Chapter 6, Problem 66QP

a)

Summary Introduction

To calculate: The required savings for each year.

Introduction:

The series of payments that are made in equal intervals is an annuity payment. The amount of annuity payments is mainly calculated based on the particular situation.

Answer to Problem 66QP

The required savings for each year is $14,019.06465.

Explanation of Solution

Given information:

Person X’s friend is celebrating her 35^th birthday currentlyas she wishes to start saving for her retirement at the age of 65. She wishes to withdraw a sum of $125,000 on each of her birthdays for 20 years that is followed by her retirement in which, the first withdrawal will fall on her 66^thbirthday. She also plans to put her money in the local credit union that offers a 7% interest for a year. She also wishes to make equivalent annual payments on each of her birthdays into the account that is established at the credit union for retirement fund.

It is assumed that she starts making the deposit on her 36^th birthday and continues to make it until her 65^th birthday.

Note: The saving possibility has the similar future value that refers to the present value of the spending on the retirement when Person X’s friend is ready for retirement.

Formula to calculate the present value annuity is as follows:

$\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{1+r^t}\right)\right]}{r}\right\}$

Note: C denotes the payments, r denotes the rate of exchange, and t denotes the period. Thus, by the present value of annuity, the amount that is essential for Person X’s friend is when she is ready for retirement and it can be calculatedas follows:

Compute the present value annuity:

$\begin{array}{c}\text{Present value annuity}=\text{C}\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$125,000\left\{\frac{\left[1-\left(\frac{1}{{\left(1+0.07\right)}^{20}}\right)\right]}{0.07}\right\}\\ =\$125,000\left\{\frac{\left[1-\left(\frac{1}{{\left(1.07\right)}^{20}}\right)\right]}{0.07}\right\}\\ =\$125,000\left\{\frac{\left[1-0.258419002\right]}{0.07}\right\}\end{array}$

$\begin{array}{c}=\$125,000\left\{\frac{0.741580}{0.07}\right\}\\ =\$125,000\left\{10.59401425\right\}\\ =\$1,324,251.781\end{array}$

Hence, the amount that is required for Person X’s friend at the time of retirement is $1,324,251.781.

Note: The present value of annuity is same for all the necessary requirements.

Formula to calculate the future value annuity is as follows:

$\text{Future value annuity}=C\left\{\frac{\left[{\left(1+r\right)}^t-1\right]}{r}\right\}$

Note:C denotes the annual cash flow or annuity payment, r denotes the rate of interest, and t denotes the number of payments. The future value of annuity represents the necessary savings for each year.

Compute the future value annuity:

$\begin{array}{c}\text{Future value annuity}=C\left\{\frac{\left[{\left(1+r\right)}^t-1\right]}{r}\right\}\\ \$1,324,251.781=C\left\{\frac{\left[{\left(1+0.07\right)}^{30}-1\right]}{0.07}\right\}\\ \$1,324,251.781=C\left\{\frac{\left[7.612255043-1\right]}{0.07}\right\}\\ \$1,324,251.781=C\left\{\frac{6.612255043}{0.07}\right\}\end{array}$

$\begin{array}{c}\$1,324,251.781=C\left\{94.46078\right\}\\ C=\frac{\$1,324,251.781}{94.46078}\\ C=\$14,019.06465\end{array}$

Hence, the required savings for each year is $14,019.06465.

b)

Summary Introduction

To calculate: The present value of the lump sum savings.

Introduction:

The series of payments that are made in equal intervals is an annuity payment. The amount of annuity payments is mainly calculated based on the particular situation.

Answer to Problem 66QP

The present value of the lump sum savings is $173,963.1388.

Explanation of Solution

Given information:

Person X’s friend has just inherited a huge sum of money. She decides to make the lump sum payment on her 35^th birthday to cover the needs of retirement rather than making equal annual payments.

