and Windows 7!

Last time, I talked about a loan I was looking at recently. It’s got a current balance of $198,984.80, a monthly P&I (Principal and Interest) payment of $1050.31, and has received 60 payments so far from the borrower. Its interest rate, as determined last time, is 4.00%.

**The question:** What was the original balance of the loan?

This is very straightforward. We already have every piece of information we need to solve it.

First things first, make sure the calculator is using 12 Payments per Year.

N: 60 (The borrower has paid 60 payments on the loan)

I/YR: 4.00 (The interest rate on the loan is 4.00%)

PV: (This is what I’m trying to find)

PMT: -1,050.31 (The monthly P&I payment is $1,050.31)

FV: -198,984.80 (The borrower owes $198,984.80 today)

The borrower originally borrowed **$220,000.22**.

It’s likely that the $0.22 is a rounding error and that they really borrowed an even $220K. These can happen because the *actual* payment should be $1,050.313650024 each month (or something similar)… but we can’t pay fractional pennies, so we round to the nearest (or next-highest) penny. Over the course of many months, the extra fractions of a penny add up, throwing off the totals ever-so-slightly. In the end, the differences usually end up being so tiny as to not be worth stressing over.

What do you think? Did you notice that over 16.67% of the time (5 years out of 30), only 9.55% of the money ($21K of $220K) has been repaid? What lessons can you take from that piece of knowledge? Let us know in the comments!

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I was looking at a mortgage this morning, and I was curious as to the interest rate the borrower is paying.

The loan has had regular monthly payments paid over the last 5 years, and $198,984.80 is owed today. The monthly payment is $1,504.59, which includes $454.28 in Escrow impounds (for property taxes and insurance). The loan amortizes fully over 30 years.

**The question:** What’s the interest rate on this mortgage?

This one is pretty straightforward. The only things we don’t know are the monthly Principal and Interest (P&I) payment and the amount of time left on the loan.

To find P&I, we subtract the impounded amount ($454.28) from the total payment ($1,504.59) to get $1,504.59 – $454.28 = $1050.31.

To find the time left, we subtract the elapsed time (5 years is 5 x 12 = 60 months) from the original 30 years = 30 x 12 = 360 months.

360 – 60 = 300 months.

First things first, make sure the calculator is using 12 Payments per Year.

N: 300 (The loan has 300 monthly payments still due)

I/YR: (This is what I’m trying to find)

PV: 198,984.80 ($198,984.80 is owed today)

PMT: -1,050.31 (The monthly payment is $1,050.31)

FV: 0 (The loan amortizes fully)

The loan has an interest rate of **4.00%**.

What do you think? Does that seem like a low rate to you? Or high, given the time frame in which it was originated? Let us know in the comments!

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Last time, I talked about a balance transfer offer I got in the mail, and I simplified the problem using an assumption. This time, I’ll change the assumption to more closely model reality.

To review, the card has a $7,000 credit limit and 0% interest on balance transfers over the first 21 months after the transfer. The fee to do the transfer is 5% of the transferred amount, which is added to the balance on the new card.

Whereas last time, we assumed that no payments were due during the 21 months, this time we’ll assume that 2% of the original balance is owed each month.

**The question:** If I were to get this card and transfer a $6,500 balance to it, what would my effective annual interest rate be if I pay it off at the end of the 21 months, assuming that I pay 2% of the original balance each month?

This one is pretty straightforward, and has a few parts.

- Find the initial balance
- Find the monthly payment
- Find the final balance due
- Find the effective interest rate

First things first, make sure the calculator is using 12 Payments per Year.

**Step 1: Initial Balance**

If we transfer $6,500 to the card and there’s a 5% fee, we’ll start out owing $6,500 x 5% = $325 more than we transferred.

$6,500 + $325 = **$6,825**. This is our initial balance.

**Step 2: Monthly Payment**

The monthly payment is 2% of the initial balance. 2% of $6,825 is $6,825 x 2% = **$136.50**.

**Step 3: Ending Balance **

N: 21 (The offer is for 21 months interest-free)

I/YR: 0 (The card charges 0% interest during this period)

PV: 6,825 (I’m transferring a $6,500 balance to the card, but I start out owing 5% more)

PMT: -136.50 (The monthly payment is $136.50)

FV: (This is what I’m trying to find)

After the 21 months, I’ll still owe the company **$3,958.50**.

