[Reposting abridged version.]

Anne,

You wrote:
Can anyone help me get a comfort level on what
the PV is for this series of cash flows?

The short answer is: if you want to compare XNPV()
and NPV() results, use RATE(p,,-1,1+r) for the
NPV() rate, where "r" is the annual rate and "p"
is the number of periods per year.

I think you are asking two questions:

1. Shouldn't XNPV() be less than NPV(), given that
XNPV() uses a daily compounding rate, whereas NPV()
uses a monthly compounding rate in this case?

2. Which compounding rate or period frequency -- daily
or monthly -- is correct for computing PV in general?

The answer to #1 is a qualified "yes": XNPV is less
than or equal to NPV __if__ you use the same assumptions
for determining the period rates. It is your varying
assumptions that cause the contradiction that you
observe. I will elaborate below.

I think a contributing factor to your discomfort is
the fact that your PV computations seem to be incorrect.
My PV computations differ from yours, but they do match
your XNPV and NPV results under varying assumptions
about the periodic rates.

I cannot infer what you did differently. My formulas
are as follows. I will explain them below.

A2:A31: date (1/25/2006, 1/25/2018, 4/25/2018,..., 1/25/2025)
B3:B31: calendar quarters (48, 49,..., 76)
C3:C31: calendar days: =A3-$A$2
G2:G31: cash flows (0, 2,593,876, 5,276,303,..., 437,573)
H3:H31: PV at 16%/4 months: =PV(16%/4,B3,,-G3)
I3:I31: PV at 16%^(1/4) months: =PV(RATE(4,,-1,1.16),B3,,-G3)
J3:J31: PV at 16%^(1/365) calendar days:
=PV(RATE(365,,-1,1.16),C3,,-G3)
L3:L31: PV at 16%/365 calendar days:
=PV(16%/365,C3,,-G3)
H32:L32: NPV, sum of the PVs: =SUM(H2:H31)

My results a
H32: 13,530,060
I32: 15,342,347
J32: 15,325,146
L32: 12,909,932

Note that H32 matches your NPV computations using
both Excel and HP12C. This is the NPV of the monthly
cash flows using 16%/4 for the periodic rate.

Also note that J32 matches your XNPV computation.
This is the NPV of the monthly cash flow using
"16%^(1/365)" for the daily rate -- that is, the
daily rate that results in a 16% annual rate.
"16%^(1/365)" is my stylistic shorthand for the actual
formula, which is (1+16%)^1/365 - 1 or equivalently
RATE(365,,-1,1.16).

Note the important difference in assumptions. Your
use of 16%/4 monthly rate results in an annual rate
of nearly 17%, not 16%. To be consistent with Excel's
XNPV -- that is, to have the same effective annual
rate -- you would want a monthly NPV rate of "16%^(1/4)"
(stylistically; see above). I32 is the NPV using
"16%^(1/4)", and it does indeed exceed J32, as you
expected.

Conversely, L32 is the NPV using 16%/365, the daily
rate computed in the same way that you determined
the monthly rate for H32. Note that L32 is indeed
less than H32, as you expected.

Conclusion: If you want to compare XNPV() and NPV()
results, use RATE(p,,-1,1+r) for the NPV() rate,
where "r" is the annual rate and "p" is the number
of periods per year.

I hope that restores your confidence in the NPV and
XNPV computations, whether you do it with a function
or by summing the PV of the cash flows.

The answer to #2 is less clear: should the NPV be
computed based on daily or periodic compounding?
As a corollary: should the daily or periodic rate
be computed as r/p or as "r^(1/p)" (stylistically;
see above)?

I would argue that there is no single correct answer.

If you are computing the PV of a real investment, I
would use the compounding frequency and method of
computing the rate that fits the investment. For
example, money market instruments compound daily,
and the daily rate is r/365. Thus, for example, if
you are comparing two investments that both yield
10% annually, but one compounds daily and that other
compounds monthly, the first will correctly have the
lower PV.

On the other hand, if the discount rate is arbritrary
(e.g, cost of capital) and especially if it is an
inflation rate, I would use a geometric rate that
preserves the annual rate, i.e. "r^(1/p)" (stylistically;
see above). Thus, for example, if the cost of an
opportunity grows at an inflation rate of 3% per year
(example: PV of the cost of materials), it should
not matter whether we consider daily or monthly cash
flows; the annual result must still be 3% per year.

But when using NPV to compare opportunities with
arbitrary discount rates and the same periodicity,
where the relative PV is important, but not the actual
number, I would use the simple rate of r/p just because
it is easier to remember and compute.

I hope that helps.