Formula to compute the future value is as follows:

$\text{Future value}=\text{PV}{\left(1+\text{r}\right)}^{\text{t}}$

Note:C denotes the annual cash flow or annuity payment, r denotes the rate of interest, and t denotes the number of payments.

Compute the future value:

$\begin{array}{c}\text{Future value}=\text{PV}{\left(1+\text{r}\right)}^{\text{t}}\\ \$1,324,251.781=\text{PV}{\left(1+0.07\right)}^{30}\\ \$1,324,251.781=\text{PV}{\left(1.07\right)}^{30}\\ \$1,324,251.781=\text{PV}\left(7.612255043\right)\end{array}$

$\begin{array}{c}\text{PV}=\frac{\$1,324,251.781}{7.612255043}\\ \text{PV}=\$173,963.1388\end{array}$

Hence, the lump sum amount is $173,963.1388.

c)

Summary Introduction

To calculate: The annual contribution of Person X’s friend.

Introduction:

The series of payments that are made in equal intervals is an annuity payment. The amount of annuity payments is mainly calculated based on the particular situation.

Answer to Problem 66QP

The annual contribution of Person X’sfriend is $6,874.68.

Explanation of Solution

Given information:

The employer of Person X’s friend contributes a sum of $3,500 into her account each year as a part of sharing the profit. In addition, Person X’s friend also expects a sum of distribution from her family trust on her 55^th birthday that amountsto $175,000.

Note: The value that is essential for retirement is known as the value of the lump sum saving at retirement and it can be subtracted to determine how much Person X’s friend is short of money.

Formula to compute the future value of the trust fund deposit is as follows:

$\text{Future value}=\text{PV}{\left(1+\text{r}\right)}^{\text{t}}$

Note:C denotes the annual cash flow or annuity payment, r denotes the rate of interest, and t denotes the number of payments.

Compute the future value of the trust fund deposit is as follows:

$\begin{array}{c}\text{Future value}=\text{PV}{\left(1+\text{r}\right)}^{\text{t}}\\ =\$175,000{\left(1+0.07\right)}^{30}\\ =\$175,000\left(1.967\right)\\ =\$344,251.49\end{array}$

Hence, the future value of the trust fund deposit is $344,251.49.

The amount that Person X’s friend needs at retirement is calculated as follows:

$\begin{array}{c}\text{Future value}=\$1,324,251.781-\$344,251.49\\ =\$980,000.29\end{array}$

Hence, the amount that Person X’sfriend needs at the time of retirement is $980,000.29.

Note: The payment can be solved by using the equation of the future value of annuity.

Formula to calculate the future value annuity is as follows:

$\text{Future value annuity}=C\left\{\frac{\left[{\left(1+r\right)}^t-1\right]}{r}\right\}$

Note:C denotes the annual cash flow or annuity payment, r denotes the rate of interest, and t denotes the number of payments.

Compute the future value annuity:

$\begin{array}{c}\text{Future value annuity}=C\left\{\frac{\left[{\left(1+r\right)}^t-1\right]}{r}\right\}\\ \$980,000.291=C\left\{\frac{\left[{\left(1+0.07\right)}^{30}-1\right]}{0.07}\right\}\\ \$980,000.291=C\left\{\frac{\left[7.612255043-1\right]}{0.07}\right\}\\ \$980,000.291=C\left\{\frac{6.612255043}{0.07}\right\}\end{array}$

$\begin{array}{c}\$980,000.291=C\left\{94.46078\right\}\\ C=\frac{\$980,000.291}{94.46078}\\ C=\$10,374.68\end{array}$

Hence, the total annual contribution is $10,374.68.

Compute the contribution that is made by Person X’s friend is as follows:

$\begin{array}{c}\text{Friend's contribution}=\$10,374.68-\$3,500\\ =\$6,874.679\end{array}$

Note: The contribution made by Person X’sfriend is calculated by subtracting the employer’s contribution from the total annual contribution.

Hence, the contribution made by Person X’s friend is $6,874.679.