Question

You are given the following information about various annuities available in the market. All the annuities pay $100 at the end of each year over the period of the investment. It is also given that the spot rate of interest for 1 year maturity is 3%.

Period of investment (in years)	Price (in $)
2	190.89
3	281.96
4	369.53

Calculate the forward rates of interest for t = 2, 3 and 4. You want to accumulate $10,000 after four years by putting a single payment into a bank account. Based on the information above, calculate the deposit required.

Answer 1

The 2 year rate can be computed by comparing the payoff of the annuities with the price. Let x be the annual rate of interest for the 2 year period which yields the following equation- 100/(1+3%) + 100/(1+x)² = 190.89. We can use Excel goal seek to solve for 'x' which gives 3.25% as the 2 year annual rate of interest rate.

The 3 year rate can be computed by comparing the payoff of the annuities with the price. Let x be the annual rate of interest for the 3 year period which yields the following equation- 100/(1+3%) + 100/(1+3.25%)² +100/(1+x)3= 281.96. We can use Excel goal seek to solve for 'x' which gives 3.1675% as the 3 year annual rate of interest rate.

The 4 year rate can be computed by comparing the payoff of the annuities with the price. Let x be the annual rate of interest for the 4 year period which yields the following equation- 100/(1+3%) + 100/(1+3.25%)² +100/(1+3.1675%)³ +100/(1+x)⁴= 369.53. We can use Excel goal seek to solve for 'x' which gives 3.3739% as the 4 year annual rate of interest rate.

To get $10,000 after 4 years, deposit needed now is - 10000/(1.033739)⁴ = $8,757.02