**Step 4: Effective Rate**

First things first, make sure the calculator is using 12 Payments per Year.

N: 21 (The offer is for 21 months interest-free)

I/YR: (This is what I’m trying to find)

PV: 6,500 (I’m transferring a $6,500 balance to the card, which means that I’m borrowing $6,500 from the card company)

PMT: -136.50 (The monthly payment is $136.50)

FV: -3,958.50 (I’ll have to pay back $3,958.50 at the end)

The effective interest rate on the ‘0% interest’ 21-month loan from the credit card company is **3.50%**. Not zero, but much lower than whatever card I’d be transferring the balance from.

To review, last time we determined that the effective rate was 2.79%, but this time the rate is higher. The reason for this is that the longer we wait before paying the money back, the lower the effective rate will be. Since last time, we didn’t pay a penny until the end of the 21 months, but this time we started paying off the balance right away, the effective rate this time is somewhat higher than last time.

What do you think? Would you prefer to pay down the balance slowly over the 21 months, or wait until the end to pay off the whole thing? Why do you feel this way? Let us know in the comments!

]]>and Windows 7!

I recently got a balance transfer credit card offer in the mail, and I was curious so I took a look at the details.

The card has a $7,000 credit limit and 0% interest on balance transfers over the first 21 months after the transfer. The fee to do the transfer is 5% of the transferred amount, which is added to the balance on the new card.

**The question:** If I were to get this card and transfer a $6,500 balance to it, what would my effective annual interest rate be if I pay it all off at the end of the 21 months? Assume that no payments are due in the meantime*.

* In reality, some payments will be due, but they’ll be fairly modest – usually around 1% – 2% of the outstanding balance. Because the payments are small and factoring them in will complicate the question, we’ll leave them out for now.

This one is pretty straightforward. The only thing we don’t know is how much our initial balance will be.

If we transfer $6,500 to the card and there’s a 5% fee, we have to pay back $6,500 x 5% = $325 more than we transferred.

$6,500 + $325 = $6,825.

First things first, make sure the calculator is using 12 Payments per Year.

N: 21 (The offer is for 21 months interest-free)

I/YR: (This is what I’m trying to find)

PV: 6,500 (I’m transferring a $6,500 balance to the card, which means that I’m borrowing $6,500 from the card company)

PMT: 0 (We’re assuming that no monthly payments are due)

FV: -6,825 (I’ll have to pay back $6,825 at the end)

The effective interest rate on the ‘0% interest’ 21-month loan from the credit card company is **2.79%**. Not zero, but not too shabby, either.

What do you think? Does this sort of balance transfer offer appeal to you? Why or why not? Let us know in the comments!

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I was having a discussion with a friend this week about a note they’re considering buying. She knows the current terms of the note, and how long it’s been since it was originated, but she doesn’t know the original owed balance. So we’ll try and help her out by determining the answer. I’ve changed all of the numbers for the sake of anonymity and simplicity.

**The question:** If there are 88 monthly payments of $528.60 left to pay and $32,872.74 is owed today, what was the original amount due? The note started out as a 120-month note.

This one has two parts.

- Find the interest rate on the note
- Find the original amount owed

First things first, make sure the calculator is using 12 Payments per Year.

**Step 1: Find the Rate**

N: 88 (There are 88 payments left)

I/YR: (This is what I’m trying to find)

PV: 32,872.74 (The borrower owes $32,872.74 today)

PMT: -528.60 (The borrower pays $528.60 each month on the note)

FV: 0 (The note amortizes fully)

The note’s interest rate is **10%**.

**Step 2: Find the Original Balance**

N: 120 (There were originally 120 payments on this note)

I/YR: 10 (From part 1)

PV: (This is what I’m trying to find)

PMT: -528.60 (The borrower pays $528.60 each month on the note)

FV: 0 (The note amortizes fully)

The borrower originally owed **$40,000.07**. I’m going to assume that the actual amount at the beginning was an even 40 grand, and that the extra 7 cents is a result of a rounding error.

Note that we could have approached Step 2 differently: we could have made N 120 – 88 = 32, and put today’s $32,872.74 into FV (from the perspective of Day One, today is 32 months in the future), making sure to make it a negative number. Solving for PV would have gotten us the same answer. Feel free to try it out if you don’t believe me.

