REAL572
Commercial Real Estate:
Investment and Analysis
The Time Value of Money Review
Time Value of Money Outline
• Basic concepts and terminology
• Single sum compounding
• Single sum discounting
• Excel Applications/Compounding
• Multiple Cash Flows/Uneven Multiple Cash Flows
• Net Present Value (NPV) and Internal Rate of Return (IRR)
REAL572
Commercial Real Estate:
Investment and Analysis
I. Basic concepts and terminology
Real estate deals almost always
involve cash amounts at different
points in time.
• Examples:
▪
▪
▪
▪
4
Buy a property now, sell it later.
Sign a lease now, pay rents monthly over time.
Take out a mortgage now, pay it back over time.
Buy land now for development, pay for
construction and sell the building later.
Basic Problem
• We need to be able to account or quantify how TIME affects
the value of the cash flows.
Basic Principles
• The earlier the cash flows, the ________ (greater/smaller)
their value.
• The more certain we are about a set of cash flows, the
_________ (greater/smaller) their value.
• The larger the discount rate, the smaller the value
• The smaller the discount rate, larger the value
Basic Principles
• Six types of problems:
▪ Future value of a lump sum
▪ Present value of a lump sum
▪ Future value of equal amounts of cash flow
(annuity)
▪ Present value of equal amounts of cash flow
(annuity)
▪ Future value of unequal amounts of cash flow
▪ Present value of equal amounts of cash flow
Why is $1 today not equivalent to $1 a
year from now?…
• Dollars at different points in time are related by the
“opportunity cost of capital” (OCC), expressed as a rate of
return.
• We will typically label this rate, “r ”.
8
Six Basic Terms
• PV = Present Value
• FV = Future Value
• R (i) (rate) = Rate of Return
• N (nper) = Time
• Pmt = Payment
• Type = Payable in arrears (0) or in advance (1)
Basic Principles
• Two major types of PV math problems:
• Single-sum problems
• Multi-period cash flow problems (equal and unequal amounts
10
REAL572
Commercial Real Estate:
Investment and Analysis
II. Single Sum Compounding
Compounding a Single-sum over
Multiple Periods
12
Discounting a Single-sum over
Multiple Periods
13
*Basic Rules* for TVM
• The following are simple rules that you should always consider no
matter what type of Time Value of Money (“TVM”) problem you
are trying to solve:
▪ Stop and think: Make sure you understand what the problem is
asking.
▪ Draw a representative timeline and label the cash flows and
time periods appropriately.
▪ Write down the information you know. Label each variable.
▪ Check your answers using a calculator if possible.
Compounding a Single-sum over
Multiple Periods
• If r = 10%, then $100 today is worth how much two years (2)
from now?
15
■
Compounding a Single-sum over
Multiple Periods
• Let’s try this in Excel
• PV = 100
• N (nper) = 2
• R (i) (rate) = 10% or .10
• FV = ??
• Pmt = Not Applicable or 0
• Type = In arrears (0)
16
Compounding a Single-sum over
Multiple Periods
• Pulling up an Excel spreadsheet
•
=FV(rate, nper, pmt, -PV, type)
• =FV(.10, 2,0, -100, 0)
• ANSWER??
• $121.00
17
■
Compounding a Single-sum over
Multiple Periods
• Your tenant owes $10,000 in rent. He wants to postpone
payment for a year. You are willing to help him but only for a
15% return. How much will your tenant have to pay one year
from now?
18
Compounding a Single-sum over
Multiple Periods
• Let’s try this in Excel
• PV = 10000
• N (nper) = 1
• R (i) (rate) = 15% or .15
• FV = ??
• Pmt = Not Applicable or 0
• Type = In arrears (0)
19
Compounding a Single-sum over
Multiple Periods
• Excel (pull up an Excel spreadsheet)
•
=FV(rate, nper, pmt, -PV, type)
• =FV(.15, 1,0, -10000, 0)
• ANSWER??
