Question

A loan of 50,000€ is to be repaid in 6 years. Solve the different options that the bank gives you to choose the one which is better.

A) Annual level payments at an effective interest rate of 5%.

B) Monthly installments that increase by 4% per year.

C) Taking into account that you have choose the best option: what is the true cost of the transaction if there is a bilateral initial expense paid by the borrower of 2,000€ and an unilateral expense also paid by the borrower at the end of the transaction of 550€?

Answer 1

1) Annual Installments

The annuity factor itself is calculated as:
AF = (1 – (1+r)-n ) ÷ r

Where:
r = interest rate per period = 0.05 (5%)
n = number of periods = 6 (years)
Applying the formula:
AF = (1 – 1.05-6 ) ÷ 0.05 = 5.0756921

Now, the equated annual instalment is given by:
Instalment = Principal ÷ annuity factor = ÷ 3.55 = euro 50,000/5.0756921=Euro 9850.8734

2) Let's consider the monthly payment to be Euro X(interest+principal)

Therfore total paid in year 1= 12X

total paid in year 2=12X*1.04

total paid in year 2=12X*1.04^2

total paid in year 2=12X*1.04^3

total paid in year 2=12X*1.04^4

total paid in year 2=12X*1.04^5

Not we know above all payments should have PV discounted at 5% to be equal to Euro50,000

NPV=50,000=12X*1.04/(1.05)+12X*1.04^2/(1.05)^2+12X*1.04^3/(1.05)^3+12X*1.04^4/(1.05)^4+12X*1.04^5/(1.05)^5

50,000=12X(0.9904762+0.981043084+0.9717+0.962446+0.953279)

X=euro 857.5251

Annual paid 1sit year=euro10290.30, ans so on

Clearly option 1 is better than option 2

3) True cost of transaction:

Option 1:

Total interest paid at the end of 6th year=Euro (9850.8734*6-50,000)=Euro 9105.2404

PV of interest cost=9105.2404/(1.05)^6=Euro 6794.4706

Upfront charge=Euro 2000

End of term charge Euro 500;, PV=500/(1.05)^6= Euro 373.1077

True cost=Euro 6794.4706+Euro 2000+Euro 373.1077=Euro 9167.5783

Option 2:

otal interest paid at the end of 6th year=Euro (10290.30+10701.91+11129.99+11575.19+12038.2+12519.73)-50,000=Euro(68255.32-50,000)=Euro 18255.3189

PV of interest @ 5%=Euro 18255.3189/(1.05)^6=Euro 13622.40

Upfront charge=Euro 2000

End of term charge Euro 500;, PV=500/(1.05)^6= Euro 373.1077

True cost=Euro 13622.40+Euro 2000+Euro 373.1077=Euro 15995.51