Time Value of Money
Review
Cash Flow Types
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Lump Sum (one time cash flow)
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Series of Even Cash Flows
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Uneven Cash Flows
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Anywhere on timeline
Examples: Set aside money for down payment in 5 years; expect to sell a property in 5
years (what is that worth today?);
Annuity – equal periodic cash flows on any set interval
Examples: Loan payment on a commercial property with a 20-yr amortization; value of a 5year lease to a landlord
Irregular amounts and/or irregular intervals
Examples: Value of a building that has income that grows over time; value of a 5-year lease
with escalating lease rates; returns an investor receives with irregular distributions from the
sponsor.
Present Value – What is the value today?
• PV comes from discounting future cash flows at an
appropriate discount rate (i.e. risk-adjusted)
• Commonly used for valuation
• What is the value of a real estate investment that produces a set of
cash flows over a 5-yr horizon?
• Commonly used with loans
• How much can I borrow if I can afford $1,500 payments each
month?
Present Value - Calculations
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Two main functions in excel – PV and NPV
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PV – use this function to find PV of a sum and/or annuity
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=pv(rate, nper, pmt, [fv], [type])
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Rate – this is the periodic rate for the calculation
Nper – this is the number of periods
Pmt – this is the amount of cash flows that occur at even intervals. It can be 0.
Fv – this is the cash flow that occurs at the very end. It is an [optional] argument.
Type – this is an [optional] dummy variable. Entering “0” (or omitting) means end-of-period cash flows.
Entering a “1” means beginning of period cash flows.
NPV – use this function to find PV of a series of uneven cash flows. Also works well for lump sum
or annuity.
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=npv(rate, cf `1’: cf `n’)
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Rate – this is the periodic rate
CF 1 – the cash flow for period 1
CF n – the last cash flow in the series
PV example 1 – lump sum
• If you think you can sell your parcel of land for $2.5
million in 5 years, what is the value today if the
appropriate discount rate is 15%?
=pv(.15,5,0,2500000)
=pv(B2, B3, B4, B5)
OK
Using cell references better
Note: The sign of the cash flows is VERY important – except for the
final answer. With a final answer, you can format it however you would
like (positive or negative).
PV example 2 - annuity
• If you will be paying $479 per month for the next 6 years on
the car you buy after graduation, how much did you borrow
assuming the interest rate is 4.9%?
=pv(.049/12, 6*12, -479) – or use cell references
=pv(.049, 6, -479*12) is wrong
Note: Any time you see an interest rate, discount rate, etc. it is
an ANNUAL rate (unless specified otherwise). You must adjust
the annual rate to a periodic rate based on the periods that are
applicable (months, weeks, days, etc.).
PV example 3 – lump sum + annuity
• The lease payment on the car you purchased calls for $479
payments for three years followed by a buyout amount of
$16,500. Assuming an interest rate of 4.9%, what is the
effective purchase price of the car?
=pv(.049/12,3*12,-479,-16500)
Note: Frequently we are concerned with making a decision
based on the calculation. In this case, assume that you could
buy the car for $29,800 in cash today? What is the better
financial decision (i.e. which has lowest cost in today’s dollars)?
NPV example 1 - annuity
• How much did you borrow for a car that will cost $479
each month for the next 6 years assuming an interest
rate of 4.9%?
=npv(.049/12,cash flows 1-72)
In this case, PV is probably the better tool!
NPV example 2 – uneven cash flows
• You are expecting to receive $10,000 in net rental income at
the end of the 1st year from your investment property.
Assuming this amount grows by $1,000 per year for the next 5
years AND that the property is sold for $150,000 at the end of
the 5th year, what is the value of this property? Assume a
12% discount rate.
Note: There are only 5 cash flows here! At the end of the 5th
year, you collect $14,000 from rental income and a simultaneous
$150,000 from selling the property.
Future Value
• To find the future value, you are compounding cash flows
at an appropriate rate of return.
• If you have been paying for 9 years on your home loan, how
much do you still owe if your payments are $1,450 per month
and the interest rate on the loan was 4.5%?
• If you purchased a rental house for $150,000 eight years ago
and the appreciation rate has been 5.5% per year, what should
the home be worth today?
• If I set aside $1,000 each month for 5 years, how much will I
have at that time if I earn 6% per year?
Terminology sidebar - rates
• Rates show up all the time in the language we use for finance and
for real estate. Here are a few terms that are synonymous and the
usage of which differ only because of context:
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Interest rate
Discount rate
Rate of return
Return
IRR
• Any time I use the language “appropriate rate”, I refer to a
fundamental notion in finance – that discount rates (required rates of
return) should have some relation to the risk involved.
Future Value - calculations
• Basically, the same arguments as the fv function – just a different
order
=fv(rate, nper, pmt, [pv], [type])
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Rate – periodic rate
Nper – number of periods
Pmt – amount of the cash flows that occur at regular intervals
[PV] – this is the lump sum that occurs once at the start. It is an [optional]
argument.
• [Type] – this is an optional “dummy variable”. 0 (or omitted) means end of
period cash flows; 1 means beginning of period cash flows.