• $11,500
20
■
Compounding a Single-sum over
Multiple Periods
• Suppose an investor deposits $1,000 today in an interest
bearing account at a local bank. The bank pays 5% interest
compounded annually and the investor expects to withdraw
the original principal plus all of the accumulated interest at
the end of five years. What is the total amount to be
withdrawn in five years?
• So we want to find the value in the future (FV)
21
Compounding a Single-sum over
Multiple Periods
• PV = $1,000
• N=5
• R = 5% (or .05)
• FV =$??
• PMT = 0 ( There are no multiple payments)
• Type = 0 (Assume at the end)
22
■
Compounding a Single-sum over
Multiple Periods
• Excel
• =FV(rate, nper, pmt, -PV, type)
• =FV(.05, 5, 0, -1000, 0)
• ANSWER??
• $1,276.28
Compounding a Single-sum over
Multiple Periods
▪ Suppose you purchase a property today that is
worth $100,000. You expect real estate prices to
appreciate at a rate of 10% per year for the next 3
years.
▪ How much will the property be worth at the end of
the 3rd year?
Compounding a Single-sum over
Multiple Periods
• PV = $100,000
• N=3
• R = 10% (or .10)
• FV =$??
25
■
Compounding a Single-sum over
Multiple Periods
• Using Excel
• =FV(rate, nper, pmt, -PV, type)
• =FV(.10, 3,0, -100000, 0)
• ANSWER??
• $133,100
REAL572
Commercial Real Estate:
Investment and Analysis
III. Single Sum Discounting
Discounting a Single-sum over
Multiple Periods
• If r = 10%, then $100 received two years from now is worth
how much today?
28
■
Discounting a Single-sum over
Multiple Periods
• Using Excel
• =PV(rate, nper, pmt, -FV, type)
• =PV(.1, 2,0, -100, 0)
• ANSWER??
• $82.64
Discounting a Single-sum over
Multiple Periods
• You have been offered an investment opportunity that is
expected to provide $1,276.28 in cash flow at the end of five
years. Assume that the discount rate is 5%, how much would
the investor pay today for this future lump sum?
30
Discounting a Single-sum over
Multiple Periods
• Using Excel
• =PV(rate, nper, pmt, -FV, type)
• =PV(.05, 5,0, -1276.28, 0)
• ANSWER??
• $1,000
Discounting a Single-sum over
Multiple Periods
• Sample Problem
▪ You are being offered an investment that will pay
you $250,000 dollars in 5 years. (FV) (nper)
▪ You know that you can get a return of 8 percent on
your existing investments. (rate)
▪ How much can you pay for the investment today
and still earn 8 percent? (PV?)
Discounting a Single-sum over
Multiple Periods
•
•
•
•
PVr,N = FV / (1 + r)N
PV8%, 5 = 250,000 / (1.08)5
= 250,000 / 1.4693
= $170,145.80
Discounting a Single-sum over
Multiple Periods
• Using Excel
• =PV(rate, nper, pmt, -FV, type)
• =PV(.08, 5,0, -250000, 0)
• ANSWER??
• $170,145.80
REAL572
Commercial Real Estate:
Investment and Analysis
IV. Single Sum (Examples)
Problem #1
• How much will a $50 deposit made today be worth in 20
years if interest is compounded annually at a rate of 10
percent?
• What kind of problem is this?
Problem #2
• How much would you pay today for the right to receive $80
at the end of 10 years if you can earn 15% percent interest
on alternative investments of similar risk?
• What kind of problem is this? Key words?
Problem #3
▪ Suppose you purchase a property today for
$250,000. You expect real estate prices to
appreciate at a rate of 5% per year for the next 5
years.
▪ How much will the property be worth at the end of
the 5th year?
Problem #4
▪ You are being offered an investment that will pay
you $319,070 dollars in 5 years. You know that you
can get a return of 5 percent on your existing
investments. How much can you pay for the
investment today and still earn 5 percent?
