8.5: Mathematics of Finance
- Page ID
- 192507
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Simple Interest
1) $600
3) $3,048
5) $1,800
7) $872
9) 11%
11) Discount $240; Proceeds $1,760
13) $7,440
15) $2,790.70
Compound Interest
1) $11,542.52
3) $5,647.77
5) Bank B pays more. Bank A \(r_\text{EFF}=0.0513\); Bank B \(r_\text{EFF}=0.0523\).
7) $12,702.00
9) 8.66 years
11) $151,257.12
13) 7.3967%
15) 17.38 million people
Future Value of Annuities and Sinking Funds
1) $13,954.01
3) $15,904.47
5) $20,578.36
7) $6,438.02
9) A lump sum of $25,000 is better; \(A\) = $38,002.64. For $400 paid monthly, \(A\) = $35,872.42.
11) $112,552.26
Present Value of Annuities and Sinking Funds
1) $1,177,953.55
3) $12,043.34
5) $2,149.21
7) $1,976.80
9) Leasing is better; \(P\) = $37,908 if payments are assumed to be made at the end of each year as is the general assumption in this book. However, if we consider payments to be made at the beginning of each year, then the present value is $41,699 = $37,908(1+0.10), which is greater than $40,000; in this case, buying would be better and this situation more realistically represents the real world.
Classification of Finance Problems
1) D
3) F
5) E
7) D
9) a) B; b) F
11) F
13) A
15) B
17) A
19) B
Additional Application Problems
1) $1,123.06
3) $171,907.63
5) $5,579.64
7) $11,680.01
9) $333.85 / month
10) $2,177.77 / month
11) $305,421.70
12) $2,479.37
13) $447.70 + $471.48 = $919.18
15) $435.48 + $806.45 = $1,241.94
17) a) $30,535.40; b) $720.63
Chapter Review
1) $872
2) 81,596
3) $1,190.05
4) $1,755.93
5) $12,156.72
6) $160,383.25
7) $289.28
8) $19,290.63
9) $2,085.33
10) $688,675.54
11) $497,897.83
12) $928.94 if \(r\) = 0.07 or $1,077.95 if \(r\) = 0.05
13) $3,447.31
14) a) $1,643.90; b) $128,451.61
15) $9,898.48
16) Present value $1,213,539.16; Future value $5,745,936.31
17) $6,669.70
18) $767,123,287.67
19) $2,375.25
20) 109,619
21) $5,805.92
22) $2,138.67
23) $1,523.33
24) $276.68
25) If leasing, $7,835.35 + $750 = $8,585.33. Purchasing for $8,000 is cheaper.
26) Cost in 5 years is $20,615.73; Sinking fund deposit is $833.79 / quarter.
27) Western Bank \(r_\text{EFF}\) = 0.597 or 5.97%. City Bank is better.
28) $404.57
29) $300 per month for 5 years is worth $14,900.82; $500 per month for 3 years is worth $16,026.59 (better).
30) 10.19 years
31) $177,692.68
32) $20,000 and $500 per month for 10 years is $61,210.74; $12000 and $1,000 per month for 5 years is $61,318.43 (better).
33) $16,384.77
34) Monthly payment $2,204.21; Total paid $793,515.60; Interest $368,515.6.
35) 15.53 years