The subject property is a commercial site under a ground lease with five level annual payments of $60,000 remaining. Each payment is due at the beginning of the lease year, and the next one is due today. The discount rate is 8%. What is the present value of the remaining rent payments?
Correct Answer
B) $258,728
Payments in advance form an annuity due, so the year-end annuity value is multiplied by 1.08: $60,000 × 3.99271 × 1.08 = $258,728. Treating the payments as year-end amounts gives $239,563, which discounts each payment one year too many. Adding the five payments gives $300,000 with no discounting, and discounting that total for one year gives $277,778.
Why This Is the Correct Answer
Option B is correct because beginning-of-period payments form an annuity due. The ordinary value of $60,000 × 3.99271 = $239,563 is multiplied by 1.08, which gives $258,728.
Why the Other Options Are Wrong
Option A: $239,563
Option A, $239,563, is the ordinary annuity value: $60,000 × 3.99271 = $239,563. It assumes the first payment comes a year from now, but it is due today, so the value is $239,563 × 1.08 = $258,728.
Option C: $277,778
Option C, $277,778, discounts the sum of the payments for a single year: $300,000 ÷ 1.08 = $277,778. Each payment has its own date, and the correct total is $258,728.
Option D: $300,000
Option D, $300,000, adds the payments at face value: $60,000 × 5 = $300,000. Four of the five payments lie in the future and must be discounted, which gives $258,728.
In advance adds one period
Rent paid in advance is received a period sooner. Multiply the ordinary annuity value by one plus the rate.
How to use: Use this adjustment when a lease or contract states that payments are due at the start of each period.
Exam Tip
Read for timing words such as due in advance, beginning of each year or first payment today. They change the answer even when the amounts and rate stay the same.
Common Mistakes to Avoid
- -Applying the ordinary annuity factor to payments in advance
- -Summing payments without discounting
- -Discounting a total as though all payments arrived on one date
Concept Deep Dive
Analysis
The timing phrase is the crux. Payments at the beginning of each year fall at today and at the ends of years 1 through 4, which is one period earlier than an ordinary annuity. Each payment is therefore worth one more period of interest, so the ordinary value is multiplied by one plus the rate.
Background Knowledge
Payments at the end of each period form an ordinary annuity, the usual assumption in appraisal, while payments at the beginning form an annuity due. The two are one period apart, so their values differ by a factor of one plus the rate.
Real-World Application
An appraiser valuing a landowner's position reads the ground lease and finds that rent is due at the start of each lease year. Modeling the rent in advance avoids understating the present value of the lease payments.
More Income Approach Questions
An undamaged but functionally inadequate system causes a supported $60,000 loss. Replacement costs $85,000 and restores only $60,000, with no other benefits. What loss classification and deduction fit?
The subject is a 10-year-old dwelling with improvement cost new of $640,000. A highly comparable 7-year-old dwelling recently sold for $580,000; at the sale date its site was worth $150,000 and its improvement cost new was $500,000. Market evidence shows depreciation accruing evenly with actual age. What is the depreciated value of the subject improvements?
Observed sale prices are $450,000, $475,000, $425,000, $500,000, $440,000. Calculate the range (maximum−minimum), to the nearest dollar.
Observed values are 100, 105, 110, 115, 120. Calculate the range (maximum−minimum).
For an arithmetic illustration only, depreciate a historical $2,000,000 construction cost on a zero-salvage straight-line 40-year schedule for 10 years. What unindexed scheduled amount remains? This is not a conclusion of current market value or current replacement cost.
For an arithmetic illustration, allocate historical construction cost $2,400,000 over 40 years on a zero-salvage equal annual schedule. After 8 years, what scheduled unindexed amount remains? This is not a current market-value conclusion.
For an arithmetic schedule only, allocate historical cost $2,500,000 over 40 years with zero residual. After 8 years, what unindexed scheduled amount remains? This is not an estimate of current market value.
The sale-price range is $45,000 and the maximum is $385,000. Find the minimum price.
A property is purchased for $1,500,000 with a loan of $1,200,000. Using the purchase price as the value, what is the loan-to-value ratio, as a percentage to one decimal?
Improvement cost new is $1,800,000. Separately supported improvement depreciation deductions of $200,000, $150,000, $100,000 are expressly nonoverlapping and not already excluded from cost new. No site is included. What depreciated improvement amount results?
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