Solutions to Problems
• #1 =FV(.1,20,0,50,0)
=$336.37
• #2 =PV(.15,10,0,80,0)
=$19.77
• #3 =FV(. 05,5,0,-250000,0) =$319,070
• #4 =FV(. 05,10,0,319070,0) =$-250,000
Review-The Present Value
of a Lump Sum
• Thus far, we have been solving for either the Future Value or
Present Value of a single sum or dollar amount.
• But with Excel, we can find any variable by simply solving for
what we don’t know.
Using Excel to Find Other Variables
▪ You can purchase a property today for $250,000
that will appreciate at 5% per year to $319,070.
How long will that take to happen?
▪ PV = $250,000
▪ Rate = .05
▪ FV = $319,070
▪ Nper = ??
▪ PMT = 0
▪ Type = end or 0
Using Excel to Find Other Variables
• =nper(rate, pmt, PV, (FV), type)
• =nper(.05,0,250000,-319070,0)
• 5.00 years
Using Excel to Find Other Variables
▪ You can purchase a property today for $250,000
that will appreciate to $319,070 in five years. What
is the rate of appreciation?
▪ PV = $250,000
▪ Nper = 5
▪ FV = $319,070
▪ Rate = ??
▪ PMT = 0
▪ Type = end or 0
Using Excel to Find Other Variables
• =Rate(nper, pmt, PV, (FV), type, guess)
• In Excel, if you need to find a ‘Rate’, you need to give it a rate
to get the program going. So you ‘Guess’ a rate.
• I will always recommend 10% or .10
• =Rate(5,0,250000,-319070,0,.1)
• 5.00%
Using Excel to Find Other Variables
• =Rate(nper, pmt, PV, (FV), type, guess)
• =Rate(5,0,250000,-319070,0,.1)
• 5.00%
• Also note that we have two whole numbers (250,000 and
319,070). Whenever that occurs, one # must be negative.
Doesn’t matter which one.
REAL572
Commercial Real Estate:
Investment and Analysis
V. Compounding
Effects of Compounding
• 8% COMPOUNDED ANNUALLY
• 8% COMPOUNDED MONTHLY
• What is the FV of a $1,000
assuming 8% compounded
annually?
• What is the FV of a $1,000
assuming 8% compounded
monthly?
• N = 1 year
• N = 12 months
• I/Y = 8%
• I/Y = 8%/12 = .0067 = .67%
• PV = 1000
• PV = 1000
• =FV(rate, nper, pmt, PV, type)
• =FV(rate, nper, pmt, PV, type)
• =FV(.08,1,0,-1000,0)
• =FV(.0067,12,0,-1000,0)
• FV = $1080.00
• FV = $1083.00
Effects of Compounding
• Important to keep the time periods consistent with rate and
time.
• For example:
Effects of Compounding
• Example:
▪ You want to remodel. It will cost you $10,000 and
you can contribute $150 at the end of each month
into a savings account that earns 7% interest
compounded monthly.
▪ How many months will it take before you have
enough money to remodel?
Effects of Compounding
• What do you know?
• Pmt = $150
• FV = $10,000
• Rate = 7% compounded monthly or .583% or .00583
• Type = End = -0• Nper = ????
Effects of Compounding
• =nper(rate/12, pmt, pv, fv, type)
• =nper(.005833,-150,0,10000,0)
• ANSWER??
• 56.48 months or 4.7 years
REAL572
Commercial Real Estate:
Investment and Analysis
VI. Excel Applications (Examples)
Problem #5
• You can buy a piece of land for $1,000,000. You think you
will be able to sell it to a developer in about 5 years for twice
that amount. You think an investment with this much risk
requires an expected return of 20% per year. What do you
think the actual return will be?
• What is this problem asking for?
54
Problem #5
• What do you know?
• PV = $1,000,000
• FV = $2,000,000
• Nper = 5 years
• Type = End = -0• Rate = ???
