Savings and Withdrawal Plans

EXERCISE 01

Deposit €15,000 at the end of each year for eight years at 8.25% effective annual interest.

A.Find the balance immediately after the eighth deposit and one year later, with no further deposit.

A.

s_{n|i}=\frac{(1+i)^n-1}{i}

After the eighth deposit · year 8

F_8=15{,}000\frac{1.0825^8-1}{0.0825}

€160,997.40

One year later · year 9, no further deposit

F_9=F_8(1.0825)

€174,279.68

Savings and Withdrawal Plans

EXERCISE 02

Build at least €41,000 with €2,900 semiannual deposits in advance. The nominal annual rate is 5%, convertible semiannually. With n deposits at times 0, 0.5, …, (n−1)/2, value the fund at t = n/2.

A.Find the number of semiannual periods required.

A.

s_{n|i}=\frac{(1+i)^n-1}{i}

i_h=\frac{0.05}{2}=0.025

41{,}000=2{,}900(1.025)s_{n|0.025}

n=\frac{\ln[1+41{,}000(0.025)/(2{,}900\cdot1.025)]}{\ln(1.025)}\approx11.998156

Round up to 12 periods (6 years): the fund is €41,007.28. After 11 periods it is only €37,107.10.

Savings and Withdrawal Plans

EXERCISE 03

Make eight equal annual deposits R at t = 1,…,8. The fund must finance ten annual withdrawals of €2,000, initially at t = 9,…,18. All balances earn 4% effective annually.

A.Find R, then recalculate if withdrawals instead occur at t = 10,…,19.

A.

a_{n|i}=\frac{1-(1+i)^{-n}}{i}

s_{n|i}=\frac{(1+i)^n-1}{i}

Rs_{8|0.04}=2{,}000a_{10|0.04}

R=\frac{2{,}000a_{10|0.04}}{s_{8|0.04}}

€1,760.52

R^{\prime}s_{8|0.04}=\frac{2{,}000a_{10|0.04}}{1.04}

R^{\prime}=\frac{R}{1.04}

€1,692.80

Savings and Withdrawal Plans

EXERCISE 04

Deposit €200 at the end of every two-month period for eight years, at 12% effective annually. Immediately after the final deposit, replace the balance with two withdrawals: C in two years and 2C in four years.

A.Find both withdrawals.

A.

s_{n|i}=\frac{(1+i)^n-1}{i}

i_{2m}=1.12^{1/6}-1,\qquad F=200s_{48|i_{2m}}

Balance at the last deposit: €15,481.35.

F=C(1.12)^{-2}+2C(1.12)^{-4}

C=\frac{F}{(1.12)^{-2}+2(1.12)^{-4}}

€7,485.32 in two years.

C_4=2C

€14,970.63 in four years.

Savings and Withdrawal Plans

EXERCISE 05

Deposit €5,000 at each year-end for twelve years at 3.5%. Immediately after the third deposit, the effective annual rate rises to 4%.

A.Find the original target fund.
B.Keep all deposits at €5,000 and find the revised fund.
C.Find the new deposit for years 4–12 that preserves the original target.

A.

s_{n|i}=\frac{(1+i)^n-1}{i}

F_{12}=5{,}000s_{12|0.035}

€73,009.81

B.

F_3=5{,}000s_{3|0.035}

F_{12}^{\prime}=F_3(1.04)^9+5{,}000s_{9|0.04}

€75,019.61

C.

R=\frac{F_{12}-F_3(1.04)^9}{s_{9|0.04}}

€4,810.09

Savings and Withdrawal Plans

EXERCISE 06

A savings plan targets €40,000 at year 8 using equal quarterly deposits in arrears at 10% effective annually. Immediately after the twelfth deposit at year 3, switch to semiannual deposits and bring the target date forward to year 7. The first revised deposit is at year 3.5; the last is at year 7.

A.Find the original quarterly deposit and revised semiannual deposit.

A.

s_{n|i}=\frac{(1+i)^n-1}{i}

i_q=1.10^{1/4}-1

i_h=1.10^{1/2}-1

R_q=\frac{40{,}000}{s_{32|i_q}}

€843.44

F_3=R_qs_{12|i_q}

Accumulated balance at year 3: €11,577.59.

40{,}000=F_3(1.10)^4+R_hs_{8|i_h}

R_h=\frac{40{,}000-F_3(1.10)^4}{s_{8|i_h}}

€2,424.06