Borrowing/ Spending annuities

(PV, Ordinary Annuity, annual)  You are offered an annuity that will pay $24,000 per year for 11 years (the first payment will occur one year from today). If you feel that the appropriate discount rate is 13%, what is the annuity worth to you today?

 $\ PV_{AD\ }=PMT\cdot\left(\frac{1}{r}-\frac{1}{r\cdot\left(1+r\right)^N}\right)\cdot\left(1+r\right)=\frac{PMT}{r}\cdot\left(1-\left(1+r\right)^{-N}\right)\cdot\left(1+r\right)$   $PV_{OA\ }=PMT\cdot\left(\frac{1}{r}-\frac{1}{r\cdot\left(1+r\right)^N}\right)=\frac{PMT}{r}\cdot\left(1-\left(1+r\right)^{-N}\right)$  

• $150,476.35

• $147,369.50

• $136,486.59

• $162,872.11

(PV,Ordinary Annuity,semiannual) What is the present value of a stream of $2,500 semiannual payments received at the end of each period for the next 10 years? The APR is 6%.

 $\ PV_{AD\ }=PMT\cdot\left(\frac{1}{r}-\frac{1}{r\cdot\left(1+r\right)^N}\right)\cdot\left(1+r\right)=\frac{PMT}{r}\cdot\left(1-\left(1+r\right)^{-N}\right)\cdot\left(1+r\right)$   $PV_{OA\ }=PMT\cdot\left(\frac{1}{r}-\frac{1}{r\cdot\left(1+r\right)^N}\right)=\frac{PMT}{r}\cdot\left(1-\left(1+r\right)^{-N}\right)$  

• $37,194

• $38,310

• $35,810

• $35,885

(PV, Ordinary Annuity,monthly) Your mortgage payment is $600 per month. There is exactly 180 payments remaining on the mortgage. The interest rate is 8.0%, compounded monthly. The first payment is due in exactly one month. What is the balance of the loan? [Balance = PV]

 $\ PV_{AD\ }=PMT\cdot\left(\frac{1}{r}-\frac{1}{r\cdot\left(1+r\right)^N}\right)\cdot\left(1+r\right)=\frac{PMT}{r}\cdot\left(1-\left(1+r\right)^{-N}\right)\cdot\left(1+r\right)$   $PV_{OA\ }=PMT\cdot\left(\frac{1}{r}-\frac{1}{r\cdot\left(1+r\right)^N}\right)=\frac{PMT}{r}\cdot\left(1-\left(1+r\right)^{-N}\right)$  

• $62,784

• $77,205

• $63,203

• $82,502

(PV, Ordinary Annuity,monthly)Your mortgage payment is $755 per month. It is a 30-year mortgage at 9.0% compounded monthly. How much did you borrow?

 $\ PV_{AD\ }=PMT\cdot\left(\frac{1}{r}-\frac{1}{r\cdot\left(1+r\right)^N}\right)\cdot\left(1+r\right)=\frac{PMT}{r}\cdot\left(1-\left(1+r\right)^{-N}\right)\cdot\left(1+r\right)$   $PV_{OA\ }=PMT\cdot\left(\frac{1}{r}-\frac{1}{r\cdot\left(1+r\right)^N}\right)=\frac{PMT}{r}\cdot\left(1-\left(1+r\right)^{-N}\right)$  

• $93,800

• $97,200

• $92,500

• $85,100

(PMT, Ordinary Annuity, annual)You plan to borrow $389,000 now and repay it in 25 equal annual installments (payments will be made at the end of each year). If the annual interest rate is 14%, how much will your annual payments be?

 $\ PV_{AD\ }=PMT\cdot\left(\frac{1}{r}-\frac{1}{r\cdot\left(1+r\right)^N}\right)\cdot\left(1+r\right)=\frac{PMT}{r}\cdot\left(1-\left(1+r\right)^{-N}\right)\cdot\left(1+r\right)$   $PV_{OA\ }=PMT\cdot\left(\frac{1}{r}-\frac{1}{r\cdot\left(1+r\right)^N}\right)=\frac{PMT}{r}\cdot\left(1-\left(1+r\right)^{-N}\right)$  

• $45,000

• $56,600

• $73,000

• $65,000

(Rate, Ordinary Annuity,annual)You are considering an investment in a 6-year annuity. At the end of each year for the next six years you will receive cash flows of $90. The initial investment is $414.30. To the nearest percent, what rate of return are you expecting from this investment?