• Given you want a 20% return, should you do this?
Problem #6
• You’re interested in a property that you think will be worth
$1,000,000 in five years. How much would you be willing to
pay assuming you need a 15% return?
• Suppose you’re wrong, and you can only really expect to sell
it for $900,000 at that time. How much would you reduce the
price to maintain the 15% required return?
56
Problem #6
• Part 1-What do you know?
• FV = $1,000,000
• Nper = 5 years
• Rate = 15%
• Type = End = -0• PV = ??
• How much should I pay for the land?
Problem #6
• Part 2-What do you know?
• FV = $900,000
• Nper = 5 years
• Rate = 15%
• Type = End = -0• PV = ??
• How much should I pay for the land?
Solutions to Problems
• #5 =rate(5,0,-1000000,2000000,0,.1) =14.8%
• No, return too low
• #6 =PV(.15,5,0,1000000,0)
=$497,177
• #6 =FV(.15,5,0,-900000,0)
=$447,459
• Reduce price by
$49,718
Problem #7
• Jim makes a deposit of $12,000 in a bank account. The
deposit is to earn interest compounded annually at the rate
of 6 percent for seven years.
• A. How much will Jim have on deposit at the end of seven
years? (Hint: What is future value?)
• B. Assuming the deposit earned a 6 percent rate of interest
compounded quarterly, how much would he have at the end
of seven years?
Problem #8
• Jones can deposit $5,000 at the end of each six-month
period for the next 12 years and earn interest at an annual
rate of 8 percent, compounded semiannually.
• A. What will the value of the investment be after 12 years?
• B. If the deposits were made at the beginning of each period,
what would the value of the investment be after 12 years?
Problem #9
• Suppose you deposit $1,250 at the end of each quarter in an
account that will earn interest at an annual rate of 10 percent
compounded quarterly. A. How much will you have at the
end of four years?
• B. If you deposit at the beginning of each quarter, how much
will you have at the end of four years?
Problem #10
• A building will require capital improvements of $1 million in
five years.
• Assuming monthly cash flow of $20,000 at the end of every
month, how much of the cash flow must the owners set aside
each month in a sinking fund assuming deposits will earn a
compounded annual return of 6%, compounded monthly?
Solutions to Problems
•
•
•
•
•
•
•
•
#7A =FV(.06,7,0,-12000,0)
=$18,043
#7B =FV(.06/4,7*4,0,-12000,0)
=$18,206
#8A =FV(.08/2,6*12,5000,0,0)
=$195,413
#8B =FV(.08/2,6*2,5000,0,1)
=$203,230
#9A =FV(. 10/4,4*4,1250,0,0)
=$113,574
#9B =FV(. 10/4,4*4,1250,0,0)
=$116,414
#10 =PMT(. 06/12,5*12,0,1000000,0)
=$14,333 per month
REAL572
Commercial Real Estate:
Investment and Analysis
VII. Multiple Cash Flows
Multiple Cash Flows
• Thus far, we have been solving for either the Future Value or
Present Value of a single sum or dollar amount.
• We also now know that we can find any variable in these
problems by simply solving for what we don’t know.
• We will now turn our attention to problems with cash flows
over multiple years
Multiple Cash Flows
• FV= (rate, nper, -pmt, PV, type)
• Type- Enter either 0 or 1
• Regular Annuity (0)
▪ an annuity in which the payments occur at the end
of each period (eg. Dividends)
• Annuity Due (1)
▪ an annuity in which the payments occur at the
beginning of each period (eg. Rent)
Multiple Cash Flows
▪ You plan to deposit $1,000 at the end of each year in
an account that pays 10 percent interest per year.
How much will the account be worth at the end of five
years?
▪ So you are looking for a future value at the end of five
years.
▪ This problem is a ‘Future Value of a Regular Annuity’
▪ Also known as ‘Future Value of an Annuity in Arrears’
Multiple Cash Flows
• Using Excel
• =FV(rate, nper, -pmt, PV, type)
• =FV(.10, 5, 1000,0, 0 )
• ANSWER??
