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EXAM FM/2 REVIEWANNUITIES
Basics Annuities are streams of payments, in our case for a
specified length Boil down to geometric series
𝑆𝑢𝑚 = ሺ𝐹𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚ሻ∗(1− 𝑟# 𝑜𝑓 𝑡𝑒𝑟𝑚𝑠)(1− 𝑟)
Two main formulas
For annuities due (double dots), simply change denominator from i to d
Once again, if unsure make a TIMELINE
𝑎𝑛 =ۀ 𝑣+ 𝑣2 + ⋯+ 𝑣𝑛 = 1− 𝑣𝑛𝑖
𝑠𝑛 =ۀ (1+ 𝑖)𝑛−1 + ⋯+ሺ1+ 𝑖ሻ+ 1 = (1+ 𝑖)𝑛 − 1𝑖
𝑠ሷ𝑛 =ۀ (1+ 𝑖)𝑠𝑛 𝑠𝑛+1=ۀ −ۀ 1 𝑎ሷ𝑛 =ۀ (1+ 𝑖)𝑎𝑛 𝑎𝑛−1+1=ۀ ۀ
Deferred Annuities Annuity with the whole series of payments pushed
back
No need to know formulas, just use TVM factors to shift
𝑚ห𝑎𝑛 =ۀ ሺ𝑚+ 1ሻห𝑎ሷ𝑛 =ۀ 𝑣𝑚𝑎𝑛 =ۀ 𝑣𝑚+1𝑎ሷ𝑛 =ۀ 𝑎𝑚+𝑛 −ۀ 𝑎𝑚 ۀ
Shortcuts Block Payments Show graph For PV, start with last payment and move back to time 0 For AV, start with first payment and move forward to
end Just an example of manipulating annuities
Prove algebraically (SS) or logically as deferred plus immediate
Another example of manipulating annuities
𝑎2𝑛 𝑎𝑛ۀ ۀ= 1+ 𝑣𝑛
Perpetuities No new formulas, just plug in infinity for n in the
originals
Interestingly, this leads to
𝑎∞ =ۀ 1𝑖 𝑎ሷ∞ =ۀ 1𝑑 1𝑑− 1𝑖 = 1
Annuities with off payments 1st method- Find equivalent interest rate for
payment period This is the easiest/quickest, so use this if possible
2nd Method- More complicated, but may have to use if you are only given symbols
Multiple payments during interest pd- mthly annuity Mthly annuity- divide by # payments and use i^(m)
Multiple interest pds per payment- split up payments (show)
1+ 𝑗= (1+ 𝑖)𝑛
𝑎(𝑚)𝑛 =ۀ 1− 𝑣𝑛𝑖(𝑚)
More Annuities If payable continuously, continue pattern and change i to δ
Double dots and upper m’s cancel
If payments vary continuously and/or interest varies continuously (unlikely)
𝑎𝑛 =ۀ 1− 𝑣𝑛𝛿 = න 𝑣𝑡𝑑𝑡𝑛0
𝑎(𝑚)𝑥 𝑎(𝑚)𝑦ۀ ۀ= 𝑎ሷ𝑥 𝑎ሷ𝑦ۀ ۀ
= 𝑎ሷ(𝑚)𝑥 𝑎ሷ(𝑚)𝑦ۀ ۀ= 𝑎𝑥 𝑎𝑦ۀ ۀ
𝑃𝑉= න 𝑓(𝑡)𝑒 𝛿𝑟𝑡0 𝑑𝑟𝑑𝑡𝑛0
Arithmetic progression General formula- annuity of first payment plus increasing
annuity of the common difference
This leads to 3 other forms by bringing through time (show) From these, you can derive all 4 increasing/decreasing
formulas (show)
𝐴= 𝑃𝑎𝑛 +ۀ 𝑄(𝑎𝑛 −ۀ 𝑛𝑣𝑛)𝑖
Geometric Progression Can usually figure out using geometric series,
without any special formulas
Calculator Highlights Beg/End option Always clear TVM values and check beg/end,
compounding, etc options
Problems A man turns 40 today and wishes to provide
supplemental retirement income of 3000 at the beginning of each month starting on his 65th birthday. Starting today, he makes monthly contributions of X to a fund for 25 years. The fund earns a nominal rate of 8% compounded monthly. Each 1000 will provide for 9.65 of income at the beginning of each month on his 65th birthday until the end of his life. Calculate X.
ASM p.109
Answer: 324.73
To accumulate 8000 at the end of 3n years, deposits of 98 are made at the end of each of the first n years and 196 at the end of each of the next 2n years.The annual effective rate of interest is i. You are given (1+i)^n=2.0Determine i. ASM pg. 123
Answer: 12.25%
Dottie receives payments of X at the end of each year for n years. The present value of her annuity is 493.Sam receives payments of 3X at the end of each year for 2n years. The present value of his annuity is 2748.Both present values are calculated at the same annual effective interest rate.Determine v^n. ASM p.136
Answer: .858
A loan of 10,000 is to be amortized in 10 annual payments beginning 6 months after the date of the loan. The first payment, X, is half as large as the other payments. Interest is calculated at an annual effective rate of 5% for the first 4.5 years and 5% thereafter. Determine X. ASM p.153
Answer: 655.70
Kathryn deposits $100 into an account at the beginning of each 4-year period for 40 years. The account credits interest at an annual effective interest rate of i. The accumulated amount in the account at the end of 40 years is X, which is 5 times the accumulated amount in the account at the end of 20 years. Calculate X. ASM p.165
Answer: 6,195
Olga buys a 5-year increasing annuity for X. Olga will receive 2 at the end of the first month, 4 at the end of the second month, and for each month therafter the payment increases by 2. The nominal interest rate is 9% convertible quarterly. Calculate X. ASM p.205
Answer: 2,729
A perpetuity-immediate pays 100 per year. Immediately after the fifth payment, the perpetuity is exchanged for a 25-year annuity-immediate that will pay X at the end of the first year. Each subsequent annual payment will be 8% greater than the preceding payment. Immediately after the 10th payment of the 25-year annuity, the annuity will be exchanged for a perpetuity-immediate paying Y per year. The annual effective rate of interest is 8%. Calculate Y. ASM p.228Answer: 54