Hi there! This is the first of what I plan to make a weekly column which will consist of a Time-Value of Money (or TVM) problem and then a brief discussion of how you’d go about solving it. The scenarios will be multi-part questions, and I’ll try to explore some of the more interesting aspects of TVM as the weeks progress.

To play along at home, you’ll need a financial calculator, and if you want the BEST financial calculator, then whip out your smart phone and download our 10bii Financial Calculator app (or buy it on Mac from the Mac App Store). It’s not free, but you won’t be disappointed in the investment you make in this great piece of software.

Okay, let’s get right into the first scenario.

**THE SCENARIO:**

I put $100,000 down and borrowed $350,000 to buy a house awhile back. The mortgage amortizes fully over 30 years, has monthly payments, and the rate is 6.625%.

I still owe $338.465.83.

a) What’s my mortgage payment (Principal and Interest)?

b) How long have I been paying on the mortgage?

Sadly, my Aunt Matilda passed away awhile back, and she left me a nice inheritance. After the dust settled, I was left with $100,000 that I want to use to pay down my mortgage balance (note, this is **not** a refinance or recast of the loan).

If I pay $100,000 on my mortgage today,

c) what’s my new payment?

d) how much time does this extra payment shave off my loan?

e) how much interest does this payment save me over the remaining life of my loan?

**THE SOLUTION:**

Whew, that’s a lot. Let’s go through it step by step.

Set up the problem this way:

P/YR |
12 | |||

N |
I/YR |
PV |
PMT |
FV |

360 | 6.625 | 350,000 | ??? | 0 |

a) Solve for PMT. My mortgage payment is (negative) **$2,241.09** per month.

b) Put the currently-owed balance of (negative) $338,465.83 into FV and solve for N. I’ve been paying for **34 months**.

c) This is a trick question. Because I just paid down principal, my payment *did not go down at all*. It did not change at all. It remained the same. The reason for this is that my payment for a fixed loan is set when the loan is created. All I’ve done by paying extra is made it so I’ll have paid it off sooner. So my payment is still (negative) $2,241.09 per month. If I wanted to reduce my monthly payment, then I’d have to create a new loan through either ‘recasting’ it or ‘refinancing’ it. We’ll go into this in greater detail in future articles.

The next part requires a bit of a change to the setup. We take my currently-owed balance of $338,465.83 and reduce it by $100,000 (we’re paying off $100,000 of principal). The resulting amount I now owe is $238,465.83. Put that into PV (make sure it’s positive, though). Leave I/YR and PMT alone. Make sure FV is 0 (the loan fully amortizes), and solve for N.

d) My new N is 160.84. Since I have to make a payment on any partial month, my new **N is effectively 161**. Since my original loan had a term of 360 months, and I’d already made 34 payments, I had 360 – 34 = 326 payments left. Now I only have 161 payments left. So paying down $100,000 on my principal reduced my number of remaining payments by 326 – 161 = **165 months saved**.

To figure out how much interest the payment saves me, we have to understand that each payment is made up of principal and interest. At the end of the loan, all of the principal is gone. And the principal for the loan starts out at the same place (I currently owe $338,465.83).

So to figure out the difference in total interest, we just add up all of the payments we make under the two scenarios (paying down $100,000 versus not doing so), and the difference is the amount of interest we didn’t have to pay.

**Scenario 1:** I **don’t** make a $100,000 payment today

326 payments of $2,241.09 makes a total of $730,594.81 coming out of my pocket.

**Scenario 2:** I **do** make a $100,000 payment today

$100,000 plus 161 payments of $2,241.09 makes $100,000 plus $360,815.23, for a total of $460,815.23 coming out of my pocket.

The difference between scenarios 1 and 2 is $730,594.81 – $460,815.23, or **$269,779.58 in interest saved**. Aunt Matilda seems to have given me a lot more than that hundred grand! Note that this number is slightly off because my last payment is going to be a little less than $2,241.09 (remember that our new N is really 160.84 and not 161), but it’s pretty close. If you want to figure it out to the penny, feel free; I’ll leave that to you.)

Note that if you wanted to cleverly just find out the value of the payments I didn’t have to make, you’d multiply the 165 payments that ‘went away’ when I paid the $100,000 by the payment, and get $369,779.58 saved. Because the numbers match when I do the problem two different ways, that gives me confidence that the answer is correct.

Okay, that’s it for me for this week. Enjoy and see you next time!