  Retail Investor .org ### 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 = \$100n = 12FV = \$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 = \$800n = 12i% = 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 = \$6,000i% = 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 = \$100,000n = 240i% = 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 = \$1000n = 50i% = 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 = \$20,077n = 248Pmt = \$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 = \$1,200n = 20i% = 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. 