What do you think? Can you come up with situations in which knowing the original value of a thing helps you to better understand it today? If so, what’s an example? Let us know in the comments!

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It’s common knowledge that when your credit score is good, lenders are more eager to lend money to you on favorable terms than when it’s not. The differences in interest rates and fees can add up to quite a lot of money over the long term – and the higher payment amounts make it harder to pay those bills.

Let’s take a look at one hypothetical borrower with two different borrowing rates. To keep things simple, we’ll focus on interest rates and ignore the extra fees that might be required of someone with poor credit.

**The question:** I want to borrow $200,000 to buy a house. How much more do I pay over the life of the loan if the lender charges 6.5% interest instead of 4.25%? Both loans are 30-year, fully amortizing loans.

This one is pretty straightforward, and has several parts.

- Find the payment at 6.5%
- Find the total paid at 6.5%
- Find the payment at 4.25%
- Find the total paid at 4.25%
- Find out how much more the higher rate costs overall.

First things first, make sure the calculator is using 12 Payments per Year.

**Step 1: Payment at 6.5%**

N: 360 (The loan lasts 30 years, which is 30 x 12 = 360 months)

I/YR: 6.5 (The interest rate on the loan is 6.5%)

PV: 200,000 (I’m borrowing $200,000)

PMT: (This is what I’m trying to find)

FV: 0 (The loan amortizes fully)

The 6.5% loan has payments of **$1,264.14** per month.

**Step 2: Total paid at 6.5%**

Overall, I’m making 360 payments of $1,264.14.

360 x $1,264.14 = **$455,090.40** in overall payments.

Note that I need to type in the payment amount before multiplying it by the number of months. This is because the $1,264.14 is actually $1,264.136046986, and multiplying *that* number by 360 will lead to a number that’s *slightly* off. Since I can’t send fractional pennies to the bank, I have to use the rounded number when I multiply.

**Step 3: Payment at 4.25%**

N: 360 (The loan lasts 30 years, which is 30 x 12 = 360 months)

I/YR: 4.25 (The interest rate on the loan is 4.25%)

PV: 200,000 (I’m borrowing $200,000)

PMT: (This is what I’m trying to find)

FV: 0 (The loan amortizes fully)

The monthly payment for the 4.25% loan is **$983.88**.

**Step 4: Total paid at 4.25%**

If I make 360 payments of $983.88, I pay a total of 360 x $983.88 = **$354,196.80**.

**Step 5: Difference in total payments**

With the 6.5% loan, I pay a total of $455,090.40. With the 4.25% loan, I only pay $354,196.80. If I get the higher-rate loan, I pay a total of $455,090.40 – $354,196.80 = **$100,893.6** more over the life of the loan. Considering I only borrowed $200,000, paying an extra $100,000 seems like a lot to me.

What do you think? Did you expect this result? Do you think I used the wrong interest rates? If so, what rates would be more realistic? Let us know in the comments!

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Back in July of 2015, I rented a storage unit to store some ‘treasures’ that wouldn’t fit once I moved into a much smaller place. At the time, the rent was $108 per month.

Yesterday, I got a letter in the mail letting me know about a rent increase (they tend to do one each year or so), effective on July 1st of 2019. The new rent will be $215 per month.

**The question:** What has the average annual percentage increase in rent been over the past 4 years? Assume monthly compounding.

This one is straightforward.

First things first, make sure the calculator is using 12 Payments per Year.

N: 48 (July 2015 to July 2019 is 4 years, or 4 x 12 = 48 months)

I/YR: (This is what I’m trying to find)

PV: -108 (The original rent was $108 per month)

PMT: 0 (This is an inflation-style question, and those don’t have money flowing in or out each month)

FV: 215 (The new rent is $215 per month)

On average, the rent has gone up **17.34%** per year.

What do they think they are, a hospital or university? That rate *significantly* outstrips inflation!

What do you think? Would you put up with that kind of rent increase? If not, what would you do if you were in my position? Let us know in the comments!

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My grandmother told me about a house that she and my grandfather used to own that she’d rented to a guy for years and years. Eventually, my grandfather died and my grandmother was getting ready to move to Florida full-time, and she offered to sell the house to the tenant before going. He was interested, and they agreed on a price of $48,000.

