EXAM FM/2 REVIEW ANNUITIES
Feb 08, 2016
EXAM FM/2 REVIEWANNUITIES
Basics Annuities are streams of payments, in our case for a
specified length Boil down to geometric series
Two main formulas
For annuities due (double dots), simply change denominator from i to d
Once again, if unsure make a TIMELINE
ππ =Ϋ π£+ π£2 + β―+ π£π = 1β π£ππ
π π =Ϋ (1+ π)πβ1 + β―+αΊ1+ πα»+ 1 = (1+ π)π β 1π
π α·π =Ϋ (1+ π)π π π π+1=Ϋ βΫ 1 πα·π =Ϋ (1+ π)ππ ππβ1+1=Ϋ Ϋ
Deferred Annuities Annuity with the whole series of payments pushed
back
No need to know formulas, just use TVM factors to shift
Perpetuities No new formulas, just plug in infinity for n in the
originals
Interestingly, this leads to
πβ =Ϋ 1π πα·β =Ϋ 1π
1πβ 1π = 1
Annuities with off payments Two Cases, assuming i is effective
rateβ¦find the effective rate per payment 1. Multiple Interest periods per payment
2. Multiple payments during interest period
Or you may need to use annuity symbols
More Annuities If payable continuously, continue pattern and change i to Ξ΄
Double dots and upper mβs cancel
If payments vary continuously and/or interest varies continuously (unlikely)
Arithmetic progression General formula β Present value where P is the first payment
and Q is the common difference between payments.
From these, you can derive all 4 increasing/decreasing formulas (show)
π΄= πππ +Ϋ π(ππ βΫ ππ£π)π
Calculator Highlights Beg/End option Always clear TVM values and check beg/end,
compounding, etc options
Problems Susan and Jeff each make deposits of 100 at the end
of each year for 40 years. Starting at the end of the 41st year, Susan makes annual withdrawals of X for 15 years and Jeff makes annual withdrawals of Y for 15 years. Both funds have a balance of 0 after the last withdrawal. Susanβs fund earns 8% annual effective. Jeffβs fund earns 10% annual effective. Calculate Y-X
ASM p.109
Answer: 2792
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 p.123
Answer: 12.25%
At an effective interest rate i, i>0, both of the following annuities have the same present value X: A 20-year annuity immediate with annual payments of 55 A 30-year annuity immediate with annual payments that pays
30 per year for the first 10 years, 60 per year for the next 10 years, and 90 per year for the final 10 years
Calculate X ASM p.136
Answer: 575
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
A perpetuity costs 77.1 and makes annual payments at the end of the year. The perpetuity pays 1 at the end of year 2, 2 at the end of year 3, β¦ , n at the end of year (n+1). After year (n+1), the payments remain constant at n. The annual effective interest rate is 10.5%. Calculate n. ASM p.205
Answer: 19
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.228
Answer: 130