36. Use the savings plan formula. Investing $100 (P=100) each month at 12% (APR=0.12) for 1 year (Y=1 and nY=12) gives 100 × [(1+0.12/12)12-1]/[0.12/12] = $1268.25.
42. Depositing $200 monthly for 18 years at 4.5% yields A = $200 × [(1+0.045/12)12 × 18-1] /[0.045/12] = $66,373.60.
44. To create a retirement fund worth $2,000,000 by making monthly deposits for 30 years, assuming an APR of 6%, you must deposit [$2,000,000 × (0.06/12)]/ [(1+0.06/12)12 × 30-1] = $1991.01 each month.
48. First, we need to figure out how much you must save by the time you reach 65. In order to draw an annual income of $200,000 from then on, without end, if we assume an APR of 6%, then the amount saved must satisfy A × 0.06 = $200,000. Hence, A = $200,000/0.06 = $3,333,333. Next, in order to save $3,333,333 after 40 years of monthly deposits at 6%, you must deposit [$3,333,333 × (0.06/12)]/[(1+0.06/12)12 × 40-1] = $1673.79 each month.
50. Since the municipal bond cost you $8000, and you receive $12,500 for it after 20 years, the total return was (12500-8000)/8000 = 0.5625 = 56.25%, and the annual return was (12500/8000)1/20-1 = 0.0226 = 2.26%.
58. Investing $2000 in small-company stocks for
a year in the best of years (1993), whose annual return
was 142.9%, would yield $2000 × (1+1.429) = $4858.
In the worst of years (1937), whose annual return was
-58.0%, we'd get $2000 × (1-0.580) = $840 (a loss).
Investing $2000 in large-company stocks for a year in the
best of years (1933), whose annual return was 54.0%,
would yield $2000 × (1+0.54) = $3080. In the worst
of years (1931), whose annual return was -43.3%, we'd
get $2000 × (1-0.433) = $1134 (a loss).
Investing $2000 in long-term corporate bonds for a year in the
best of years (1982), whose annual return was 42.6%,
would yield $2000 × (1+0.426) = $2852. In the worst
of years (1969), whose annual return was -8.1%, we'd
get $2000 × (1-0.081) = $1838 (a loss).
Investing $2000 in U.S. Treasury bills for a year in the
best of years (1981), whose annual return was 14.7%,
would yield $2000 × (1+0.147) = $2294. In the worst
of years (1938), whose annual return was -0.02%, we'd
get $2000 × (1-0.0002) = $1999.60 (a tiny loss!).
74. The current yield of a $1000 Treasury Bond with a coupon rate of 2.5% and a market value of $1050 is [(0.025 × $1000)/$1050] = 0.0238 = 2.38%.
80. The annual interest on a $10,000 Treasury Bond with a current yield of 3.6% that is quoted at 102.5 points is 3.6% × (102.5% × $10,000) = 0.036 × $10,250 = $369.
84. Polly deposits $50 monthly for 10 years at 6%,
yielding A = $50 × [(1+0.06/12)12 × 10-1]
/[0.06/12] = $8,193.97. Overall, she deposits $50 × 12
× 10 = $6000.
Quint deposits $40 monthly for 10 years at 6.5%,
yielding A = $40 × [(1+0.065/12)12 × 10-1]
/[0.065/12] = $6,736.13. Overall, he deposits $40 × 12
× 10 = $4800.
Thus we see that although Polly has a lower APR than Quint,
she comes out ahead because her monthly payments are
significantly higher than his.
Section 4C:
16.
(a) A 15-year loan of $12,000 at an APR of 8% requires monthly payments of
[$12,000 × (0.08/12)]/[1-(1+0.08/12)(-12 × 12)]
= $114.68.
(b) The total payments over the term of the loan will be
$114.68 × 12 × 15 = $20,642.40.
(c) $12,000 of the payments pays off the principal, and the
remaining $20,642.40 - $12,000 = $8,642.40 is all interest.
18.
(a) A 15-year mortgage of $150,000
at an APR of 7.5% requires monthly payments of
[$150,000 × (0.075/12)]/[1-(1+0.075/12)(-12 × 15)]
= $1390.52.
(b) The total payments over the term of the mortgage will be
$1390.52 × 12 × 15 = $250,293.60.
(c) $150,000 of the payments pays off the principal, and the
remaining $250,293.60 - $150,000 = $100,293.60 is all interest.
22.
(a) A 5-year loan of $10,000 at an APR of 10% requires monthly payments of
[$10,000 × (0.010/12)]/[1-(1+0.010/12)(-12 × 5)]
= $212.47.
(b) The total payments over the term of the loan will be
$212.47 × 12 × 5 = $12,748.20.
(c) $10,000 of the payments pays off the principal, and the
remaining $12,748.20 - $10,000 = $2748.20 is all interest.
26.
A 15-year student loan of $12,000 at an APR of 8% requires monthly payments of
[$12,000 × (0.008/12)]/[1-(1+0.08/12)(-12 × 15)]
= $114.68.
Note that the monthly interest rate here is 0.08/12 = 0.0066667.
For a $12,000 starting loan principal, the interest due at the end
of the first month is 0.0066667 × $12,000 = $80. So of the
$114.68 first monthly payment, $114.68-$80=$34.68 goes toward
the principal. This effectively reduces the loan to $12,000.00
-$34.68 - $11,965.32.
The interest due at the end
of the second month is 0.0066667 × $11,965.32 = $79.77.
So of the $114.68 second monthly payment, $114.68-$79.77=$34.91 goes toward
the principal. This effectively reduces the loan to $11,965.32
-$34.91 - $11,930.41.
A similar argument shows that an additional $35.14 is paid towards
the principal at the end of the third month, and the loan is thereby
reduced to $11,895.27.
| Payment period | Interest paid | Toward principal | New principal |
| 1 | $80.00 | $34.68 | $11,965.32 |
| 2 | $79.77 | $34.91 | $11,930.41 |
| 3 | $79.54 | $35.14 | $11,895.27 |
30. If you pay off a credit card debt of $2500 in 2 years at an APR of 20%, your monthly payments will be [$2500 × (0.20/12)]/[1-(1+0.20/12)(-12 × 2)] = $127.24, and the total payments will be 24 × $127.24 = $3053.76.
36.
(a) For the first six months, with an APR of 6%, which comes out
to 0.5% per month, the interest payments are $4000 × 0.005
= $20 per month.
(b) After the first six months, the APR jumps to 24%
(four times as large as before), which is 2% per month. The interest
payments go up to $4000 × 0.02 = $80 per month (four times
as large as before).