a local government is about to run a lottery, but does not want to be involved in the payoff if a winner…

a local government is about to run a lottery, but does not want to be involved in the payoff if a winner picks an annuity payoff. the government contracts with a trust to pay the lump - sum payout to the trust and have the trust (probably a local bank) pay the annuity payments. the first winner of the lottery chooses the annuity and will receive $150,000 a year for the next twenty - five years. the local government will give the trust $2,000,000 to pay for this annuity. what investment rate must the trust earn to break even without the percentage sign. for example enter 12.34 for 12.34%. (note: enter your answer as percent to the nearest basis point or hundredth percent.)

a local government is about to run a lottery, but does not want to be involved in the payoff if a winner picks an annuity payoff. the government contracts with a trust to pay the lump - sum payout to the trust and have the trust (probably a local bank) pay the annuity payments. the first winner of the lottery chooses the annuity and will receive $150,000 a year for the next twenty - five years. the local government will give the trust $2,000,000 to pay for this annuity. what investment rate must the trust earn to break even without the percentage sign. for example enter 12.34 for 12.34%. (note: enter your answer as percent to the nearest basis point or hundredth percent.)

Answer

Explanation:

Step1: Identify the present - value of the annuity

The present - value of an ordinary annuity formula is $PV = A\times\frac{1-(1 + r)^{-n}}{r}$, where $A=$150000$, $n = 25$ years, and $PV=$2000000$. We need to solve for $r$. $$2000000=150000\times\frac{1-(1 + r)^{-25}}{r}$$ This is a non - linear equation. We can use a financial calculator or an iterative trial - and - error method. Let's use the trial - and - error method.

Step2: First trial

Let's start with $r = 6%$ or $r=0.06$. $$\frac{1-(1 + 0.06)^{-25}}{0.06}=\frac{1 - 0.2329977}{0.06}=\frac{0.7670023}{0.06}\approx12.7834$$ And $150000\times12.7834 = 1917510<2000000$

Step3: Second trial

Let's try $r = 5%$ or $r = 0.05$. $$\frac{1-(1 + 0.05)^{-25}}{0.05}=\frac{1-0.295302777}{0.05}=\frac{0.704697223}{0.05}=14.09394446$$ And $150000\times14.09394446=2114091.67>2000000$

Step4: Interpolation

We know that when $r_1 = 5%$, $PV_1=2114091.67$ and when $r_2 = 6%$, $PV_2 = 1917510$. The formula for interpolation is $r=r_1+\frac{PV_1 - PV}{PV_1 - PV_2}(r_2 - r_1)$ Substitute $PV = 2000000$, $PV_1 = 2114091.67$, $PV_2=1917510$, $r_1 = 0.05$ and $r_2 = 0.06$ $$r=0.05+\frac{2114091.67 - 2000000}{2114091.67 - 1917510}(0.06 - 0.05)$$ $$r=0.05+\frac{114091.67}{196581.67}\times0.01$$ $$r=0.05 + 0.005803\approx0.0558$$

Answer:

$5.58%$