An investment of 1000 accumulates to 1360.86 at the end of five years. If the force of interest is δ during the first year and 1.5δ in each subsequent year. How to find δ?
Asked
Active
Viewed 348 times
3
-
Is "the force of interest" the interest rate? – Patricio May 06 '21 at 08:01
-
@fae The result is 4.5%. If you are interested in a (right) answer give a reply. Btw, isn´t the number 1360.86? – callculus42 May 06 '21 at 12:49
-
@callculus yess it is 1360.86 – fae May 06 '21 at 13:50
-
@callculus this is my calculation 1000(e^delta)(e^1.5delta)^4 = 1360.86 and the delta value i got is 0.04401669... – fae May 06 '21 at 13:52
2 Answers
3
You need to solve the equation
$$1000 (1+\delta)(1+1.5\delta)^4=1360.80$$
(I get $4.54225\%$)
Patricio
- 1,604
-
-
@callculus I've just assumed that the interest is annually compounded rather than continuously compounded as you do – Patricio Jul 30 '21 at 17:41
1
You missed the integral at the exponent.
$$1000\cdot e^{\int_0^1 \delta \ ds +4\cdot \int_0^1 1.5\cdot \delta \ ds}=1360.86$$
Dividing the equation by 1000 the equation becomes
$$e^{\int_0^1 \delta \ ds +4\cdot \int_0^1 1.5\cdot \delta \ ds}=1.36086$$
$$e^{7\cdot \delta}=1.36086$$ $$7\cdot \delta=\ln(1.36086)$$ $$\delta=\frac{1}{7}\cdot \ln(1.36086)\approx 0.04401669$$
It seems that you haven´t omitted the integrals at your calculation if I compare our results. So I can confirm your result. Can you now find the equivalent annual effective interest rate in the first year?
callculus42
- 30,550
-
(1+i)^5 = e^(0.04401669326)(5) and the effective rate i got is 0.04499979914 – fae May 06 '21 at 16:04
-
Only for the fiirst year. But at the end your result is right. $$(1+i)=e^{\Large{\int_0^1 0.0440166932 \ d \delta } }=1.045$$ – callculus42 May 06 '21 at 16:20
-
-
@fae This is $1+i$. Therefore the annual interest rate is $i=0.045=4.5%$. Don´t forget to mark this answer as accepted. This is necessary since the totally wrong answer of Patricio has 3 upvotes. Very misleading. – callculus42 May 06 '21 at 17:00