Question

Problem #4: (a) You put $200 in a bank today, and the bank says that you will get $300 in four years. What is the implied interest rate per year? (b) You invest $560 in an asset that promises to pay an interest rate of 6% annually. How long will it take for your investment to grow to $1400? (c) If the stated APR is 12%, then what is the EAR with daily compounding? (d) What APR, compounded weekly, would give an EAR of 10%?

Answer 1

(a) Implied Interest rate = (future value/present value)^{1/no. of years} - 1 = ($300/$200)^1/4 - 1 = 1.5^0.25 - 1 = 1.1067 - 1 = 0.1067 or 10.67%

(b) We can use financial calculator for calculation of no. of years as below:

PV = investment = -560; I/Y = interest rate = 6%; FV = future value = 1400; PMT = payment = 0 > CPT = compute > N = no. of years = 15.73

PV needs to be entered as negative value as it's a cash outflow.

It will take 15.73 years for your investment to grow to $1400.

(c) EAR = (1+APR/365)³⁶⁵ - 1

EAR = (1+0.12/365)³⁶⁵ - 1 = (1+3.2876712328767123287671232876712e-4)³⁶⁵ - 1 = 1.0003287671232876712328767123288³⁶⁵ - 1 = 1.1275 - 1 = 0.1275 or 12.75%

(d) EAR = (1+APR/52)⁵² - 1

0.10 = (1+APR/52)⁵² - 1

0.10 + 1 = (1+APR/52)⁵²

1.1 = (1+APR/52)⁵²

1.1^1/52 = 1+APR/52

1.1^{0.01923076923076923076923076923077} = 1+APR/52

1.0018345688392329563061188633346 = 1+APR/52

APR/52 =  1.0018345688392329563061188633346 - 1 = 0.00183456883923295630611886333463

APR = 0.00183456883923295630611886333463*52 = 0.0954 or 9.54%