Questions on loan payment - Broverman's book.

[pde]

Question 1(Problem 3.1.5 page 200). Bettey borrows $ 19,800 from bank X. Betty repays the loan by making 36 equal payments pf principle at the end of each month. She also pays interests on unpaid balance each month at a nominal rate of 12% compounded monthly. Immediately after the 16th payment is made, bank X sells the rights of future payments to bank Y. Bank Y wishes to yield a nominal rate of 14% compounded semi-annually, on its investment. What price does bank Y receive?
Book's answer: 10,857.28

Question 2 (Problem 3.1.7 page 200). A 30-year loan of 1000 is repaid with payments at the end of each year. Each of the first ten payments equals the amount of interest due. Each of the next ten payments equals 150% of the amount of interest due.Each of the last ten payments is X. The lender charges interest at an effective annual rate of 10%. Calculate X.
Book's answer: 97.44

I do not get the same answers that the book has.

[Junkmonkey]

I have posted a solution to question 2 here:

[url]http://evo-games.com/music/Loan_Repayment_01-26-2007.pdf[/url]

I hope that helps.

- junk

[pde]

Thank you for your answer. I could not agree more. Any idea on this first question?

[Sam Broverman]

pde,


For the first 10 years only interest and no princiapal is paid,
so at the end of the 10th year just after the payment,
the balance owing is still 1000. Then the key observation
is that since 150% of interest paid each year, the principal
paid each year is 50% of the interest. The interest due is 10%
of the previous balance, so the princiaple repaid is
50% of 10% of the previous balance, which is 5% of the
previous balance. For the 2nd 10 years, each year 5% of
the previous balance is paid as principal, so the new balance is
always 95% of the previous balance. This goes on for 10 years,
so at the end of the 2nd 10 years, the balance owed is
1000 x .95 x .95 ... x .95 = 1000 x (.95)^10 = 598.74 .
This amount is repaid with 10 level payment over the final
10 years, so the level annual payment for the final 10
years is K from the equation 598.74 = K a-angle-10 @10%.
Solving for K results in 97.44 .

Sam Broverman

[pde]

Thank you Dr. Broverman. Could you help me with question 1? Best regards.

[Sam Broverman]

The key point in solving this problem is identifying
the pattern of payment that Betty is making.
Since she is making 36 equal payments of principal,
and since the initial loan is 19,800 , each month the
principal repaid is 19.800/36 = 550 . So each month
she pays 550 plus interest at 1% on the previous
month's balance. Her schedule of payments is
1st month 550 + 198 (the 198 is 1% of 19,800),
2nd month 550 + 192.5 (192.5 is 1% of 19,250
because the balance is 19,800 - 550 = 19,250 after the
principal payment in the 1st month),
3rd month 550 + 187 (187 is 1% of 18,700) , etc.

Bank Y is buying Betty's final 20 payments (just after
the 16th payment), so we need to determine those
final 20 payments. Looking at the pattern of the first 3 payments
we see that the first payment is 550 + (36 x 5.5) ,
the 2nd payment is 550 + (35 x 5.5) ,
the 3rd payment is 550 + (34 x 5.5) , etc.
Following this pattern, the final 20 payments will be:
17th payment is 550 + (20 x 5.5) ,
18th payment is 550 + (19 x 5.5) , , . . . ,
36th payment is 550 + (1 x 5.5) .
Bank Y is buying a combination of a level annuity
with present value 550 a-angle-20,
plus a decreasing annuity with pv 5.5 x (Da)-angle-20 .
All we need to find what Bank Y pays is Bank Y's monthly valuation
interest rate. We are told that Bank Y has a rate of 14% compounded
semi-annually so we can find the equivalent 1 month interest rate
for Bank Y and then calculate the pv of the two annuities.

[pde]

Thank you again. Dr. Broverman

[Gagan]

Sorry if this is old
but wouldn't it be easier to use makeham's formula?
Outstanding balance at time 16 is 11000.

K + i/j (L - K)

bank Y receives the PV of all the remaining principal payments
so K is 550 a angle 20
i = 1%
j is the equivalent monthly interest that bank Y wants

L is the remaining "loan" to be paid to bank Y so it's 11000.

[Sam Broverman]

Gagan,

Makeham's formula works well for the solutions.
Thank you for that post.