Financial Mathematics - how to solve?

Question

Winston deposits $10 into a fund today and $20 fifteen years later. Interested is credited at a nominal discount rate of 𝑑 compounded quarterly for the first 10 years, and a nominal interest rate of 6% compounded semiannually thereafter. The accumulated balance in the fund at the end of 30 years is $100. Find 𝑑.

Answer 1

The 10 dollar deposit will be worth 10(1/(1-d)) in 1 quarter. 10 years ie 40 quarters. In 10 years

the $10 will be worth 10*(1/(1-d))⁴⁰ Let's call this Number K.

In 10 years K will earn 6% compounded semiannually for another 20 years or 40 compounding periods at 3% per compounding period.or K(1+3%)⁴⁰ In 15 years we make another deposit of $20 which will accrue at 3% for Every 6 months for 15 years or 30 periods. So that piece will be worth 20*(1+3%)³⁰

We Have K(1.03)⁴⁰ + 20(1.03)³⁰=100

20*(1.03)³⁰ =20*2.4272 =48.545

so

K(1.03)⁴⁰ = 100-48.545 =51.4575

divide both sides by (1.03)⁴⁰ recognizing that 1.03⁴⁰=3.262. and get

K= 51.4575/3.262 =15.7738. 15.7738 = 10*(1/(1-d))⁴⁰ if we multiply both sides by (1-d)⁴⁰ and divide both sides by 15.7738 we get (1-d)⁴⁰= 10/15.7738 = .66396. so, taking the 40th root we get

1-d = (0.66396)^(1/40)= .98867 so d = 1-.98867 . 0.01133 quarterly discount rate.

Hope this helps.

If you need help with this stuff. I am available and know it quite well.

Lenny