# Calculate APY revolving loan with offset/staggered payment schedule?

I want to buy a weekly meal ticket, 7 lunches for \$7, but do not have \$7 today. Lender says, "I will loan you a consecutive series of 5 meal tickets for 5 consecutive weeks (35 consecutive lunches as you receive a new ticket on day 1, 8, 15, 22, and 29) if you pay me \$6 every 5 days (you pay on day 5, 10, 15, 20, 25, 30, and 35)."

• What is the APY of the loan Lender offers me?

I can see that over 35 days, I will make 7 payments of \$6 = \$42 using Lender (vs. 5 payments of \$7 = \$35 using cash). Yet, it seems incorrect (underestimating APY) to calculate that Lender offers me \$35 loan on day 1 and that I make periodic payments of \$42 over 35 days because Lender never puts \$35 at risk. That is, this is not a simple home or auto loan problem.

• What is simplest formula to correctly calculate APY on first \$7 loan?
• What is simplest formula to correctly calculate APY on series of five \$7 loans?

There is no simple formula for this scenario.

There are two sets of payments, each made on a different regular schedule. There is no simple way to separate the repayments as to the loan they apply to.

The "simplest" way is to do a day by day rolling balance in a spreadsheet:

Here I'm assuming an interest rate of 0.5% per day. So, on Day 1 you get \$7, and that's what you owe. On Day 2, you add 0.5% to that, so you owe \$7.0350.

Each day, you add one day's interest to the previous day's principle. On Day 6 you do the same, but subtract the payment you make against your debt.

This goes on day by day, adding a day's interest, adding any additional loan you get, and subtracting any payment you make, until Day 35. (I've hidden Day 11 to Day 28; it's just more of the same)

Note that On Day 35, you're more than \$6 ahead; the interest rate guess is too low.

Now, I could just guess at different interest rates until I get a zero balance on Day 35. But Excel has a Goal Seek operation; I can tell Excel to fiddle with the interest rate value until the Day 35 balance is very close to zero.

When I do this the result is :

So your lender is charging you 3.7% a day!!!

• I think this assumes you pay no interest on the day you get the money and on the day you pay it back. If you get the money at noon and pay it back at noon, I think that adds a day's worth of interest, as per my answer. – barrycarter Jan 10 '16 at 21:31
• I made no such assumption.... Each Day N, I add the day's interest that the Day (N-1) balance has incurred, and then subtract/add any payments/repayments made on Day N – DJohnM Jan 10 '16 at 21:40
• You're right and I'm wrong. I flipped one of my equations below, now fixing. – barrycarter Jan 10 '16 at 21:49

The surprising answer: your daily interest rate is 3.74%, for an APY of about 67 million %

Let d be the daily interest rate, compounded daily.

I can't think of an easy way to do this, so below is a difficult way.

I'm sure there is an easier way, and would appreciate if someone posted it.

``````
(* Day 1: you borrow \$7 *)

owed = 7

(* Days 1-5: you pay 4 days interest *)

owed = owed*(1+d)^4

(* Day 5: you pay back \$6 *)

owed = owed - 6

(* Days 5-8: you pay 3 days interest *)

owed = owed*(1+d)^3

(* Day 8: you borrow \$7 more *)

owed = owed + 7

(* Days 8-10: you pay 2 days interest *)

owed = owed*(1+d)^2

(* Day 10: you pay back \$6 *)

owed = owed - 6

(* Days 10-15: you pay 5 days interest *)

owed = owed*(1+d)^5

(* Day 15: you pay back \$6, but borrow \$7 *)

owed = owed + 7 - 6

(* Days 15-20: you pay 5 days interest *)

owed = owed*(1+d)^5

(* Day 20: you pay back \$6 *)

owed = owed - 6

(* Days 20-22: you pay 2 days interest *)

owed = owed*(1+d)^2

(* Day 22: you borrow \$7 more *)

owed = owed + 7

(* Days 22-25: you pay 3 days interest *)

owed = owed*(1+d)^3

(* Day 25: you pay back \$6 *)

owed = owed - 6

(* Days 25-29: you pay 4 days interest *)

owed = owed*(1+d)^4

(* Day 29: you borrow \$7 more *)

owed = owed + 7

(* Days 29-30: you pay 1 day interest *)

owed = owed*(1+d)

(* Day 30: you pay back \$6 *)

owed = owed - 6

(* Days 30-35: you pay 5 days interest *)

owed = owed*(1+d)^5

(* Day 35: you pay back \$6 *)

owed = owed - 6
``````

We know you now owe \$0. In terms of d, from the computations above, this is:

(this site apparently doesn't support TeX so that's an image)

We now solve for d (using numerical methods), getting 0.037417, or 3.74% daily interest, or right around 67,118,717 percent annual interest.

Note that I'm not solving the "first loan" problem, but you can use a technique similar to this to find it.

EDIT: Personal story which really doesn't belong here, but ...

If you make a credit card purchase and later return it, you are retroactively not charged interest on the amount, even if the return occurs after the grace period. I did this several times when trading FOREX without any problems.

One month, however, I made a \$1 purchase between the purchase and return, leading to interest charges that prompted me to write these paragraphs as part of a letter to them (they did take care of it).

On 2010-07-28, I paid GFT FOREX \$10,000, but this was later returned (on 2010-09-22), and would therefore not be subject to interest [...]

The only other charge I made was a \$1 payment to #PROTECTMYID.COM on 2010-08-16. On 2010-09-14, I was charged \$107.37 in interest, which equates to a 10,737% interest rate per month, or an APR of just over 234,744,506,302,795,641,364,507,200% (over 234 trillion trillion percent), which seems a little high.

so I suppose you got lucky...