Loan Amortisation

EXERCISE 01

A €12,000 loan is repaid with ten equal annual payments in arrears at 15% effective annually.

A.Find the payment, split the sixth payment into interest and principal, and find debt after payment 6.

A.

a_{n|i}=\frac{1-(1+i)^{-n}}{i}

C_k=R_k-I_k,\quad I_k=iD_{k-1},\quad D_k=D_{k-1}-C_k

R=\frac{12{,}000}{a_{10|0.15}}

€2,391.02

D_5=Ra_{5|0.15}

I_6=0.15D_5

€1,202.26

C_6=R-I_6

€1,188.76

D_6=Ra_{4|0.15}

€6,826.32

Principal repaid through year 6: €5,173.68.

Loan Amortisation

EXERCISE 02

A €9,000 loan is repaid over twelve years with equal annual principal instalments at 10%. Payments are at year-end.

A.Find the principal instalment, interest and payment in year 5, and debt immediately after that payment.

A.

C_k=R_k-I_k,\quad I_k=iD_{k-1},\quad D_k=D_{k-1}-C_k

C=\frac{9{,}000}{12}=750

D_4=9{,}000-4(750)=6{,}000

I_5=0.10(6{,}000)=600

R_5=750+600=1{,}350

D_5=9{,}000-5(750)

Remaining debt €5,250; principal already repaid €3,750.

Loan Amortisation

EXERCISE 03

A €100,000 loan is repaid over ten years with monthly payments and constant principal instalments. Treat 4% as an effective annual rate. Payments occur at month-end.

A.Calculate the first two and last two schedule rows.

A.

C_k=R_k-I_k,\quad I_k=iD_{k-1},\quad D_k=D_{k-1}-C_k

i_m=1.04^{1/12}-1,\qquad C=\frac{100{,}000}{120}

D_{k-1}=100{,}000-(k-1)C,\qquad R_k=C+i_mD_{k-1}

Month	Opening debt	Interest	Principal	Payment	Closing debt
1	100,000.00	327.37	833.33	1,160.71	99,166.67
2	99,166.67	324.65	833.33	1,157.98	98,333.33
119	1,666.67	5.46	833.33	838.79	833.33
120	833.33	2.73	833.33	836.06	0.00

Loan Amortisation

EXERCISE 04

A loan has twenty annual payments with equal principal instalments. Payment 15 is €12,240 and payment 16 is €11,700. Interest is charged on opening debt.

A.Infer the annual rate and initial principal.

A.

R_k=C[1+(n-k+1)i]

12{,}240=C(1+6i)

11{,}700=C(1+5i)

Ci=12{,}240-11{,}700=540

C=11{,}700-5(540)=9{,}000

i=\frac{540}{9{,}000}

6% annually.

D_0=20(9{,}000)

€180,000.00

Loan Amortisation

EXERCISE 05

A €10,000 loan is repaid over 15 years at 12% effective annually. Payments occur at year-end. The first ten payments are twice each of the last five payments.

A.Calculate both payment amounts.

A.

a_{n|i}=\frac{1-(1+i)^{-n}}{i}

10{,}000=2R a_{10|0.12}+R(1.12)^{-10}a_{5|0.12}

R=\frac{10{,}000}{2a_{10|0.12}+(1.12)^{-10}a_{5|0.12}}

€802.50 for years 11–15.

R_{1:10}=2R

€1,605.00 for years 1–10.