# force of interest, δ

by Tharindu
(Colombo)

Calculate the unknown interest rate, δ

Bruce deposits 100 into a bank account.

His account is credited interest at a nominal rate of interest of 4% convertible semiannually.

At the same time, Peter deposits 100 into a separate account.

Peter’s account is credited interest at a force of interest of δ .

After 7.25 years, the value of each account is the same.

Calculate δ.

(A) 0.0388

(B) 0.0392

(C) 0.0396

(D) 0.0404

(E) 0.0414

### Comments for force of interest, δ

 May 11, 2014 Calculate δ by: Staff Answer Part I Bruce’s account is credited interest at a nominal rate of interest of 4% convertible semiannually. The nominal interest rate is the stated annual interest rate. For Bruce’s 100 deposit, the nominal interest rate is 4%. Convertible semiannually means the interest is compounded every six months. The future value of Bruce’s deposit after 7.25 years is: FV = PV(1 + i)ⁿ FV = future value = unknown PV = present value = 100 i = interest rate per compounding period in decimal form = 4% per year ÷ 2 compounding periods per year = 2% = .02 n = number of time periods = 7.25 years * 2 time periods per year = 14.5 FV = 100(1 + .04/2)^(2*7.25) FV = 100(1 + .04/2)^(14.5) FV = 100(1 + .02)^(14.5) FV = 133.2608228361737 ---------------------------------------------------

 May 11, 2014 Calculate δ by: Staff --------------------------------------------------- Part II At the same time, Peter deposits 100 into a separate account. Peter’s account is credited interest at a force of interest of δ . The “force of interest” is that interest rate that compounds continuously, rather than compounding after a fixed time period (such as compounding every six months). The relationship between force of interest, the future value, and the present value is: PVe^(yδ) = FV FV = future value = The future value of Peter’s deposit after 7.25 years is exactly the same as Peter’s = 133.2608228361737 PV = present value = 100 e = base of the natural logarithm ≈ 2.71828 y = years = 7.25 δ = constant force of interest = unknown Solve for the force of interest, δ 100e^(7.25δ) = 133.2608228361737 100e^(7.25δ) / 100 = 133.2608228361737 / 100 e^(7.25δ) = 1.332608228361737 7.25δ = ln(1.332608228361737) 7.25δ / 7.25 = ln(1.332608228361737) / 7.25 δ = ln(1.332608228361737) / 7.25 δ = 0.2871380957946 / 7.25 δ = 0.0396052545924 Final Answer: δ = 0.0396, answer (C) 0.0396 ---------------------------------------------------

 May 11, 2014 Calculate δ by: Staff --------------------------------------------------- Part III Thanks for writing. Staff www.solving-math-problems.com

 Sep 21, 2018 Another way to find the value of the force of interest. NEW by: Anonymous δ = ln[ ( 1+0.04/2)^2 ] δ = ln ( 1.0404 ) δ = 0.039605255