• $6,105.10
• Now let’s assume deposits are at the BEGINNING
Multiple Cash Flows
• Using Excel
• =FV(rate, nper, -pmt, PV, type)
• =FV(.10, 5,-1000,0, 1 )
• ANSWER??
• $6,715.61
Multiple Cash Flows
• As before with single sum problems, we can also solve for
any variable via Excel
• Assume you earned $6,715.51 by placing $1,000 at the
beginning of each year into an account bearing interest at a
rate of 10% per year. How many years does it take to
accumulate the amount?
Multiple Cash Flows
• nper= (rate, pmt, PV, FV type)
• =nper(10%,-$1000,0,$6715.61,1)
• ANSWER??
• = 5 years
Multiple Cash Flows
• Example
▪ You want to have $20,000 available at the end of
five years.
▪ How much would you need to deposit at the end of
each year in an account earning 8 percent interest
to make this possible if the interest is compounded
annually?
▪ Called a “sinking fund” problem
Multiple Cash Flows
• PMT= (rate, nper, -pv, fv, type)
• PMT= (8%, 5, 0, 20,000, 0)
• ANSWER??
• $3409.13
• We must deposit $3,409.13 at the end of each year to have
$20,000 if we earn 8%/year
Multiple Cash Flows
• Example
▪ Suppose you are offered an opportunity to receive
$1,600 at the end of every year for 10 years.
▪ If you require a 7 percent rate of return, how much
would you be willing to pay to get in on this
opportunity?
Multiple Cash Flows
• PV= (rate, nper, pmt, fv, type)
• PV= (7%, 10,$1600, 0, 0)
• ANSWER??
• = $11,237.73
REAL572
Commercial Real Estate:
Investment and Analysis
VIII. Uneven Cash Flows
Uneven Cash Flows
• While real estate may produce even cash flows over a period
of time (Eg. I will receive annual rent of $2,000/year for the
next five years to house an antique car in my garage); most
time they are not. Consider the following cash flows:
• $1,000 at the end of year 1
• $2,000 at the end of year 2
• $12,000 at the end of year 3
Uneven Cash Flows
• Excel uses the NPV function to discount cash flows
• =NPV(rate, Value 1, Value 2,….Value n)
• =NPV(.05,1000:12000)
• =$13,132.49
Uneven Cash Flows
• Let’s keep the cash flows but change the discount rate to
10%
• =NPV(rate, Value 1, Value 2,….Value n)
• =NPV(.1,1000:12000)
• =$11,557.76
REAL572
Commercial Real Estate:
Investment and Analysis
IX. Net Present Value (NPV) and Internal Rate of Return (IRR)
Uneven Cash Flows
• NET PRESENT VALUE= PV of Benefits minus the PV of
Costs
• NPV Decision Rule:
▪ If the NPV is ≥ 0, then make the investment
▪ If the NPV is < 0, do not make the investment
Uneven Cash Flows
• NET PRESENT VALUE= PV of Benefits minus the PV of
Costs
• Let’s return to our cash flows and 5% discount rate
• Assume you can purchase this property for $12,000.
• Would you make this investment?
Uneven Cash Flows
• We still use the NPV function to discount cash flows but we
add one more factor
• We subtract the investment
• =NPV(rate, Value 1, Value 2,….Value n) – Investment
• =NPV(.05,1000:12000) - 12000
• =$1,132.49
▪ If the NPV is ≥ 0, then make the investment
▪ If the NPV is < 0, do not make the investment
Uneven Cash Flows
• Now let’s change the discount rate to 10%
• =NPV(rate, Value 1, Value 2,….Value n) – Investment
• =NPV(.1,1000:12000) - 12000
• =($442.24)
▪ If the NPV is ≥ 0, then make the investment
▪ If the NPV is < 0, do not make the investment
Comparing Investments using NPV
▪
▪
▪
▪
Project #1:
o Pay $10,000 today for an investment that pays:
• $1,000 at the end of year 1
• $2,000 at the end of year 2
• $12,000 at the end of year 3
Project #2:
o Pay $10,000 today for an investment that pays:
• $1,000 at the end of year 1
• $12,000 at the end of year 2
• $1,800 at the end of year 3
Compute the NPV for each project using a 16% discount rate
Which project is the better investment?
