Chapter 4, Problem 125P

To determine

i.

Whether given statement is true or not” Waiting for 5 years to start saving doubles what you deposit each year”.

Answer to Problem 125P

The statement is false.

Explanation of Solution

Given:

Retirement period from now: 40 years

Amount required: $ 2 M

Interest Rate = 10%.

Calculation:

If savings are done for amount S immediately:

$\begin{array}{l}\$2000000=S\times FVIFA(10\%,40)\\ \$2000000=S\times 443.5926\\ S=\$4518.83\end{array}$

If I start saving after 5 years later with annual saving being S, then:

$\begin{array}{l}\$2000000=S\times FVIFA(10\%,35)\\ \$2000000=S\times 271.0244\\ S=\$7379.41\end{array}$.

Conclusion:

Since $7379.41 is much less than the double of $4518.33 (which is equal to $9036.66), the statement is false.

To determine

ii.

Whether required multiplier is higher or lower than 10% rate.

Answer to Problem 125P

Required multiplier is higher.

Explanation of Solution

Given:

Interest Rate = 12%

Retirement period from now: 40 years

Amount required: $ 2 M.

Calculation:

When interest rate is 12%

Required multiplier (FVIFA factor) =

$\text{FVIFA}\text{(12\%,40)=767}\text{.0914>442}\text{.5926}\text{(Higher)}$.

Conclusion:

Required multiplier is higher.

To determine

iii.

Interest Rate.

Answer to Problem 125P

Interest rate is 0% or 14.78%.

Explanation of Solution

Given:

Savings are done of double amount after 5 years

Retirement period from now: 40 years

Amount required: $ 2 M.

Calculation:

Let required interest rate per year be R%

$\frac{\$2000000}{FVIFA(R\%,40)}=\frac{2\times \$2000000}{FVIFA(R\%,40)}$

$FVIFA(R\%,40)=2\times FVIFA(R\%,35)$

$\begin{array}{l}\frac{\left[{(1+R)}^{40}-1\right]}{R}=\frac{2\times \left[{(1+R)}^{35}-1\right]}{R}\\ \left[{(1+R)}^{40}-1\right]=2\times \left[{(1+R)}^{35}-1\right]\end{array}$

$\begin{array}{l}{(}^1-1=2\times \left[{(1+R)}^{35}-2\right]\\ 2\times \left[{(1+R)}^{35}\right]={(}^1+1\end{array}$

If 1+R = Z, then,

$\begin{array}{l}2Z^{35}=Z^{40}+1\\ Z^{40}-2Z^{35}+1=0\end{array}$

Solving this polynomial equation,

$\begin{array}{l}Z=1orZ=1.1478(theonlytworealroots)\\ 1+R=1or1+R=1.1478\\ R=0orR=0.1478\end{array}$.

Conclusion:

R = 0% or R = 14.78%.