Varying annuities and perpetuities

2 free worked examples

Solved in full, free to everyone, and never rotated. Read these before deciding whether the rest of the drill is worth paying for.

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. $(Ia)_{\overline{n}|} = \dfrac{\ddot a_{\overline{n}|} - nv^{n}}{i}$ - note the DUE factor in the numerator.
   3. $\ddot a_{\overline{10}|} = 7.801692$ and $10v^{10} = 5.583948$.
   4. $(Ia)_{\overline{10}|} = \dfrac{2.217744}{0.06} = 36.9624$, 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. $(Ia)_{\overline{8}|} = \dfrac{6.786373-5.414715}{0.05} = 27.43316$, 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.