 in 7 years.
    Interest Rate?

Solutions:

  1.  1420.46
    PV = \frac{FV}{(1+i)^{n}} = \frac {10,000}{1.05^{40}}
  2. 1532.83
    = \frac {2,000}{1.03^{9}}
    PV = \frac{FV}{(1+i)^{n}}
  3. 9514.66
    = \frac {10,000}{1.01^{5}}
    PV = \frac{FV}{(1+i)^{n}}
  4. 0.9524
    v = (1+i)^{-1}
  5. 0.9174
    v = (1+i)^{-1}
  6. 0.945
    v = 1-d
  7. 0.9975
    v = 1-d
  8. 0.04
    d = 1-v
    v = 1-d
    both sides are present value of 1 paid at end of period
  9. 0.15
    d = 1-v
    v = 1-d
    both sides are present value of 1 paid at end of period
  10. 50000 (1-0.05)30= 10731.94
    PV=FV vn
  11. 1000 (1-0.04)15= 542.09
    PV=FV vn
  12. 1.02^4 -1 =0.0824
    (1+ \frac {0.08}{4} ) ^{4} -1 = 0.082
    = (1+ \frac {i^{(m)}}{m} ) ^{m} -1
  13. 12%
  14. 1.0025^{12 }-1 = 0.0304
    (1+ \frac {0.03}{12} ) ^{12} -1 = 0.0304
    = (1+ \frac {i^{(m)}}{m} ) ^{m} -1
  15. 3%
  16. (1+ \frac {0.06}{2} ) ^{2} -1 = 0.0609
    = (1+ \frac {i^{(m)}}{m} ) ^{m} -1
  17. 1%
  18. 0.9174
    v = \frac{1}{1+i}
  19. 0.05660
    d = \frac{i}{(1+i)}
  20. 6%
    =12[(1.061679)^{\frac{1}{12}}-1]
    i^{(m)} = m[(1+i)^{\frac{1}{m}}-1]
  21. 2.25%
    = 365[(1.0227543)^{\frac{1}{365}}-1]
    i^{(m)} = m[(1+i)^{\frac{1}{m}}-1 ]
  22. 0.0638
    i = \frac{d}{(1-d)}
  23. 5.9%
    i = (\frac{FV}{PV})^{\frac{1}{n}}-1
  24. 10.4%
    i = (\frac{FV}{PV})^{\frac{1}{n}}-1

Oh, by the way.  If you have been reading these posts for a while, I am sure that you realize that all of these exercises, plus about 200 more, are part of my Anki deck for exam FM.