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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


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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



Compound interest calculation:  the future value of Bruce’s deposit

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May 11, 2014
Calculate δ
by: Staff


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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




Force of interest:  relationship to future value and present value of Peter’s deposit 

*** Click to enlarge image ***







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May 11, 2014
Calculate δ
by: Staff


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Part III


Solve for the force of interest, δ 

*** Click to enlarge image ***




Force of interest for Peter’s deposit 

*** Click to enlarge image ***







Thanks for writing.

Staff
www.solving-math-problems.com


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