MATH EXAMPLES USING PV of an ANNUITY
Here are some examples of the use of TimeValueOfMoney calculations using the "PV of an annuity" function. This page relates to the discussion at Rates of Return. The inputs here refer to inputs into a financial calculator.
Problem #1: You want to prepay 12 months of your $800 monthly rent and earn a 5% return on the 'investment'. What $amount should you offer your landlord? 
Inputs PV$: PV = $100 n = 12 FV = $105 
STEP ONE : is to determine what rate of return would be applied each month in order that a true yearly return equals 5% ($105 at yearend). Use the "PV of a Dollar" calculation to solve for i% = 0.407%/month. 
Solve for: i% = 0.407% 
STEP TWO : is to find the PV of the $800 monthly rents using the "PV of an Annuity" calculation. The payments happen at the beginning of the month, so choose that option on your calculator. 
Inputs PVannuity: Pmt = $800 n = 12 i% = 0.407% 
CONCLUSION : You can earn a true 5% return by paying $9,388.84 now instead of $9,600 (800*12) monthly. 
Solve for: PV = $9,388.84 
Problem #2: You are purchasing an annuity for $100,000. It will pay you $6,000 a year. How long must you live in order to realize a 5% return?  Inputs: PV = $100,000 Pmt = $6,000 i% = 5% 
CONCLUSION : You must live at least 36.7 more years. 
Solve for: n = 36.7 
Problem #3:
You have a $100,000 US mortgage at 4% with a 20 year amortization. What percent of the monthly payments are interest expense?  Inputs: PV = $100,000 n = 240 i% = 0.3333% 
DISCUSSION : US mortgages are quoted using simple interest, so the interest rate applied each month is ( ( 4%/12 =) 0.3333%. The number of montly periods in 20 years is ( 20*12=) 240 months. 
Solve for: Pmt = $605.98 
CONCLUSION : The interest charged this month is $100,000 * 0.3333% = $333. As a percent of the $605.98 payment that is ( 333/605.98=) 55%. 

Problem #4: You are purchasing a leasehold property. It will cost $1,000 more to maintain each year than a feesimple property. There are 50 years remaining on the lease. You want a 4% operating return. How much less is the L/H worth than the feesimple?  Inputs: Pmt = $1000 n = 50 i% = 4% 
CONCLUSION : The difference in value between the properties is $21,482.
Remember that you would also discount the leasehold ppty by the necessary investment today to buy a replacement property at the end of the lease. 
Solve for: PV = $21,482 
Problem #5: You are buying an oil company. It has 10,000 BOE of reserves. It is producing at 2,000 BOE per year. The net profit from each BOE produced is $25. You want a 10% return. How much is the company worth?  Inputs: Pmt = $50,000 n = 5 i% = 10% 
DISCUSSION : The total profit each year will be (2,000*25=) $50,000. The reserves will last (10,000/2000=) 5 years. 
Solve for: PV = $189,539 
CONCLUSION : The whole company is worth $189,539 today, as long as the cash flow is measured net of financing and taxes.
Note: analysts frequently say that a 'Price to Cashflow' equal to 4 times is appropriate. In this example 189,539 divided by 50,000 equals 3.8 times cashflow. But the metric is essentially meaningless because each reserve has a different lifespan and a different net cash profit. 

Problem #6:
Find the interest rate being charged on a 4year auto lease. The purchase price today is $30,000 and the value of the lease was found to be $20,077 from Problem #9 (PVdollar). The montly lease payments are $505.  Inputs: PV = $20,077 n = 248 Pmt = $505 
DISCUSSION : The number of payments is 4yrs * 12 = 48 pmt. 
Solve for: i% = 0.7974% 
CONCLUSION : The 0.7974% interest charged per month must now be converted to a yearly rate. Problem #10 (PVdollar)will show this to be 10%. 

Problem #7:
Find the extra price you would be willing to pay for a house whose upkeep was cheaper by $1200 a year. 
Inputs: Pmt = $1,200 n = 20 i% = 6% 
DISCUSSION : The number of year is chosen from the expected life of the property = 20 years . Your expected rate of return is 6 percent (including operating savings and capital gains. 
Solve for: PV = $13,764 
CONCLUSION : The present value of the yearly savings is $13,764. 


