FIN 21 Calculating Payments in an Annuity

Let's have a look at problem 311A. This problem has us calculating payments based on APR, that interest rate that we don't love so much in corporate finance class. So you decide to buy a new car. The car's cost is 50 grand and you pay $10,000 today, meaning you're going to have to borrow $40,000, financing the rest with a car loan. So you're borrowing $40,000. The terms of the loan, 4.5% APR for 60 months, and interest is compounded monthly on this loan. First question is what are the monthly payments? We're going to need to use PV of an annuity to solve that. And the second is what's the effective annual rate of the loan? Okay. Let's figure out the payments. So the first thing we say is, okay, APR is simply the interest rate per period times by the number of periods in a year. So if the interest is compounded monthly, this 4.5% is actually, we just divide by 12 to figure out the monthly rate of interest. So 4.5% divided by 12, 4.5 divided by 12 is 0.375% interest per month. That's our true interest rate. It's really 0.375% per month. It's not really 4.5% per year as we'll show with our effective annual rate when we at last calculate that. Okay. So we need to figure out the payment. Now we'll use the present value of an annuity formula. I'll also use the financial calculator, solve this in a few minutes, but let's do it with the formula. I think the formula is a good way to do it. So we know the present value of this loan. We borrowed $40,000 today. What we don't know is our monthly payment. And this C is your cash payment. So we're solving for C, but we have everything else. 1 plus our rate of interest, monthly rate of interest, 0.375%, which translates to 0.00375 as a decimal, raised to the power of T, which is the number of months, 60 months, all divided by 0.00375. Let me zoom in on this. I think I'm getting a little far away. Okay. So we start with what's in the brackets inside 1.00375 to the power of 61.00375 Y to the X 60 is 1.25. Now I go one divided by that number. So I just go one over X in my calculator to get 0.79, one minus 0.79. It's going to be like 0.2 roughly. The way I do this is I go minus one and then I hit the plus minus button. So it's 0.201. I divide by 0.00375 divided by 0.00375 and I get 53.64. That's the number in the big brackets. So I have C times 53.64 equals $40,000. I just have to divide 40,000 by 53 and I get the answer. So again, having the number in my calculator, so I don't round, I just go one over X times 40,000 and I get $745.72. C equals 745.72. It asked me what the monthly payment is. That's my monthly payment. Monthly payment. I put an S. Okay. So there we have it. We have calculated the monthly payment. I'll just show you quickly what this looks like in the financial calculator for you financial calculator havers. So again, second function, clear out any time value money information that's in there. We know our N is 60. That's the number of periods. I put 60 for my N. I know my interest rate is, okay, this one's tricky. It's 0.375%. So I put in 0.375, not 0.00375. I put in 0.375, the actual percentage. My PV is 40,000 positive PV because I'm getting a $40,000 loan. I don't know my payment. My future value is going to be zero. So I can, I can put it in zero. You don't have to, I have to compute the PMT and there it is 745.72 and that's exactly what we had 745.72. So the calculator gets us the same place a little bit more quickly with these present value of an annuity. But you know, if I was told I had to do it by formula, no problem. If I was told I had to do it with calculator, no problem. I don't use the tables, but the tables are also no problem. Okay. What's the effective annual rate of this loan? This should be very straightforward. If it's 0.375 per month, you add one to that, you go 1.00375. And now with APR, you just multiply that by 12 with effective annual rate, you compound it. So you compound it 12 times. You raise it to the power of 12, 1.00375 to the power of 12 is 1.0459, which converts to 4. You just take away the 1, 4.59%. So as a finance person, we think 4.59% is a more accurate number to be quoting than 4.5%. But there we have it. We've solved 311A. If this video helped, it always is kind of you if you can help me to give a thumbs up or a sub. Thanks again for watching. Have a great day. Bye bye.