Finding present value of growing annuity when growth doesn't coincide with payments

[DexterOnline]

Help me find present value of an annuity in amount of 1000 dollars coming in at the end of each month for 3.5 years

Payments grow at a nominal rate of 6% compounded monthly each year starting at the start of 2nd year and stay at the same growth rate for the whole year

Nominal rate of return on the monthly payments is 9% compounded monthly

 

[stapel]

Help me find present value of an annuity in amount of 1000 dollars coming in at the end of each month for 3.5 years

Payments grow at a nominal rate of 6% compounded monthly each year starting at the start of 2nd year and stay at the same growth rate for the whole year

Nominal rate of return on the monthly payments is 9% compounded monthly

What formula(s) did they give you for this? How far have you gotten in applying that information?

Please be complete. Thank you!

 

[DexterOnline]

@Staples

I am sorry, my mentors asked me to find the present value using a formula. I showed them the calculations by creating a schedule of transactions in Excel for all 42 months but those old folks weren't happy with my performance. According to them, such time value of "no money no honey" problems must always be solved using closed form formulas

@Wilmer

Sorry for my bad English as there is no language institute out here in the "wailing wall street" bay area

You are right on with my intent

The rates are annual and compounded monthly

And yes the payments increase at start of year then stay constant for the whole year till the next year when then increase by the rate once more

Thank You

 

[DexterOnline]

Well, assuming you mean payments increase once annually:

FV = [pu^0(w)(v^30) + pu^1(w)(v^18) + pu^2(w)(v^6) + pu^3(v^6 - 1)(v^0)] / i

PV = FV / (1 + i)^42

Now Dexter, what's the REAL reason why you posted that

The mentors of old on the board are demanding a closed form formula not a sum of compounded or discounted cash flows

 

[DexterOnline]

I did figure out a closed form formula; I'll post it tomorrow....if you're nice

It's just that out of 6080 Time value of no money no honey formulas, its hard to find the one that is fit for the task at hand

 

[jonah2.0]

Well, assuming you mean payments increase once annually:

g = payment increase = .06
i = .09 / 12 = .0075
p = initial payment = 1000

u = 1 + g
v = 1 + i
w = v^12 - 1

FV = [pu^0(w)(v^30) + pu^1(w)(v^18) + pu^2(w)(v^6) + pu^3(v^6 - 1)(v^0)] / i

PV = FV / (1 + i)^42

FV = ~52,796 and PV = ~38,576

Beautiful Sir Denis, simply beautiful!!!
Wouldst thou agree that if g = .06/12 then FV be 49,448.13?

 

[DexterOnline]

Perhaps Sir Jonah will Absinthedly astound us with a way to handle
incomplete years, like your 3 years 6 months or as example 4 years 7 months.

You mentioned someone I love dearly our friend Sir Jonah v2.0 (can't wait for the next upgrade v3.0 )

Please Sir Jonah, lets put together the last piece of the problem to allow for fractional part of year such as 3 yr. 6 mo. or 4 yr. 7 mo.

 

[DexterOnline]

Welcome.
And you can tell your old folks mentors that the formula has a bonus:
not only present value, but also provides future values at last-year-end
plus at last-month-end: they'll stick a red star to your forehead

Just received a Red Star from the mentors of old :smile:

The mentors of old have now changed the original problem slightly and promised me a Red Carpet reception :!: if I could offer them the solution to same problem where the 1st growth begins at the 7th month and stays that way for the remaining 6 months and then 2nd year onwards, the growth changes as before

So, now rather the first growth starting at the start of second year, it begins earlier at the mid of first year. The rest of the problem is the same

Sir Wilmer, can I still use your last formula to find present value of this investment

I be honored to walk on the Red carpet at the reception

 

[DexterOnline]

OK, here's the closed form formula including the part months;
introducing new variable "m", to represent the number of months in
the final year: hope that's not too much work for you to calculate that!

FV = {p*f*[u^n * v^(-12) - v^(12*(n - 1))] / [i*(u*v^(-12) - 1)]} = 43,519.50685...
That's to the end of the complete years.

This updates to the end of the extra months:
FV = FV * v^m + p * u^n *(v^m - 1) / i = 52,796.40626...

PV = FV / [(1 + i)^(n*12 + m) = 38,575.54405...

Sir Wilmer

The mentors of old called me in and charged me with plagiarism for submitting copied work taken from Sir Wilmer

They demanded me to show my own work

I resubmitted the following to the board

PV = [{y^(3)-1} / {y-1}] [{1-x^(-12)} / {x-1}] + y^(3) [{1-x^(-6)} / {x-1}]

Do you think I will get the Red Wine for this one

 

[DexterOnline]

Thank You Sir Wilmer

Did you review my last submission to the board showing my own work

Would that be acceptable, should I be expecting Red Wine

 

[DexterOnline]

Hope you do...so you can leave me alone :cool:

However, doesn't look like a closed form formula; only handles your 3.5 years problem.

Yes that was given for example data

Since I can't seem to find the Latex editor on this forum thus was unable to post as I do on MHF

Let me if I could retry

PV = [{y^(floor(n))-1} / {y-1}] [{1-x^(-1/p)} / {x-1}] + y^(floor(n)) [{1-x^(-[n-floor(n)]/p)} / {x-1}]

where

n is the number of years

p is the length of the period such as 1/12 for months, 1/4 for quarters, 1/2 for half-years, 1 for years

floor(n) denotes the integer part of the number of years

n-floor(n) denotes the fractional part of the number of years

 

[DexterOnline]

Completely absurd!
All that to end up with n (years) and m (months).
Keep it simple: make n and m GIVENS.

Like the i say in FV: FV = A(1 + i)^n
The i is the PERIODIC rate.
It's conversion to such is not part of formula.
It is calculated separately, to become a GIVEN.
"FV = A(1 + i)^n where i is the periodic rate" is STANDARD.

Very Sorry, Sir Wilmer

It is that I write these formulas in a computer program thus when a user inputs a value of N for the year (decimal value), I leave it upto the code to find n (complete years) and m (months)

And when user inputs interest rate i and growth g, the code takes these values and finds two set of annualized values x for interest rate and another y for ratio of annualized growth rate over the annualized interest rate raised to a certain power that happens to be the length of the period

[B]y[/B] = x_g * x_p

[B]x_p[/B] = x^(-1/p)

[B]x[/B] = (1+aey(i,c))^(1/p)
aey(i,c) = (1+i*c)^(1/c) - 1

[B]x_g[/B] = (1+aey(g,g_c))^(1/g_p)
aey(g,g_c) = (1+g*g_c)^(1/g_c) - 1

[B]PV[/B] = [B]A[/B] * [{y^([B]floor(n)[/B])-1} / {y-1}] [{1-x^(-[B]1/p[/B])} / {x-1}] + y^([B]floor(n)[/B]) [{1-x^([B]-[n-floor(n)]/p[/B])} / {x-1}]

[B]i[/B]=9%
[B]g[/B]=6%
[B]A[/B]=1000
[B]n[/B]=3.5
[B]p[/B]=1/12
[B]c[/B]=1/12
[B]g_p[/B]=1
[B]g_c[/B]=1

[B]x[/B]=?
[B]y[/B]=?

[B]PV[/B] = [B]A[/B] * [{y^([B]3[/B])-1} / {y-1}] [{1-x^(-[B]12[/B])} / {x-1}] + y^([B]3[/B]) [{1-x^([B]-6[/B])} / {x-1}]

I am not a social person thus unaware how to tender my formulas for humanoids, I like those androids