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### MATH EXAMPLES USING PV and FV of a DOLLAR

Here are some examples of the use of Time-Value-Of-Money calculations using the "PV of a Dollar" function. This page relates to the discussion at Rates of Return. At the bottom of that page there is an explanation of the inputs for a financial calculator.

 Problem #1:A Canadian mortgage is quoted at 6%. What is the interest rate? Inputs:PV = \$100n = 2i% = 3% DISCUSSION : Canadian mortgages are quoted using simple interest based on semi-annual compounding. So the interest rate applied every six months is 6/2 = 3%. Allocate \$100 to be the PV because it allows you to see the interest rate included in the resulting FV. The number of six-month compounding periods in a year is 2. Solve for:FV = \$106.09 CONCLUSION : The true interest rate is 6.09%. You can see this from the FV [106.09 - 100]/100. This analysis would be the same for a quote on bond coupons. These are also quoted using simple interest compounded semi-annually.

 Problem #2:You have a bi-monthly-pay Canadian mortgage quoted as 6%. What is the interest rate that will be used to calculate the interest portion of each payment? Inputs:PV = \$100n = 12FV = \$103 DISCUSSION : From problem #1 above you know that the 6% quote really means that 3% is charged over a six-month period. It is this six-month period that we now look at. The added complication here is that the 3% is a 'true' interest rate per six months. Each payment's interest is considered to compound such that the total after six months equals that 3%. Assign the PV and FV to be \$100 and \$103 to reflect the 3% rate over six months. The number of compounding bi-monthly periods within the six-months = 12. Solve for:i% = 0.2466% CONCLUSION : The interest rate that will be multiplied by the mortgage balance each payment is 0.2466%.

 Problem #3:You have a bi-monthly-pay US mortgage quoted as 6%. What is the interest rate that will be used to calculate the interest portion of each payment? And what is the true interest rate? Inputs:PV = \$100n = 24i% = 0.25% DISCUSSION : Unlike Canadian mortgages, US mortgages are quoted using simple interest compounded as frequently as the payments. The interest applied at each payment equals the quoted rate divided by the payment frequency. In this case 6% divided by 24 = 0.25%. Solve for:FV = \$106.176 CONCLUSION : The interest rate that will be multiplied by the mortgage balance each payment is 0.25%. The true interest rate is 6.176%. This analysis is exactly the same as you would use to analyse a GIC that pays (or compounds) interest more frequently than yearly. The GIC will probably pay monthly so divided the quoted rate by 12, and use 12 periods.

 Problem #4:You borrow \$100 from a pay-day-loan outfit. When you receive your paycheque 30 days later, they deduct \$110 from it. What was the interest rate? Inputs:PV = \$100n = 12.2i% = 10% DISCUSSION : The interest rate over the 30 days was 110-100/100 = 10%. The number of compounding period in a year is 365/30 = 12.2. Solve for:FV = \$320 CONCLUSION : The true interest rate is 220%. You can see this from the FV: [320 - 100]/100. This is the interest rate used by lobby groups trying to shut down these companies. Their arguments are false because the \$10 cost pays for the transaction processing, not interest. You would think it well worth it.

 Problem #5:You want to invest in a strip bond that will cost you \$150 today. Its face value will be \$200 in 5 years. What is its interest rate? Inputs:PV = \$150n = 5FV = \$200 CONCLUSION : The true interest rate is 5.9%. Solve for:i% = 5.9%

 Problem #6:You bought a home 10 years ago for \$100,000. It just sold for \$150,000. What was the rate of return over the period? Inputs:PV = \$100,000FV = \$150,000n = 10 CONCLUSION : The annual return was 4.14%. Remember that this problem measures only the capital gains and ignores the operating returns (rents saved). Solve for:i% = 4.14%

 Problem #7:You are offered a 91 day T-bill, costing \$98, with a quoted 8.19% return. What is the true return? Inputs:PV = \$100i% = 2.0408%n = 4 DISCUSSION : No interest is paid on treasury bills. They are sold at a discount from \$100 par value. The difference between the discount and par represents the lender's income. The formula for calculating the quoted yield is: Yield = (100-price)/price * (365*100)/term. In this case: (100-98)/100 * (365*100)/91 = 8.19%. To find the true return consider that the income compounds with maturity which is 365/91 = 4 times per year. The income at maturity is 100-98/100 = 2.0408%. Solve for:FV = \$108.42 CONCLUSION : The true interest rate from the FV is 8.42%. (108.42-100)/100 Of course you only get this rate for three months. You face reinvestment risk at that point. You must consider the probable trend in rates.

 Problem #8:You want to set aside a sum of money today, to be used to buy a replacement property in 50 years. The property is worth \$100,000 in today's dollars. The money will be invested at 5%. How much needs to be set aside now? Inputs:FV = \$100,000i% = 3%n = 50 DISCUSSION : Inflation will increase the value of the replacement property. Assume inflation of 2%. Only the remaining 3% of the investment returns will actually grow the value of the portfolio. Solve for:PV = \$22,811 CONCLUSION : You need to set aside \$22,811 today.

 Problem #9:What is the value of a lease for a capital asset? E.g. an auto lease. It has a 4-year term and the cost to buy the asset outright is \$30,000. Your projected after-tax investment return is 8%. The expected value of the asset at the end of 4 years is 45% of today's value. Inputs:FV = \$13,500n = 4i% = 8% DISCUSSION : This can be determined directly by measuring the income you could derive from the asset. But when the asset will not be generating income (like a car) you find the value of the lease by comparing it to the cost to purchase. Find the present value of the residual value (45% of \$30,000 = \$13,500). Then subtract it from the purchase price. Solve for:PV = \$9,923 CONCLUSION : The value today (PV) of the expected resale value is \$9,923. So the value of the lease is \$30,000 - \$9,923 = \$20,077. This example is the first step in analysing the true rate of interest charged on an auto lease. It continues in Problem #6 (PVannuity).

 Problem #10:Your auto lease is costing you 0.7974% per month. What is the equivalent yearly interest rate? This example is a continuation from Problem #6 (PVannuity). Inputs:PV = \$100n = 12i% = 0.7974% CONCLUSION : The yearly lease rate is 10%. (\$110/\$100)-1 Since this is larger than your personal investing return (8% from the start of the example directly above), you would do better to buy for cash. Solve for:FV = \$110

 Problem #11:You must chose between a GIC that pays out 6% interest monthly vs. one that pays (or reinvests) yearly. What yearly interest rate would be the economic equivalent? Inputs:PV = \$100n = 12i% = 0.5% CONCLUSION : The equivalent yearly rate would be 6.17% - \$106.17 less \$100. Solve for:FV = \$106.17