Balance Schedule
When you pay-off a loan using a sequence of regular payments (“amortize” the loan), the interest It for the current period t, is subtracted from the loan payment Kt and the remaining portion PRt = Kt – It is the amount of principal repaid.
For a 30-year loan of $80000 at an interest rate of with monthly payments of 479.64, the first few lines of the balance schedule are given below.
t Kt It PRt OBt
0 80000.00
1 479.64 400.00 79.64 79920.36
2 479.64 399.60 80.04 79840.32
3 479.64 399.20 80.44 79759.88
and so on...
The Outstanding Balance
The outstanding balance OBt (i.e., portion of the principal) remaining to be repaid at time t can be found by comparing the amount of the loan to the amount of payments already made or by considering the amount of payments remaining. But to find the balance at a given point in time we must determine the value of the payments at this same point in time.
The following two methods may be used to determine the current value of the remaining debt at time t. This is the amount you would pay to settle the debt immediately following the tth payment. Of course, if you continue making payments, you will end up paying more than this amount since this unpaid debt continues to accumulate interest charges.
The Retrospective Method: (compare the loan size versus the payments already made)
The value of the original loan L considered upon the time of the tth payment is .
At this same time, the first t payments have a total value of
.
Thus, the current value of the remaining debt at time t is the difference
.
If the payment size K is constant, this geometric sum simplifies to
.
The Prospective Method: (finding the value of the remaining payments)
The value of the remaining debt at time t may also be considered as the current value of the remaining payments. At this time, the final payments have a present value of
.
If the payment size K is constant, this present value simplifies to
.
Example:
Consider a loan of $80000 at an interest rate of . If the loan is to be repaid in equal monthly payments over a period of 30 years, how large is the unpaid balance at the beginning of the last 10 years of payments?
Since the loan has a present value of $80000, we can determine the size of the monthly payment to be $479.64. Thus, with 10 years (120 payments) remaining to be made, we can use either method above to determine the outstanding balance.
Using the retrospective method, based on the 240 payments already made, we find the remaining debt to equal
. Using the prospective method, the present value of the final 120 payments is found to equal .
Why the difference in the two totals? Mathematically the two approaches are equivalent but we rounded the value of the payment size. If we kept better precision and used
K = 479.6404201, the two computations would both yield 43202.869 for the remaining debt.
Level Payments of Principal
If you wish to reduce the principal by the same amount each period, say PRt = X, then the payments Kt = It + PRt = It + X are not constant but decrease as the interest charged per period decreases. That is, the reduction in the outstanding balance OBt-1 – OBt = PRt = Kt – It = X for each period t. Compare this to the usual situation where for constant payment size K, the change in the outstanding balance OBt-1 – OBt = PRt = K – It = K – j ( OBt-1 ) increases as the outstanding balance decreases.
For a loan of $80000 at an interest rate of , suppose we wish to reduce the debt by $500 each month over a period of 160 months (i.e., 160($500) = $80000). The first few lines of the balance schedule are given below.
t Kt It PRt OBt
0 80000.00
1 900.00 400.00 500.00 79500.00
2 897.50 397.50 500.00 79000.00
3 895.00 395.00 500.00 78500.00
and so on...
Annual Percentage Rate (APR)
The APR is a measure of the cost of a loan that takes into account any fees that must be paid. We’ve seen that at an interest rate j per period, the loan payment K for a loan of L is given by . If fees totaling F are paid at the time of the loan, then the amount borrowed is essentially L – F. To determine the effective rate of the loan, determine the rate j* for which n payments of K settles a loan of L – F. That is, find the APR j* such that , where K is still the payment size determined originally for the loan of L. Note that when fees are taken into account, the APR j* is larger than the quoted rate j . This is to be expected because if the same size payments K are being used to pay-off the smaller total amount L – F, it is on account of a higher interest rate.
For a 30-year loan of $80000 at an interest rate of with monthly payments of 479.64, find the APR if the loan requires fees totaling $4000. That is, find the rate j* such that using the “unknown interest” procedure included on your business calculator. (Using a graphical approach, I found the rate j* = 0.0054, or 5.4%.)
Sinking Fund
Consider a loan where the entire principal is paid in one lump sum at the end of the term of the loan. Instead of paying off the principal in periodic payments, suppose we only payoff the interest charge each period.
How will you have enough to make the large lump sum payment when the loan comes due? To save enough to make the lump sum payment, establish a “sinking fund” where instead of making loan payments, you make a deposit each period into an interest-bearing account. These deposits form an annuity whose future value can be used to fund the lump sum payment and settle the loan. For a loan of L, we need to make periodic deposits of X such that . In paying the periodic interest charges as well as making deposits into the sinking fund, our expenditure each period is . Thus, for this sinking fund method, we note Kt = It + PRt = iL + X is considered the periodic payment.
Since we are accumulating our savings toward funding our lump sum payment, how should we view our outstanding balance at a given point in time? The outstanding balance OBt equals , the difference between what is in our sinking fund and the size of the loan.
The “principal repaid” PRt is the growth in our sinking fund .
That is, . Also, the net interest It for a given period is , the difference between the interest charge on paid and the interest earned by the sinking fund. Note that the interest rate on the loan is typically expected to be higher that your savings rate .
Balance Schedule (Sinking Fund Method)
Consider constructing a balance schedule based on this sinking fund method. Consider again a 30-year loan of $80000 at an interest rate of assuming the annuity earns . In order for our sinking fund to accumulate the necessary $80000 over the 30-year period, we need to make monthly deposits X such that . Solving for X yields X = $99.73 per month. Note upon our first deposit at t = 1, the sinking fund hasn’t earned any interest and so I1 = 0.005L – 0 = 400 and PR1 = 99.73.
At time t = 2, our first deposit has earned interest of (0.004)(99.73) = 0.40 and so the net interest for the period is I2 = 0.005L – 0.40 = 399.60. Also, the increase in our sinking fund is PR2 = and K2 = 399.60 + 100.13 = 499.73.
At time t = 3, our net interest is I3 = = 399.60. Also, the increase in our sinking fund is PR3 = .
Continuing in this way, we construct the balance schedule shown below.
t Kt It PRt OBt
0 80000.00
1 499.73 400.00 99.73 79900.27
2 499.73 399.60 100.13 79800.14
3 499.73 399.20 100.53 79699.61
4 499.73 398.80 100.93 79598.68
5 499.73 398.40 101.34 79497.34
6 499.73 397.99 101.74 79395.60
and so on...
Look carefully! Over the first few months, the sequence of PRt may appear to be an arithmetic sequence but the balance of the sinking fund is a geometric sum and so the increase is not linear.
What is the total net interest paid over the 30 years? Compute the difference between the interest paid and the interest earned. The total interest paid equals 360($400) = $144,000. The total interest earned by the sinking fund equals . Hence, the total net interest is $99904.63. But we also know the total interest is simply the difference between what you paid and what you borrowed. That is, total interest equals 360($499.73) – $80000 = $99902.80. The discrepancy between these two answers is due to the round-off error in X = 99.73. Using a more precise X = 99.7322835, results in total interest of $99903.622 using either approach.