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These documents are finance problem set answers.
Typology: Exercises
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Class 2
a. The present value of Mike Polanski’s future salary payments can be written as an annuity with growth. Employing the fomula that we have on the slides, the PV of this annuity with growth is;
〡 䚁 =
⡱⡨ 䚁 = $760,662.
b. The present value of all future savings are equal to PV(salary) x 0.05 = $38,033.13. Since these savings are invested at an interest rate of 8%, at age 60, Mike will have accumulated:
Future value = $38,033.13 x (1.08)^30 = $382,714.
c. During Mike’s retirement, from age 60 to age 80, Mike plans to consume the same amount every year. Thus, Mike will use the $382,714.30 that he has accumulated to create a 20-year annuity cashflow stream. The question is asking you to find the annual cashflow in that annuity. We have;
〡 䚁
So;
⡰⡨ 䚁 ⇒ ᠩ = $38.980.
a. Stated annual rate of 6% = 0.5% per month. Using the same technique as in part (c) of question 4 we can find the monthly annuity cashflow that delivers a present value for the annuity stream of $500k
b. At the end of the first year of payments, you owe the mortgage company an annuity stream with 348 monthly payments and a monthly cashflow of $2997.75. The present value of this stream is the value of your outstanding debt to the mortgage company. It is;
⡱⡲⡶ 䚁 = $493,859.
c. The total annual payment you made in the first year of the mortgage was 2997.75 × 12 = $35,973. The loan balance has declined by 500,000 - 493,860 = $6,140. Therefore the remainder of the payments you have made are interest payments;
Interest = 35,973 – 6,140 = $29,
the twentieth year. In 19 years we have 132 months remaining so the balance is;
⡩⡱⡰ 䚁 = $289,
⡩⡰⡨ 䚁 = $270,
Therefore, there was 289,162 – 270,018 = $19,144 in principal repaid, and $35,973 – 19, = $16,829 in interest repaid.
This is an example of an amortization loan, which is a loan that is repaid with regular payments. In each payment, part is used to pay interest and part is used to reduce the loan balance (amortization). Over the loan’s lifetime, the outstanding balance is reduced therefore the part of the regular payment that goes to interest payment is reduced and the amortization amount increases.
1+EAR = (1+0.05/2)^2 =1.
Then we can convert the annual rate to a monthly rate:
(1.050625)1/12^ = 1.
[Alternatively one could directly convert the six month rate into a one month rate: (1+0.05/2)1/6^ = 1.004124.]
Given this discount rate, we can use the annuity formula to compute the present value of the monthly payments:
Thus paying $4,000 a month for 48 months is equivalent to paying a present value of $173,867 today. This cost is $23,867 higher than the cost of purchasing the system outright, so it is better to pay $150,000 for the system rather than lease it.