Question

Suppose you will receive $17,000 in 11 months and another $15,000 in 23 months. If the discount rate is 4% per annum (compounding monthly) for the first 14 months, and 13% per annum (compounding monthly) for the next 9 months, what single amount received today would be equal to the two proposed payments? (answer to the nearest whole dollar; don’t include the $ sign or commas)

Answer 1

Suppose you will receive $17,000 in 11 months and another $15,000 in 23 months. If the discount rate is 4% per annum (compounding monthly) for the first 14 months, and 13% per annum (compounding monthly) for the next 9 months, what single amount received today would be equal to the two proposed payments?

Actually this question is regarding Present value of single sum in future.

But for solve this problem, Should maintain two steps. And also calculate the present value of two payment separately.

The first proposed amount, there is no any problem to calculate its present value.

First calculate the preset value of 17000 on this date. While considering discount rate, It is yearly rate, but here compounding monthly. So should convert annual discount rate to monthly discount rate

PV calculation of first proposed amount of $ 17000

Formula for calculation

PV of single sum = FV * ( 1 / 1 + r )n

Here,

FV = future value = 17000

R = monthly discount rate = 4% / 12 = 0.3333%

N = number of period compounding = 11

PV = 17000 * ( 1 / 1 + 0.3333 )11

PV = 17000 * 0.96405 = 16389.01     (a)

Next calculate the present value of the 2nd proposed amount of $ 15000 in 23 months after

But here has a complication, that is two discount rate are there in among those 22 months. First 14 months 4% and after 11 months 13%.

For this situation, we calculate it through 2 steps

Steps 1 : pretend the the end of 14th month is present date and calculate the PV of that amount in that time. For using the same above formula. And discount rate of 13% should use here. Also remember that this discount rate is annual, so should convert monthly because of compounding. And here only the number of periods are 9 days( 14th months to 23rd months )

Here,

FV = future value = 15000 after 23 months

R = monthly discount rate = 13% / 12 = 1.08333%

N = number of period compounding = 9

PV on 14th month = 15000 * ( 1 / 1 + 1.08333%)9-

PV on 14th month = 15000 * 0.9076 = 13613.67

2nd steps : convert this amount of 13613.67 to its today value (0th month)

This is the period of 0th month to 14th month total of 14 months and discount rate in this period is 4% annually

Here,

FV = PV on 14th month = 13613.67

R = discount rate monthly = 4% / 12 = 0.3333%

N = 14 periods

PV on today (0th period ) = 13613.67 * ( 1 + 1/ 0.3333% )14

PV on today (0th period ) =   13613.67 * 0.95448 = 12994.03 (b)

After calculating each amount to its present value, jus add together to find what single amount received today would be equal to the two proposed payments

Just add a and b

single amount received today = 16389.01  + 12994.03 = 29383