Varying annuities and perpetuities

2 free worked examples

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1. Exam FMVarying annuities and perpetuitiesExam level

   A scholarship pays 1,000 at the end of year 1, 2,000 at the end of year 2, and so on up to 10,000 at the end of year 10. At 6% effective, find the present value.

   1. A$36,962.41$
   2. B$38,000.00$
   3. C$40,000.00$
   4. D$43,998.55$
   5. E$55,000.00$

   Solution

   1. The payments are $1,000\times (1,2,\dots ,10)$, so this is $1,000(Ia{)}_{\overline{10}|0.06}$.
   2. $(Ia{)}_{\overline{n}|}=\frac{{\ddot{a}}_{\overline{n}|}-n{v}^n}{i}$ - note the DUE factor in the numerator.
   3. ${\ddot{a}}_{\overline{10}|}=7.801692$ and $10v^{10}=5.583948$.
   4. $(Ia{)}_{\overline{10}|}=\frac{2.217744}{0.06}=36.9624$, so the present value is 36,962.41.

   Trap. Using a rather than ä in the numerator, which is the signature error on increasing annuities.

2. Exam FMVarying annuities and perpetuitiesExam level

   At 5%, find the present value of payments of 100 at the end of year 1 increasing by 100 a year for 8 years.

   1. A$2,500.00$
   2. B$2,743.32$
   3. C$3,116.44$
   4. D$3,500.00$
   5. E$4,000.00$

   Solution

   1. The payments are $100\times (1,2,\dots ,8)$, so this is $100(Ia{)}_{\overline{8}|0.05}$.
   2. $a_{\overline{8}|0.05}=6.463213$, so ${\ddot{a}}_{\overline{8}|}=6.786373$.
   3. $8v^8=8(0.676839)=5.414715$.
   4. $(Ia{)}_{\overline{8}|}=\frac{6.786373-5.414715}{0.05}=27.43316$, so $PV=2,743.32$.
   5. The average payment is 450 across eight years, an undiscounted total of 3,600 - discounting removes about a quarter of it.