35. Loan Repayment method - Part 4

Added: 2026-05-12

[00:00:15]Hello everyone, welcome to the course financial mathematics for actuaries.

[00:00:21]We are learning the loan repayment methods.

[00:00:26]We have already learned the loan amotization schedule and in this session we are going to learn the other method that is called the syncing fund method.

[00:00:40]Let's do that.

[00:00:43]Let us learn another method of loan repayment that is the syncing fund method. So what is a syncing fund?

[00:00:53]Note that the borrower needs to repay the loan principle along with the interest amount. So in the sinking fund method, the borrower deposit an amount periodically into an interestbearing account to accumulate the principle at time n. Now you recall the interest only loan where we have understood that the borrower make a periodic payment of interest only during the term and a final payment of the full principle at the end of the term. So for such a loan the borrower would have to make a series of n interest payments each of an amount that is l multiplied by i along with a payment of l at time n. So the borrower has an obligation to pay a single lum at or a single lump sum of l at time n. For this purpose, the borrower wish to make a periodic deposit into a fund to accumulate the principle. This fund is basically known as the syncing fund.

[00:02:04]Right? So the method the method of loan payment is basically known as the syncing fund method. So the simple method of loan repayment is mostly useful when the borrower believes that it is very difficult for him to have a or to make a large lumpsum amount at the end of the term of loan. So basically what he want to do is from the start he wants to put or set aside some fund so that it will accumulate the principle when the term got up. Let us take one example here.

[00:02:49]Suppose a loan of five lak rupees by a six payment annuity immediate at an effective interest rate of 6% per year.

[00:02:59]Now let us now calculate the syncing fund deposit.

[00:03:05]So here as we have understood that the borrower has to make a periodic interest only payment that is the 6% of 5 lak rupees every time till sixth payment and he likes to set aside some fund so that it will accumulate up you know at the end of sixth years it should five lakh rupees. So the solution here the interest payment is therefore 6% of five lak rupees that is 30,000 rupees.

[00:03:45]So every time the borrower needs to make a 30,000 rupees for six figure and the syncing fund deposit therefore will be the initial loan amount that is this one divided by the accumulated value of the deposit or the installment that is s angle n at i. So here in our problem five lakhs divided by s angle 6 at 0.06 which will be equal to 5 lakh divided by 6.975.

[00:04:24]You can use your calculator to get this value and then finally you will get that it will become the syncing fund deposit will become 71,684.58.

[00:04:39]Okay. So let us now understand it more formally.

[00:04:44]So for this consider a loan amount of a angle n i. This can also been seen as the initial loan amount that is l. And by the syncing fund method the borrower pays an amount that is interest rate times this one that is I time this amount which will be equal to 1 minus v to the power n as we have already seen in the amotization schedule each period to serve the interest. In addition he deposit an installment into the sinking fund.

[00:05:21]Suppose the sinking fund accumulate the interest rate I. So I is the interest rate what the borrower will get if he deposit in the sinking fund. To accumulate a angle N at I this amount at time n each sinking phone installment is therefore okay this the initial loan divided by the accumulated value here.

[00:05:49]So a angle N I divided by S angle N I which is equal to V ^ N.

[00:06:00]Okay. Thus the total amount the borrower has to pay each period is therefore 1 minus VN 1 - V ^ n plus this one. So it will be equal to 1 in our example here.

[00:06:20]which is the same as the amount paid in the loan amotization schedule.

[00:06:27]Now let's put it in a generic form. So syncing fund schedule of a loan of a angle n at i by an n payment unit annuity at interest rate i. Okay. So the generic form of loan syncing fund schedule look like this. So we have here 1 to n period of time. The installment amount is the unit annity and therefore the installment amount you can think as one rupee or $1 whatever you can say.

[00:07:09]The interest payment what we have understood just now uh the interest amount or interest at i time the loan amount that is 1 - v ^ n and the syncing fund deposit therefore this minus 1 sorry this + one and this will give you the v to the power n. Okay. And the syncing fund valance therefore will be equal to what? This is the loan amount divided by the accumulated value in the first year.

[00:07:51]That will be equal to same as v to the power n which is equal to s angle 1.

[00:08:02]Okay. Same you will go on. Okay. Tilt t period of time. Then we will have I * 8 that is this one. Notice here that this interest payment is fixed for all the period. Same as the syncing fund al syncing fund deposit also fixed for all the period. Right? On the other hand, in the loan amotization schedule, the interest payment was actually decreasing and the loan deduction will be increasing in the loan amotization schedule.

[00:08:43]But here in the case the interest payment is constant every time you are making the same interest payment and the and the borrower want to deposit same amount every time till the end of the period to accumulate the total loan balance or the principle.

