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MATH EXAMPLES USING PV of an ANNUITY

Here are some examples of the use of Time-Value-Of-Money 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 lease-hold property. It will cost $1,000 more to maintain each year than a fee-simple 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 fee-simple?
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 4-year 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.