Comparing Investments using NPV
• Set up the cash flows in Excel
• 2. =NPV(disc rate, cf1, cf2, ….,cfn) MINUS initial investment
Determining Rates of Return
• So far, we have been focused on how much you should be
willing to pay to achieve a particular rate of return.
• We have also now determined whether our investment will
result in a Positive or Negative Net Present Value given cash
flows and a discount rate.
• But sometimes, you know the cost and the expected benefits
and want to know the rate of return provided by the investment.
• This is known as the investment yield or internal rate of return
(“IRR”).
Determining Rates of Return
• CF0 = -$3,000
• CF1= $500
• CF2 = $1,000
• CF3 = $2,000
• =IRR(Values, Guess)
Determining Rates of Return
• How can we use the IRR to make investment decisions?
▪ If the IRR is greater than our required rate, then we
should……..
o Make the investment
▪ If the IRR is less than our required rate, then we
should……
o Avoid the investment
REAL572
Commercial Real Estate:
Investment and Analysis
X. Net Present Value (NPV) and Internal Rate of Return (IRR)
Examples
Problem #11
• Assume you want to invest in a property that will produce
NOI for year one of $35,000. NOI will increase by 3% each
year thereafter. Assume you pay $325,000 for this property
and you will sell it for $400,000 in year 5. You want to earn at
least a 12% return on your investment.
▪ What is your NPV?
Problem #12
• Assume you want to invest in a property that will produce
NOI for year of $35,000. NOI will increase by 3% each year
thereafter. Assume you pay $325,000 for this property and
you will sell it for $400,000 in year 5.
▪ What is your internal rate of return?
Problem #13
• Walt is evaluating an investment that will provide the following
returns at the end of each of the following years: year l, $12,500;
year 2, $10,000; year 3, $7,500; year 4, $5,000; year 5, $2,500;
year 6, $0; and year 7, $12,500. Walt believes that he should
earn 12 percent compounded annually on this investment.
• How much should he pay for this investment?
• What if he expects to earn an annual return of 9 percent
compounded monthly?
• How much should he pay?
Problem #14
• You are considering the purchase of some raw land. If the
property is expected to be worth $50,000 in 15 years, what is
the present value of this investment? Assume there are no
intermittent cash inflows or outflows and that the investor
expects to earn a 10% annual return on such investments.
95
Problem #15
• What is the maximum price you should pay today for the
right to receive $10,000 per year for 20 years from a piece of
rental real estate if the series of rental payments are
discounted at a 10% annual rate? Assume also that the
property will be worth $50,000 when you sell it at the end of
the 20-year period.
96
Problem #16
• Assume a $400,000 investment in a small shopping center is
expected to produce the following annual cash flows over a
five- year holding period. CF1, $37,000; CF2, $38,100; CF3,
$39,253; CF4, $40,431; CF5, $504,000. What is the IRR on
this investment?
97
Problem #17
• You purchased a parcel of land today for $50,000.For how
much will you have to sell the property in 15 years to earn a
10% annual return on both your initial $50,000 outlay and the
expected annual payments of $1,000 for property taxes and
insurance? Assume these funds could be invested at
comparable risk to earn a 10% annual return.
98
WHAT’S IMPORTANT TO TAKE FROM
THESE SLIDES?
1. Being able to solve TVM functions in Excel
a. FV Compounding
b. PV Discounting
c. Single Period
d. Multiple Period
e. Unveven Cashflows
2. Being able to INTERPRET the TVM solutions meaning