Topic 7 - Loan Schedules Flashcards
What is a loan?
A loan is a common transaction involving compound
interest
State the features of a loan
Repaid via regular instalments
At a fixed rate of interest
For a predetermined term
Give two examples of loans
Repayment mortgage Endowment mortgage (repayments interest only, still need to repay capital)
Repayments are comprised of what two components?
Capital and interest components
Identify Capital and Interest Components
Example
Bank lends you £5,000 to be repaid annually in arrears over 5 years at an effective annual interest of 6% pa.
Calculate the repayment
5000 = X 𝑎5¬ @ 6%
5000 = 4.2124X
X = £1,186.97
Each payment covers both capital and interest
Identify Capital and Interest Components
Example
Bank lends you £5,000 to be repaid annually in arrears over 5 years at an effective annual interest of 6% pa.
Calculate the repayment
5000 = X 𝑎5¬ @ 6%
5000 = 4.2124X
X = £1,186.97
Each payment covers both capital and interest
Evaluate the cashflows
Annual repayment £1,186.97
At t = 1 the interest due is 5000x6% = £300
Leaving £886.97 to reduce capital
So outstanding capital is 5000-886.97 = £4,113.03
At t = 2 the interest due is 4113.03x6% = £246.78
Leaving £940.19 to reduce capital
So outstanding capital is 4113.03-940.19 = £3,172.84 etc
So cashflows in full:
Time Repayment Interest Capital Capital Outstanding
0 0 0 0 £5,000
1 £1,186.97 £300 £886.97 £4,113.03
2 £1,186.97 £246.78 £940.19 £3,172.84
3 £1,186.97 £190.37 £996.60 £2,176.24
4 £1,186.97 £130.57 £1,056.40 £1,119.84
5 £1,186.97 £67.19 £1,119.78 -
Rolling the loan forward to identify capital outstanding is acceptable for short term loans however this is time consuming and inefficient for long term loans.
What are the two methods to calculate capital outstanding?
- Calculate accumulated value of loan less accumulated
value of payments to date - Calculate the present value of future payments.
Time Repayment Interest Capital Capital Outstanding
0 0 0 0 £5,000
1 £1,186.97 £300 £886.97 £4,113.03
2 £1,186.97 £246.78 £940.19 £3,172.84
3 £1,186.97 £190.37 £996.60 £2,176.24
4 £1,186.97 £130.57 £1,056.40 £1,119.84
5 £1,186.97 £67.19 £1,119.78 -
Verify that the capital outstanding after the 3rd payment in the example is £2,176.24.
Method 1 (accumulation method)
5000x1. 06^3 –1186.97𝑠3¬
5000x1. 06^3 –1186.97x3.1836 = £2,176.24
Method 2 (PV method)
- 97𝑎2¬
- 97x1.8334 = £2,176.19
Using theory, state the equation of value for the loan at t=0
L0 = X1v + X2v^2 +..+ Xnv^n
Lt is amount of loan outstanding at t = 0,1,2..,n immediately after the payment at t
Xt is repayment amounts at t = 1,2..,n
i is effect rate of interest per unit time
What are the two ways to find the loan outstanding at any time t?
Prospectively (analogous to PV method, method 2)
Retrospectively (analogous to Accumulation method, method 1)
What is the Prospective Method of calculating the loan outstanding at any time t?
Method looks forward and calculates PV of future
cashflows
Consider position at t=n, after the final instalment of
capital and interest the loan is exactly repaid.
So Xn must cover remaining capital after instalment at n-1, together with interest due
bn = iLn-1 and fn = Ln-1
f,t is the capital repaid at t
b,t is the interest paid at t
Xn = iLn-1 + Ln-1 = (1+i)Ln-1 Ln-1 = X,n x v [Xn = bn + fn]
Similarly at any time t+1, the capital repaid is Lt –Lt+1
Xt+1 = iLt + (Lt –Lt+1)
Lt = (Xt+1 + Lt+1)v
Similarly …
Lt+1 = (Xt+2 + Lt+2)v and so on until we get to Ln = 0
If we work forward and substitute for Lt+r, we get…
Lt = (Xt+1 + Lt+1)v
Lt = (Xt+1 + (Xt+2 + Lt+2)v)v = Xt+1v+ Xt+2v^2 + Lt+2v^2
Lt = Xt+1v+ Xt+2v^2 + Xt+3v^3 + Lt+3v^3
Lt = Xt+1v+ Xt+2v^2 + Xt+3v^3 +…+Xnv^(n-t)
So for the prospective method we adapt the formula
Lt = Xt+1v+ Xt+2v^2 + Xt+3v^3 +…+Xnv^(n-t) to
Lt = Xt 𝑎n-t¬
n = original term
t = time of last payment
NB the present value must be calculated at a repayment date
Prospective Method Question Bank lends you £50,000 to be repaid annually in arrears over 10 years at an effective annual interest of 6% pa. Calculate the capital outstanding after: The 4th payment has been made The 8th payment has been made
Solution Step 1 –calculate the repayment first 50000 = X𝑎10¬ @ 6% 50000 = 7.3601X X = £6,793.39
Step 2 –calculate the capital outstanding after 4th payment L4 = 6,793.39𝑎6¬ L4 = 6,793.39 x 4.9173 = £33,405.14 Step 3 –calculate the capital outstanding after 8th payment L8 = 6,793.39𝑎2¬ L8 = 6,793.39 x 1.8334 = £12,455.00
What is the Retrospective Method of calculating the loan outstanding at any time t?
Method looks backwards and calculated the accumulated value of past cashflows Consider position at t=1 b1 = iL0 f1 = X1 - iL0 L1 = L0 –(X1 –iL0) = (1+i)L0 –X1
For t ≥ 1 the interest due and capital repaid are
bt = iLt-1 and ft = Xt –iLt-1 giving……
Lt = (1+i)Lt-1 –Xt
Similarly….
Lt-1 = (1+i)Lt-2 –Xt-1
If we work back from t to 0….
Lt = (1+i)Lt-1 –Xt
Lt = (1+i)((1+i)Lt-2 –Xt-1) –Xt = (1+i)^2Lt-2 –(1+i)Xt-1 –Xt
Lt = (1+i)^tL0 –((1+i)^(t-1)X1 + (1+i)^(t-2)X2 +…+ (1+i)Xt-1 + Xt)
So for retrospective method we adapt the formula
Lt = (1+i)^tL0 –((1+i)^(t-1)X1 + (1+i)^(t-2)X2 +…+ (1+i)Xt-1 + Xt) to
Lt = (1+i)^tL0 - X 𝑠t¬
L0 is initial loan amount
t is time of last payment
𝑠t¬ is an accumulation function NOT an increasing annuity!
Retrospective Method Question Bank lends you £50,000 to be repaid annually in arrears over 10 years at an effective annual interest of 6% pa. Calculate the capital outstanding after: The 4th payment has been made The 8th payment has been made
Solution
Calculate the repayment first
50000 = X𝑎10¬ @ 6%
50000 = 7.3601X
X = £6,793.39 (repayment £6,793.39)
Step 1 –calculate the capital outstanding after 4th
payment
L4 = 50000x(1.06)^4 - 6793.39𝑠4¬
L4 = 50000x(1.06)^4 - 6793.39 x 4.3746 = £33,405.48
Step 2 –calculate the capital outstanding after 8th
payment
L8 = 50000x(1.06)^8 - 6793.39𝑠8¬
L8 = 50000x(1.06)^8 - 6793.39 x 9.8975 = £12,454.83
Calculating Interest and Capital Elements
Given the outstanding capital at any time what can we calculate?
The interest & capital element of an instalment
