Mathematics: Compound Interest + Periodic Payment of an annuity?

Question

Hey there, Maths exam next week and I'm studying for it. Would like an answer to this that I can work from.

Very much appreciated.

(b) A staff member inherits a lump sum of €160,000. He is considering investing it at 3.6% per annum, compounded annually. How much will he have after ten years? If the interest is compounded quarterly, how much of a difference will this mean to him after ten years?

c) Alternatively, the staff member considers buying an apartment for his daughters to use while attending college. The apartment will cost €325,000. If he makes a down payment of €100,000 and borrows the rest over 20 years, what monthly repayment will he have to make? Interest at 5.4% is compounded monthly.

Answer 1

(b)

Annually:

E160,000(1 + 0.036)^10 = E227,885.94.

Quarterly:

E160,000(1 + 0.036 / 4)^(4 * 10) = E228,963.70.

Difference: E1077.76.

(c)

Let:

L be the loan,

p be the monthly payment,

Ea be the amount of E1 after 1 month,

r be the fractional interest rate,

n be the number of months.

L a^n = p(a^(n - 1) + a^(n - 2) + ... + a^2 + a + 1)

= p(a^n - 1) / (a - 1)

p = L a^n (a - 1) / (a^n - 1)

= L(a - 1) /  [ 1 - a^(- n) ]

L = E(3.25 - 1.00) * 10^5

= E 2.25 * 10^5.

a = 1 + 0.054 / 12

n = 240

p = 2.25 * 0.054 * 10^5 / [ 12 { (1 - a^(- 240) } ]

= E 1535.07.