Question

Just won a lottery jackpot of $11,000,000. You will be paid in 26 equal annual installments beginning immediately. If you had the money now, you could invest it in an account with a quoted annual interest rate of 9% with monthly compounding interest. What is the present value of the payments you will receive?

Answer 1

The formula for the present value of an anuallity is:

PV = A * [ 1 - (1 + r) ^ (-n) ] / r

Here:

A = 11000000 / 26 = 423076.92

r is the effective rate of 9% monthly compounded

r = (1 + 0.09/12) ^ (12) - 1 = 1.0938 - 1 = 0.0938

n = 26

Then, PV =  423,076.92 * [1 - (1+0.0938)^(-26) ] / (0.0938)

PV = 4,072,055.25

Answer 2

The present value of the payments you will receive is $\boxed{\bf\ \ 4,454,027.7329}$.

Further explanation:

Given:

The jackpot amount is $\ \ 11,000,000$.

The annual installment is for $26\text{ years}$.

The annual interest rate is $9\%$ with monthly compounding of interest.

Formula used:

The present value of payment can be calculated by the formula given below.

$\boxed{P_{v}=P\left(\dfrac{1-(1+r)^{-n}}{r}\right)}$    .....(1)

Here, $P_{v}$ is the present value of payment in $25\text{ years}$, $P$ is the cash flow monthly, $r$ is the interest rate.

The effective rate of interest can be calculated by the formula given below.

$\boxed{r^{'}=\left(1+\dfrac{\text{interest rate}}{\text{compounding frequency}}\right)^{\text{compounding frequency}}-1}$

Here, $r^{'}$ is the effective rate of interest.

Calculation:

The cash flow per month is calculated as follows:

$\dfrac{{11,000,000}}{{26}} = 423,077$

The effective rate of interest is obtained as follows:

$\begin{aligned}r^{'}&=\left(1+\dfrac{9\%}{12}\right)^{12}-1\\&=\left(1+\dfrac{0.09}{12}\right)^{12}-1\\&=(1.0045)^{12}-1\\&=0.0938\end{aligned}$

Substitute $0.0938$ for $r^{'}$ and $423077$ for $P$ in equation (1) to obtain the present value.

$\begin{aligned}P_{v}&=423077\left(\dfrac{1-(1+0.0938)^{-25}}{0.0938}\right)\\&=423077\left(\dfrac{1-(1.0938)^{-25}}{0.0938}\right)\\&=423077\left(\dfrac{0.8936954082}{0.0938}\right)\\&=423077\times9.5277\\&=4030950.7329\end{aligned}$

Therefore the present value for $26\text{years}$ can be obtained as sum of the amount of present value obtained for $25\text{years}$ and the amount annually.

$4030950.7329+423077=4454027.7329$

Thus, the present value of the payments you will receive is $\boxed{\bf \ \ 4,454,027.7329}$.

Learn more:

1. Solution of linear equation ://brainly.com/question/1682776

2. Interest rate https://brainly.com/question/558692

Answer details:

Grade: Senior school

Subject: Mathematics

Chapter: Compound interest

Keywords:  Equations, annually, present value , 26 years, effective rate of interest, rate of interest, installment, money, $1100000, $423077, lottery, jackpot, invest, compounding interest, payments.