She found out recently that the tenant had finally needed to move, himself, and sold the house for $79,000. He owned the house for 28 years.

**The question:** What was the average annual appreciation of the house over the 28 years the man owned the house? Assume monthly compounding.

This one is a very straightforward question. The only thing we need to know is how many months 28 years is. 28 x 12 = 336 months.

First things first, make sure the calculator is using 12 Payments per Year.

N: 336 (The man owned the house for 336 months)

I/YR: (This is what I’m trying to find)

PV: -48,000 (He bought the house for $48,000)

PMT: 0 (Inflation calculations don’t include monthly payments, in or out)

FV: 79,000 (He sold the house for $79,000)

The house appreciated **1.78%** per year over the 28 years the man owned the house. That’s not a huge amount of appreciation, but he had a place to live while he owned it.

What do you think? Did you expect the answer to be higher or lower? Why? Do you think houses in your area have higher or lower appreciation than this house? Let us know in the comments!

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One of the most common problems people have when working with a financial calculator is in entering the wrong number for N. N means ‘number of periods’, but some people hear that as ‘number of months’ and some as ‘number of years’. Sadly, both are incorrect, as periods can take up any amount of time (though months and years are the most common period-lengths I’ve seen used).

Let’s see the results of this type of misunderstanding in action, using the same initial deposit amount and monthly contribution to an investment account, but changing the length of time that elapses.

**The question:** If I initially put $10,000 in an investment account that earns 6.5% interest, and then contribute $250 per month to that account, how much more do I have in the account if I do so for 50 years rather than 50 months?

This one is pretty straightforward, and has a few parts:

- Find the answer after 50 years
- Find the answer after 50 months
- Find the difference

First things first, make sure the calculator is using 12 Payments per Year.

**Part 1: 50 years**

N: 600 (I’m doing the investment for 50 years, which is 50 x 12 = 600 months)

I/YR: 6.5 (The account yields 6.5% interest)

PV: -10,000 (My initial deposit is $10,000)

PMT: -250 (I’m investing $250 per month)

FV: (This is what I’m trying to find)

After 50 years, I’ll have **$3,690,289.20** in the account.

**Part 2: 50 months**

N: 50 (I’m doing the investment for 50 months)

I/YR: 6.5 (The account yields 6.5% interest)

PV: -10,000 (My initial deposit is $10,000)

PMT: -250 (I’m investing $250 per month)

FV: (This is what I’m trying to find)

After 50 months, I’ll only have **$145,322.12** in the account.

**Part 3: The difference**

Continuing the investment for 50 years rather than only 50 months gets me $3,690,289.20 – $145,322.12 = **$3,544,967.08** more. In other words, almost all of the money accrues after the first 50 months.

Because answers can vary *wildly* based on different inputs, it’s critical that we double-check our numbers and our scenarios when modeling them with the calculator.

What do you think? Does this answer surprise you? Why or why not? Let us know in the comments!

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A while back, I talked about a note that’s available for purchase. As with all notes, however, there’s the possibility of the borrower paying it off early.

**The question:** If the borrower owes $44,948.85 on a fully-amortizing note with 99 monthly payments left of $717.35, but they decide to pay it off in 2 years, what will their payoff amount be?

This one has two parts, which are both straightforward.

- Find the interest rate on the loan
- Find the payoff amount

First things first, make sure the calculator is using 12 Payments per Year.

**Part 1: Interest Rate**

N: 99 (The loan has 99 payments left)

I/YR: (This is what I’m trying to find)

PV: 44,948.85 (The borrower currently owes $44,948.85)

PMT: -717.35 (The borrower pays $717.35 per month on the loan)

FV: 0 (The loan amortizes fully)

The borrower’s interest rate is **12.00%**.

**Part 2: Payoff Amount**

N: 24 (The borrower will pay off the loan in 2 years, which is 2 x 12 = 24 months)

I/YR: 12.00 (From part 1)

PV: 44,948.85 (The borrower currently owes $44,948.85)

PMT: -717.35 (The borrower pays $717.35 per month on the loan)

FV: (This is what I’m trying to find)

After two years, the borrower will have to pay **$37,723.52** to satisfy (pay off) the loan.

What do you think? If you were paying on this loan, would you try to pay it off early, or would you ride it out for the whole 99 months? Why or why not? Let us know in the comments!

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