In March 2018, Daniela Motor Financing (DMF), offered some securities for sale to the public. Under the terms of the deal, DMF promised to repay the owner of one of these securities $5,000 in March 2048, but investors would receive nothing until then. Investors paid DMF $850 for each of these securities; so they gave up $850 in March 2018, for the promise of a $5,000 payment 30 years later.

Required:
a. Assuming you purchased the bond for $850, what rate of return would you earn if you held the bond for 30 years until it matured with a value $5,000?
b. Suppose under the terms of the bond you could redeem the bond in 2025. DMF agreed to pay an annual interest rate of 1.3 percent until that date. How much would the bond be worth at that time?
c. In 2025, instead of cashing in the bond for its then current value, you decide to hold the bond until it matures in 2048. What annual rate of return will you earn over the last 23 years?

a. Assuming you purchased the bond for $850, what rate of return would you earn if you held the bond for 30 years until it matured with a value $5,000?

future value = present value x (1 + r)ⁿ

future value = $5,000
present value = $850
n = 30

5,000 = 850 x (1 + r)³⁰

(1 + r)³⁰ = 5,000 / 850 = 5.882652

³⁰√(1 + r)³⁰ = ³⁰√5.882652

1 + r = 1.0608444

r = 0.0608444

r = 6.08%

b. Suppose under the terms of the bond you could redeem the bond in 2025. DMF agreed to pay an annual interest rate of 1.3 percent until that date. How much would the bond be worth at that time?

future value = present value x (1 + r)ⁿ

future value = 850 x 1.013⁷ = $930.43

c. In 2025, instead of cashing in the bond for its then current value, you decide to hold the bond until it matures in 2048. What annual rate of return will you earn over the last 23 years?

5,000 = 930.43 x (1 + r)²³

(1 + r)²³ = 5,000 / 930.43 = 5.373859398

²³√(1 + r)²³ = ²³√5.373859398

1 + r = 1.075849638

r = 0.0758

r = 7.58%