Question

1. A 20 year annuity has annual payments which increase by $500 each year. The first payment is $10,500 on Jan. 1, 2018. The annual effective interest is 1%. What is the value of the annuity on Oct. 1, 2017?

The answer should be $274,250.53 And I don't have more information about this...this is how the problem looks like.

2. A loan is repaid with level installments payable at the end of each half-year for 3(1/2) years, at a nominal rate of interest of 8% convertible semiannually. After the 4th payment, the outstanding loan balance is $5000. Find the amount of the loan.

3. Perpetuities in arithmetic progression. If a perpetuity has first payment P and each payment increases by Q, then its present value, one period before the first payment, is P/i + Q/i^2 Using this formula, find the present value of a perpetuity-immediate which has annual payments with first payment $360 and each subsequent payment increasing by $40, at annual interest rate 1.3%.

4. Filip buys a perpetuity-immediate with varying annual payments. During the first 5 years, the payment is constant and equal to 10. Beginning in year 6, the payments start to increase. For year 6 and all future years, the current year’s payment is K% larger than the previous year’s payment. At an annual effective interest rate of 9.2%, the perpetuity has a present value of 167.50. Calculate K, given that K < 9.2.

Answer 1

	1	Present Value (PV) of Cash Flow:
		(Cash Flow)/((1+i)^N)
		i=Discount Rate=annual effective interest=1%=0.01
		N=Year of Cash Flow
		CASH FLOW
		N	A	P=A/(1.01^N)
	Date	Year	Cash Flow	Present Value
	Jan.1 2018	0	$10,500	$10,500.00
	Jan.1 2019	1	$11,000	$10,891.09
	Jan.1 2020	2	$11,500	$11,273.40
	Jan.1 2021	3	$12,000	$11,647.08
	Jan.1 2022	4	$12,500	$12,012.25
	Jan.1 2023	5	$13,000	$12,369.05
	Jan.1 2024	6	$13,500	$12,717.61
	Jan.1 2025	7	$14,000	$13,058.05
	Jan.1 2026	8	$14,500	$13,390.51
	Jan.1 2027	9	$15,000	$13,715.10
	Jan.1 2028	10	$15,500	$14,031.95
	Jan.1 2029	11	$16,000	$14,341.18
	Jan.1 2030	12	$16,500	$14,642.91
	Jan.1 2031	13	$17,000	$14,937.26
	Jan.1 2032	14	$17,500	$15,224.35
	Jan.1 2033	15	$18,000	$15,504.29
	Jan.1 2034	16	$18,500	$15,777.19
	Jan.1 2035	17	$19,000	$16,043.17
	Jan.1 2036	18	$19,500	$16,302.34
	Jan.1 2037	19	$20,000	$16,554.80
	Jan.1 2038	20	$20,500	$16,800.66
			SUM	$291,734.26
		Value of annuity on Jan. 1, 2018	$291,734.26
		Number of months between Oct1, 2017 and Jan1 2018	3
		Number of years=(3/12)=	0.25
		Value of annuity on Oct. 1, 2018	$291,009.45	291734.26/(1.01^0.25)