Debt Restructuring and Financial Equivalence

EXERCISE 01

A borrower received €10,000 six years ago and €6,000 three years ago and has made no repayments. All amounts accrue at 12% effective annually. Today the debts are replaced by 48 equal quarterly payments starting immediately (t = 0, 0.25, …, 11.75), plus a payment of ten times the quarterly amount at t = 15.

A.Find the quarterly payment and the final lump sum.

A.

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

D_0=10{,}000(1.12)^6+6{,}000(1.12)^3

Debt today: €28,167.79.

i_q=1.12^{1/4}-1

D_0=R(1+i_q)a_{48|i_q}+10R(1.12)^{-15}

R=\frac{D_0}{(1+i_q)a_{48|i_q}+10(1.12)^{-15}}

€990.55 per quarter.

C_{15}=10R

€9,905.52

Debt Restructuring and Financial Equivalence

EXERCISE 02

An €84,000 loan has three annual principal instalments C, 2C and 4C, with 5% annual interest on outstanding debt. Immediately after the first payment it is refinanced into four equal annual payments at 6%. In the holiday alternative, skip year 2 and pay at years 3–6; 6% also applies during the holiday.

A.Build the original repayment schedule.
B.Find the new payment without a holiday.
C.Find the payment with one year of capitalised interest, then with interest paid during that year.

A.

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

C+2C+4C=84{,}000\quad\Rightarrow\quad C=12{,}000

Period	Opening debt	Interest	Principal	Payment	Closing debt
1	84,000.00	4,200.00	12,000.00	16,200.00	72,000.00
2	72,000.00	3,600.00	24,000.00	27,600.00	48,000.00
3	48,000.00	2,400.00	48,000.00	50,400.00	0.00

B.

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

D_1=84{,}000-12{,}000=72{,}000

R=\frac{72{,}000}{a_{4|0.06}}

€20,778.59

C.

D_2=72{,}000(1.06)=76{,}320

R_{capitalised}=\frac{76{,}320}{a_{4|0.06}}

€22,025.30

I_{holiday}=72{,}000(0.06)=4{,}320

R_{interest\ paid}=\frac{72{,}000}{a_{4|0.06}}

€20,778.59

Debt Restructuring and Financial Equivalence

EXERCISE 03

A €60,000 machine is financed by a five-year loan with equal monthly payments, using a 15% nominal annual rate convertible monthly. Immediately after payment 24, refinance the residual debt into six equal annual payments, first due one year later, at 16% effective annually.

A.Calculate the monthly payment, residual debt and new annual payment.

A.

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

i_m=0.15/12=0.0125

R_m=\frac{60{,}000}{a_{60|0.0125}}

€1,427.40

D_{24}=R_ma_{36|0.0125}

€41,176.47

R_a=\frac{D_{24}}{a_{6|0.16}}

€11,174.88

Debt Restructuring and Financial Equivalence

EXERCISE 04

An €80,000 loan is repaid with ten equal annual payments at 5%. Immediately after payment 7, suspend payments in years 8 and 9. Then replace the remaining debt with four equal payments at years 10–13, at the same 5% rate.

A.Calculate the original payment and the balance after payment 7.
B.Build the schedule during and after the holiday, with interest capitalised.
C.Instead pay interest during both holiday years. Find the final four payments.

A.

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

R=\frac{80{,}000}{a_{10|0.05}}

€10,360.37

D_7=Ra_{3|0.05}

€28,213.85

B.

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

D_8=D_7(1.05),\qquad D_9=D_7(1.05)^2

Year	Opening debt	Capitalised interest	Payment	Closing debt
8	28,213.85	1,410.69	0.00	29,624.54
9	29,624.54	1,481.23	0.00	31,105.77

R_{new}=\frac{D_9}{a_{4|0.05}}

€8,772.19

Period	Opening debt	Interest	Principal	Payment	Closing debt
10	31,105.77	1,555.29	7,216.91	8,772.19	23,888.86
11	23,888.86	1,194.44	7,577.75	8,772.19	16,311.11
12	16,311.11	815.56	7,956.64	8,772.19	8,354.47
13	8,354.47	417.72	8,354.47	8,772.19	0.00

C.

I_8=I_9=0.05D_7

Pay €1,410.69 in each holiday year; principal remains €28,213.85.

R_{new}=\frac{D_7}{a_{4|0.05}}

€7,956.64