Equivalence of cash flows
Construction Engineering 221
Equivalence of cash flows
• Non- annual compounding (discounting)
– A sum of money invested at 4% annual
interest is compounded semi-annually. How
many years will it take for the invested sum to
double in value:
•
•
•
•
i = 2%, F = 2P, n = ?
Go to the 2% tables (page 100)
Find n where F/P = 2 (n= 35)
35 semiannual periods equal 17.5 years
Equivalence of cash flows
• If a 5 year balloon lease has a 10% annual
rate, but interest is charged monthly (like a
credit card), what will be the buyout
amount on a $2500 purchase?
– F = 2500(1 + .10/12)60 or
• 2500(1.6453) = 4113.25
•
Equivalence of cash flows
• What if the lease caps are compounded
annually:
– F = 2500 (n=5, i=.10) or1.6105
– Buyout is 4026.25
– Lender makes an additional $87 merely by
compounding the interest monthly instead of
annually
Equivalence of cash flows
Cash flow factors and symbols
take the formula and solve using table
factors. Expressed as:
F = P(F/P, i, n) or
P = F(P/F, i, n)
Other equivalencies use the same
procedure
Equivalence of cash flows
• Uniform series equivalence
– Repeating cash flow for a known number of
periods is called and annual amount (like a
mortgage payment). Can be estimated (like
maintenance) or known (like rental income)
– Future worth of a annual payment (called an
annuity) is F = A(F/A, i, n) where A is the size
of the annual amount
Equivalence of cash flows
• Sinking fund is an account that you must
make deposits in each year in order to
have F dollars at n periods in the future.
Therefore, the A/F factor is called the
sinking fund factor, and A is the sinking
fund allocation
Equivalence of cash flows
• An annuity is a series of equal payments
made over a period of time (for instance, a
bond that pays interest annually).
Therefore, the P/A factor is called the
annuity factor and A is the annuity amount
• When the stream of payments is
“perpetual” (like a building generating
rents), the term capitalized cost is
sometimes used.
• Capitalized cost is P = A/i
Equivalence of cash flows
• Assume you have a client who wants to net
lease the project on a 25 year lease. They can
afford $20,000 a month in lease payments
($240,000 per year using end of year
convention)
• What should the capital budget be for the new
building if you are the turnkey contractor and
your cap rate is 8%?
• P = 240,000(10.6748 {p. 124}), or $2,562,000
Equivalence of cash flows
• If the building could be re-leased for
another 25 years at the same rate, the
initial cost budget would be:
– 240,000 (12.2335) = 2,936,000
– An infinite series of 240,000 annual payments
has a present worth of 240,000/.08 or
3,000,000, so you can see that for many
buildings, the capitalized cost formula is a
viable option
Equivalence of cash flows
• Can calculate “past worth” by setting t=o at
some arbitrary point in the cash flow series.
Sometimes this calculation is needed in lawsuits
to award damages or determine remedies for
loss of use
• Can also calculate periods needed to earn a
multiple of investment. Table 2 on Page 16 give
double and triple earnings.
– n= log X/ log (1 + i), where X is the multiple desired
on the initial investment
Equivalence of cash flows
• Variable (non-standard flows)
– Gradient cash flows are better
representations of many equipment
maintenance schedules than uniform series
(gets more expensive to maintain as the
equipment gets older)
– The gradient starts in year 2 because it is the
INCREASE in annualized payments that are
of interest
Equivalence of cash flows
• P/G factor calculates the present worth of
the ESCALATING costs, not the base
costs
• Base costs must be calculated using the
P/A factor
• Maintenance costs are the most typical
example of gradient series
Equivalence of cash flows
• Stepped cash flows are handled as two
separate cash flows using the
superpositioning technique (see example
on page 19). This would apply to a rental
contract with an escalation clause
• Missing or extra cash flows can be
handled similarly (major overhaul
example), as can beginning of period
payments
Equivalence of cash flows