Note: this can't be solve scientifically using calculus, try-and-error is the only way.

 $\ PV_{AD\ }=PMT\cdot\left(\frac{1}{r}-\frac{1}{r\cdot\left(1+r\right)^N}\right)\cdot\left(1+r\right)=\frac{PMT}{r}\cdot\left(1-\left(1+r\right)^{-N}\right)\cdot\left(1+r\right)$   $PV_{OA\ }=PMT\cdot\left(\frac{1}{r}-\frac{1}{r\cdot\left(1+r\right)^N}\right)=\frac{PMT}{r}\cdot\left(1-\left(1+r\right)^{-N}\right)$  

• 8%
  

• 9%

• 12%

• 21%

(Rate, Ordinary Annuity, annual)Suppose an annuity costs $40,000 and produces cash flows of $10,000 over each of the following eight years. What is the rate of return on the annuity?

Note: this can't be solve scientifically using calculus, try-and-error is the only way.

 $\ PV_{AD\ }=PMT\cdot\left(\frac{1}{r}-\frac{1}{r\cdot\left(1+r\right)^N}\right)\cdot\left(1+r\right)=\frac{PMT}{r}\cdot\left(1-\left(1+r\right)^{-N}\right)\cdot\left(1+r\right)$   $PV_{OA\ }=PMT\cdot\left(\frac{1}{r}-\frac{1}{r\cdot\left(1+r\right)^N}\right)=\frac{PMT}{r}\cdot\left(1-\left(1+r\right)^{-N}\right)$  

• 0%
  

• 10.5%

• 18.6%

• 25%

(PV, Annuity Due, semiannual) What is the present value of $2,500 semiannual payments received at the beginning of each period for the next 10 years? The APR is 6%.

Note: pay attention to keywords for annuity due - starting immediately, starting now, starting today, at the beginning of each.

 $\ PV_{AD\ }=PMT\cdot\left(\frac{1}{r}-\frac{1}{r\cdot\left(1+r\right)^N}\right)\cdot\left(1+r\right)=\frac{PMT}{r}\cdot\left(1-\left(1+r\right)^{-N}\right)\cdot\left(1+r\right)$   $PV_{OA\ }=PMT\cdot\left(\frac{1}{r}-\frac{1}{r\cdot\left(1+r\right)^N}\right)=\frac{PMT}{r}\cdot\left(1-\left(1+r\right)^{-N}\right)$  

• $37,194.70
  

• $38,309.50
  

• $35,809.50

• $36,884.80

(PV, annuity due, monthly) Your mortgage payment is $600 per month. There are exactly 180 payments remaining on the mortgage. The interest rate is 8.0%, compounded monthly. The next payment is due immediately. What is the balance of the loan? [Balance = PV]

Note: pay attention to keywords for annuity due - starting immediately, starting now, starting today, at the beginning of each.

 $\ PV_{AD\ }=PMT\cdot\left(\frac{1}{r}-\frac{1}{r\cdot\left(1+r\right)^N}\right)\cdot\left(1+r\right)=\frac{PMT}{r}\cdot\left(1-\left(1+r\right)^{-N}\right)\cdot\left(1+r\right)$   $PV_{OA\ }=PMT\cdot\left(\frac{1}{r}-\frac{1}{r\cdot\left(1+r\right)^N}\right)=\frac{PMT}{r}\cdot\left(1-\left(1+r\right)^{-N}\right)$  

• $63,203
  

• $77,205
  

• $62,784

• $82,502

Day 1 example before

Imagine you just won a $150 million Powerball. What is the implied interest rate that makes the two payment options indifferent? (Hint: They should be the equivalent, use value in B as present value of A)

A. Annuity of $5 million per year in 30 years.

B. Cash lump sum of $100 million today.

Keep 2 decimal points