[00:09:03]Let us understand this clearly by putting some number in this and therefore we construct a syncing fund schedule for this problem for a loan of $5,000 by a six payment annuity with six% interest rate. Now construct a syncing fund schedule. For this purpose we need to calculate for the first row. So let us now calculate the first row. So the interest payment will be 5,000 multiplied by the 6% interested that will be amount be $300 and the syncing fund deposit for is L divided by a single NI. So 5,000 divided by a single 6 at 6% interest rate which will be equal to 716.8.

[00:10:08]So for the first row we will write here year and the installment that is the annuity installment and we will write the interest payment interest payment and we will write the syncing fund deposit syncing ing fund deposit.

[00:10:41]Okay. And we will write the syncing fund interest and the syncing fund outstanding balances. Okay. So this sinking fond once you understand the first row we can easily understand the rest of the rows.

[00:11:18]Okay. So first row will be one then we will we can easily compute it as okay one. So installment will become this plus this. So it will become 1 0 1 6 81. Okay. Or you can simply say this is you know like calculated to be 716.81.

[00:11:57]So this plus this will be 1016.81 81 interest payment as you have already computed here is 300. The syncing fund deposit is 71 6 8 or 81 we can say syncing fund interest rate initially will be zero and the sinking point balances is same as 81 right for the second period for the second period you can simply have see the interest payment will be same. Okay.

[00:12:38]And the sinking point deposit also will be same. So that installment also will be same. Okay. This plus this will become 1016 81. Okay. This is the same installment what you are make you are making in the loan amortization. So now we are just understanding the syncing phone method as compared to the loan amortization method. So we are now seeing that these two will yield the loan installment.

[00:13:10]Okay, the installment amount. So now here the syncing fund interest of the in the two year the sinking fund generate one interest rate to the 6% of this amount right 6% of 716.81.

[00:13:31]So you just multiply okay 716.81 * 0.06 06 which will give us a value of 43 01. That value we can just you know write it here 43 0 1.

[00:14:00]Okay. And then the sinking point balances therefore this one plus this one plus this one which will become 1476 4 763.

[00:14:17]Okay. So now you can compute the similarly for rest of the row. We'll go on computing in similar procedure. The rest of the row we can have is this right?

[00:14:39]Okay. So you can see here that in the first year okay as we have already computed that the interest payment will be 300. The syncing fund deposit is 716.81.

[00:14:53]You have already understood how to computed this. Excuse me.

[00:14:58]And this plus this will be amounted to be this one. Okay. The similarly we have computed this second row also. Okay. In the second time al here notice here that the interest payment is same for all the six period. Similarly the syncing fund deposit also same for all the six period. So the total deposit you have made is 4,300.86 and the total interest you have pay paid is 1,800.

[00:15:33]Right? And the syncing fund and the syncing fund interest your deposit in your syncing fund has earned an interest amount of $699.14.

[00:15:48]Okay? And therefore you can able to deposit your principal amount that is 5,000 rupees.

[00:15:58]So this is how the syncing fund schedule look like. Now one important thing you have to notice here is that the rate of interest rate you earned from the sinking fund that is 6% here is same as the rate of interest rate for your loan amount.

[00:16:27]But you know like sometime these two rates are not exactly same. Most of the time the loan the interest rate you are going to pay is higher than the interest rate you are going to earn. Okay. So when the interest rate of or sorry so when the rate of interest charged to the loan and earned by the syncing fund are same the amount of payment computed using the amortization and syncing fund methods are exactly same. However, there is no guarantee that the rate earned in the syncing fund, okay, say J is the same as the periodic interest rate on the loan that is I. And we know that most of the time the interest rate the borrower has to make is lower than the interest rate he usually pay. So I is basically greater than J. In this situation, we now consider the case where the syncing fund earns a rate of interest different from that charge to the loan.

[00:17:47]So specifically suppose the syncing fund earn interest rate of JP J per payment period for a loan amount a angle N I to accumulate to the principle in the sinking fund over n period the periodic installment is a angle N I divided by S angle NJ okay hence the total installment ment therefore will become I* A angle N at I plus A angle N at I divided by S angle N at J or you can rewrite this upon solving you can simply take this term out we will have a angle N I time I + 1 upon S angle N at J this is the rate of interest. This is the rate of interest you earned from your syncing fund. But this is the rate of interest you you are going to pay for the loan. Okay, this can also be written as L. L being the initial loan. So this will be equal to this. Okay. Then we can rewrite that the total installment therefore will become L * I + 1 upon S angle N at J.

[00:19:24]Okay.

[00:19:26]So only thing we need to remember is that when the rate of interest are same then the loan amotization method and the syncing fund methods are quite similar. However, if the rate of interest are different then they may be different.

[00:19:50]I believe you have now understood the loan repayment method where we have understood the loan amotization schedule and the syncing fund method.

[00:20:01]Thank you. We will be meeting in the next week to understand the bond and bond valuation techniques.

[00:20:11]See you in next week.