Net Present Value questions - I know there are some finance geeks here

[Daisyedwards800]

How do you find the net present value at 03/15/2016 of a stream of 37 annual payments at an annual discount rate of 3.5%, when the first payment starts on 12/31/2016?

[Proud Foot]

In order to give the correct answer there are a few things that you did not outline here.  What is the amount of the annual payment and what is the value at the end of the payments?

The Net Present Value (NPV) of the annual payments(P) at 12/31/2016 would be calculated by this formula: P*((1-(1.035^-36))/ 0.035)

You would then calculate the NPV of the previous answer (V) at 3/15/2016 using this formula: V*(1.035^-(291/365))

[Vanguards and Lentils]

So I don't know if this is actually the finance-y to do this, but here goes... We can first find the net present value at 12/31/2016, and then make a correction afterwards. Call the annual payment N.

the first payment on 12/31/2016 is worth N in 12/31/2016 dollars
the second is worth .965*N in 12/31/2016 dollars
...
the 37th is worth .965^36*N in 12/31/2016 dollars

so the total is worth N(1+.965+.965^2+...+.965^36) = (using a geometric sum) N(1-.965^37)/(1-.965) or roughly 20.93N.

To find the 3/15/2016 present value, you might first compute a "daily" discount rate which is equivalent to the annual discount rate, assuming daily compounding.

(1+daily_rate)^365=1.035, so the daily_rate is roughly .00943%, the amount lost due to inflation each day.

Then the 3/15/2016 present value of 20.93N in 12/31/2016 dollars (which is 291 days later) would be found from

MarchPV(1.0000943^291)=20.93N
MarchPV = 20.36N

[Vanguards and Lentils]

I was a minute late but yes, exactly what he said.

[Daisyedwards800]

$5,000,000 for each annual payment

[Daisyedwards800]

and yearly compounding.... actually not sure about this part.

[Daisyedwards800]

In order to give the correct answer there are a few things that you did not outline here.  What is the amount of the annual payment and what is the value at the end of the payments?

The Net Present Value (NPV) of the annual payments(P) at 12/31/2016 would be calculated by this formula: P*((1-(1.035^-36))/ 0.35)

You would then calculate the NPV of the previous answer (V) at 3/15/2016 using this formula: V*(1.035^(291/365))

Curious where you get .35 from ?

[Proud Foot]

In order to give the correct answer there are a few things that you did not outline here.  What is the amount of the annual payment and what is the value at the end of the payments?

The Net Present Value (NPV) of the annual payments(P) at 12/31/2016 would be calculated by this formula: P*((1-(1.035^-36))/ 0.035)

You would then calculate the NPV of the previous answer (V) at 3/15/2016 using this formula: V*(1.035^(291/365))

Curious where you get .35 from ?

Sorry that was a typo.  I missed the extra 0 to make it 0.035.  I corrected it above and will correct my original post.

[Daisyedwards800]

Where do you get 36 and .035 from?

[Proud Foot]

Where do you get 36 and .035 from?

Good question.  I got the 36 as you are calculating the Present Value as of the first payment date.  Because of this the first payment is not discounted is not counted in the number of payments.  I did leave out the part of what to do with that first payment. 

Here is the updated formula with the payment information you gave below as well as correcting my mistake above.

PV@12/31/16 = 5,000,000+(5,000,000*((1-(1.035^-36))/0.035)
or PV@12/13/16 = 106,452,469

PV@3/15/16 = 106,452,469 * (1.035^-(291/365))
or PV@3/15/16 = 103,572,483

eta: I updated my original post to correctly note that the formula should have a negative exponent where I missed the - when typing it in.

[Daisyedwards800]

Where do you get .035?

[Vanguards and Lentils]

That's how percentages work... percent comes from "percentum" which means "by 100". So 3.5% = 3.5/100 = .035, and N% is N/100

[Proud Foot]

It is the 3.5% discount rate.


I didn't really explain where the numbers came from in my first post so here are the formulas without numbers
 P*(1-(1+r)^-(n-1))/r)+P 
 and V*(1+r)^-n2

Where P = Annual Payment
r = Discount rate
n = number of payments
V = known future value (in this case the PV of the annual payments discounted to 12/31/2016)
n2 = number of periods until the first payment.