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
How would one calculate the interest & capital element of an instalment
If we consider a single instalment Xt at time t, we can
identify the elements using the following three steps
- Calculate loan outstanding after previous instalment at t-1, Lt-1
- Calculate interest using iLt-1
- Calculate capital using Xt - iLt-1 or Lt-1 – Lt
How would one calculate the interest paid & capital repaid at any time t
If we consider a series of instalments Xt+1 to Xt+r
Calculate loan outstanding before the payments (Lt)
Calculate loan outstanding after the payments (Lt+r)
Capital paid is Lt - Lt+r
Total interest and capital is total payment i.e. Xt+1 + Xt+2 +..+ Xt+r
Interest paid is then (Xt+1 + Xt+2 +..+ Xt+r) –(Lt - Lt+r)
Loan of £20,000 is repayable by 8 equal instalments in arrears, calculate the following with i = 5% pa
- Annual repayment
- Interest element of the 3rd payment
- Capital element of 5th payment
- Capital repaid in last 3 years of loan
- Total interest paid over the loan
Repayment
20000 = X 𝑎8¬
X = 20000/6.4632 = £3,094.44
Interest element of the 3rd payment
iL2 = 5% x 3094.44 𝑎6¬
iL2 = 5% x 3094.44 x 5.0757 = £785.32
Capital element of 5th payment
= X –iL4 = 3094.44 –5% x 3094.44 𝑎4¬
= 3094.44 –5% x 3094.44 x 3.546 = £2,545.80
Capital repaid in last 3 years of loan
L5 = 3094.44 𝑎3¬
L5 = 3094.44 x 2.7232 = £8,426.78
Total interest paid over the loan
8 x 3094.44 –20000 = £4,755.52
Produce a loan schedule table using the headings: Time, Instalment, Interest due, Capital repaid, and Loan outstanding.
Table entries should be variable notation as discussed in Topic 7
Time Instalment Interest due Capital repaid Loan
outstanding
0 0 0 0 L0
1 X1 iL0 X1-iL0 L1=L0 –(X1-iL0)
2 X2 iL1 X2-iL1 L2=L1 –(X2-iL1)
……
n Xn iLn-1 Xn-iLn-1 0
What is the reality regarding the timeframe of most loan repayments?
In reality most loans are payable more frequently than
annually usually monthly (using annual effective rate)
Instalments payable more frequently than annually
We have payments of Xt at t = 1/p, 2/p, 3/p, … , n
- State L0
- State Lt (the loan outstanding at any time t) using prospective and retrospective method
L0 = X1/p x v^(1/p) + X2/p x v^(2/p) + X3/p x v^(3/p) +...+Xnv^n L0 = pX𝑎n¬(p)
Loan outstanding prospectively
Lt = pX𝑎n-t¬(p)
Loan outstanding retrospectively
Lt = (1+i)^(t)L0 - pX𝑠t¬ (p)
Instalments payable more frequently than annually
Loan of £20,000 is repayable by monthly instalments in arrears for 8 years, calculate the following with i = 5%pa
- Monthly repayment
- Interest element of the 25th payment
- Capital element of the 25th payment
Monthly repayment
20000 = 12X 𝑎8¬(12)
X = 20000/(12x((1-1.05^-8)/0.048889)) = £252.14 pm
Interest element of the 25th payment
i(12)/12L2 = 0.048889/12x 12x252.14 𝑎6¬(12)
= 0.048889/12 x 3025.68 x (1-1.05^-6)/0.048889 = £63.99
Capital element of 25th payment
252.14 –63.99 = £188.15
What measure of interest charge is commonly used when borrowing?
When borrowing common to use flat rate of interest as a measure of interest charge
What is flat rate?
Total interest paid over whole transaction, per unit of initial loan, per year of the loan
State the general formula for calculating the flat rate of interest per annum for a 3 year loan repayable in monthly installments
If a loan of L0 is repaid monthly (£X pm) over 3 years
Total capital and interest is 36X (total amount repaid)
Total capital is L0, so total interest is 36X –L0
Flat rate of interest per annum is
F = (36X−L0)/3L0
F = (pxnxX−L0)/nL0
NB –this ignores gradual repayment of capital
[Page 35 week 7 lecture notes shows correct notation. x subscript on both p and n on numerator]
What do the letters in the acronym APR stand for?
Annual Percentage Rate
What is the APR of a loan?
- APR is the effective annual rate of interest charged on a loan rounded to nearest 1/10th of 1% (1 decimal place)
- APR is very roughly twice the flat rate.
Flat Rates & APRs Example Loan of £6,000 repayable by 24 monthly instalments in arrears. Flat rate of interest is 10% pa. - What is the monthly repayment? - What is the APR?
Monthly repayment F = (pxnxX−L0)/nL0 10% = (12x2xX−6000)/(2x6000) rearrange (6000 + 2x10%x6000)/24 = X = £300
APR
Solve 6000 = 300x12x𝑎2¬ (12)
Calc P1 = 3600 𝑎2¬ (12) @ i = 20% as APR approx twice flat rate
3600 x (1-1.2^-2)/0.18371 = £5,987.70 (too low try 19%)
Calc P2 = 3600 𝑎2¬ (12) @ i = 19%
3600 x (1-1.19^-2)/0.17522 = £6,037.02
i ≈0.19 + (6000−6037.02)/(5987.70−6037.02)x(0.20 –0.19) = 19.75%, 19.8%
