True fact: Ancient Roman contractors who built roads were paid half when the road was completed,

Question

 and half 40 years after the road had been completed if the road was still in good working order. Assume the annual yield is 2%. A contractor receives a contract to build a small road and will be paid a total 12,000 denarii. It will take 2 years to complete the road, so the contractor will be paid 6,000 denarii in 2 years and another 6,000 denarii in 42 years. What (to the Ancient Roman contractor) is the present valueof this contract?Round your answer to the nearest denarii.

Answer 1

Starting at the beginning of the construction of the road, there are two future values: 6000 denarii in 2 years and 6000 denarii in 42 years. To find the present value of the contract these two future values must be converted to their present value with 2% A.P.R.

 

PV = A1 * P(i,n1) + A2 * P(i,n2), where A1 and A2 are the two lump sums, n1 and n2 are the time in years to the future payments and i is the annual interest rate.

 

 

 

PV = 6000 * P(2%,2) +6000 * P(2%,42)) 

 

We can look up the present values in a financial table, P(2%,2) = 0.96117, P(2%,42) = 0.4353.

 

Plugging in these values, gives:

 

6000 * 0.96117 + 6000 * .4353 = 5767.02 + 2611.80 = 8379 denarii rounded to the nearest whole number.

 

The two P values can be also be found from the formula: 1/(1+i)^n. -> P(2,2) = 1/(1 + 0.02)^2 = 0.96117, P(2,42) = 1/(1 + 0.02)^42 = 0.4353. This formula is the growth of compound interest in reverse. 

 

What this means is that 5767 denarri earning 2% interest will yield 6000 denarii in two years and 2611.80 denarii earning 2% will yield 6000 denarri in 42 years. Therefore, the present value of the contract is these two values added.