Question

Sam borrows 100 from Peter now, and 190 a year from now. He pays back 130 two years from now, and 220 3 years from now.

A) write down the resulting equation of value at year 3, and the corresponding third degree polynomial equation.

B) Find Sam’s dollar-weighted yield rate.

Answer 1

a. Here we calculate the future value at eah point of time and assume that "r" is the interest rate

FV of $100 borrowed now = 100*(1+r)^3

FV of $190 borrowed a year from now = 190*(1+r)^2

FV of $130 paid pack 2 years from now = -130*(1+r)

FV of 220 paid back three years from now = -220

We frame the equation as follows:

100*(1+r)^3 + 190*(1+r)^2 -130*(1+r) - 220 =0

100*(1 + 3*(1)*r^2 + 3*(1)^2*r + r^3) + 190*(1+r^2 + 2r) -130 -130r -220 =0

100 + 300r^2 + 300r + 300r^3 + 190 +190r^2 + 380r -130-130r -220 =0

300r^3 +490r^2 +550r-60 =0

This is the third degree polynominal equation

b. To find the dollar weighted yield rate, we have to find r such that the equation in a is satisfied

When we plug in r = 0.10, we approximate the equation

So the dollar weighted yield rate = 0.10 or 10%

So the dollar weighted yield rate =10%