FV example 1 – lump sum
• If you invested $5,000 today at an average annual rate
of 8%, how much money would you have in 35 years?
=fv(.08,35,0,-5000)
Note: The (-) sign in front of the $5,000 is important. A
purchase of an investment will be a cash outflow.
FV example 2 - annuity
• If you invest $500 per month into your 401(k), how much
will you have in 20 years if you earn an average of 9%
per year on the investments?
=fv(.09/12,20*12,-500) or use cell references
Make sure you take into account the periods stated in the
problem. There are cash flows occurring every month.
FV example 3 – mixed cash flows
• This is a VERY useful example – how much do you have left on your
loan if you originally borrowed $250,000 at a rate of 4.75% with a
30-yr term and you have made 7 years of payments?
Solution: This is a two-part question. The first part is to figure out the
payment (it’s $1304.12). Then use that to figure out the loan amount
remaining.
=fv(.0475/12, 7*12, -1304.12, 250000)
Make sure you pay attention to the signs!
Rates, Returns, IRR
• Can solve for interest rate, rate of return or some other rate.
• If you purchase a house in California 33 years ago for $125,000 and
it is not worth $1.5 million, what has been the average annual
appreciation rate over that time?
• If you purchased a car for $20,000 and you have 6 years of
payments at $350 per month, what interest rate are you borrowing
at?
• If you buy a property for $1 million and the cash flows for a 3-year
hold are expected to be $100K, $125K and $1.3 million respectively,
what is your return on this investment?
Rate, IRR functions
• Two main functions in excel to solve for interest rate (discount
rate, rate of return, etc.)
• Rate
=rate(nper, pmt, pv, [fv], [type], [guess])
• IRR
=irr(cash flow 0: cash flow “n”) – the CF0 is generally a negative
number representing your investment cash outflow
Note that the answer you get in both cases is a periodic rate. So, if the
problem you are working on has periods that are months, you need to
take the answer you get and “annualize” it. For our purposes, just
multiply by 12 (if monthly) or 52 (if weekly) or 365 (if daily).
Rate example 1 – annual compounding
• You purchase a rental house for $125,000 and sell it 13 years later
for $300,000. What was the annual appreciation rate for the house?
=rate(13, 0, -125000, 300000)
• You have saved and invested $10,000 to this point in your life. If
you add $6,000 each year to your investments for the next 15 years,
what rate of return will you need to reach your goal of $500,000?
=rate(15, -6000, -10000, 500000)
Rate example 2 – monthly compounding
• You borrowed $300,000 for a rental home purchase.
The payments are $1,500.07 for principal and interest
each month for the next 30 years. What is the interest
rate on the loan?
=rate(30*12,-1500.07,300000,0) * 12
Note: The answer needs to be reported as an annual rate.
IRR example 1 – annual cash flows
• Assume you purchase a property for $1 million. For the first year,
the net operating income (NOI) for the property is $80,000. Each
year the NOI grows by 4%. The property value increases by 5% per
year and is sold at the end of the 10th year. What is the IRR on this
investment? Alternatively, what is the return on this investment?
=irr(-1000000, 80,000, 83200 . . . , 1742750) – but you should be using
cell references with these numbers!
Note: There is an FV calculation necessary to get the value of the
property at sale: =fv(.05, 10, 0, -1000000).
IRR example 2 – monthly cash flows
• You made an initial investment of $250,000 in a rental
house. For the first year, you receive $1,500 each
month in net income and sell the property for $260,000
at the end of the year. What was the annual rate of
return on the investment?
=irr(-250000,1500, 1500, ...) * 12
Determining how long
• On occasion, you will need to determine how long it will
take to achieve a certain goal or target.
• If you are saving $1,000 per month and earning 8% (annual rate
of return), how long will it be until you have $90,000?
• If you are paying $2,500 each month on your $300,000 mortgage
(5.125%, 30-year term), how long will it be until the balance is
down to $100,000?
NPER
• The function to use is the NPER (number of periods)
function
=nper(rate, pmt, pv, [fv], [type])
NPER example 1
• How long will it take you to save up $100,000 if you start
today with $2,500 and put in an extra $400 each week?
Assume a 6% rate of return.
=nper (.06/52, -400, -2500, 100000)
Note: The result will be the number of weeks (because that
is the period length in view in this problem.
NPER example 2
• How long will it take to pay down the $400,000 loan on
my duplex to $200,000 if the required payment is $2,200
per month and the interest rate is 3.9%?
=nper(.039/12, -2200, 400000, -200000)
PMT function
• The function to find a payment amount (equal amount at
equal interval, i.e. annuity payment) is the PMT function
=pmt(rate, nper, pv, [fv], [type])
• What is the required payment on a 30-yr mortgage if the loan
amount is $250,000 and the interest rate is 5.25%
• How much do I need to set aside each month to reach
$1,000,000 in retirement savings assuming I have 30 years and
will earn 8% per year?
PMT example 1
• What is the required payment on my $30,000 car loan if
the term is 6 years and the interest rate is 4.9%?
=pmt(.049/12, 6*12, 30000, 0)