Solutions to Chapter 3 Knowledge Checks

Knowledge Check 3.1

1. P  = $3,000;  r  =  6%  = 0.06;  t  =180/365 years [latex]I=Prt=$3,000\times0.06\times\frac{180}{365}=$88.77[/latex]
2. I=$55; r =5.5%=0.055; t =125/365 years [latex]P=\frac{I}{rt}=\frac{$55}{0.055\left(\frac{125}{365}\right)} =$2920.00[/latex]
3. P = $900,   I = $65,   r = 7.5% = 0.075

[latex]t=\frac{I}{Pr}=\frac{$65}{$900\times 0.075}=0.962963\text{ years}[/latex]

To convert this answer to days, you must multiply by 365. To eliminate a rounding error, be sure to use the exact value from your calculator, i.e., just multiply the above value by 365 without re-entering the displayed value .

[latex]∴ t =\frac{$65}{$900\times0.075}\text{years}\times \frac{365\text{ days}}{1\text{ years}} = 351.48\text{ days}[/latex]

This should now be rounded up to 352 days.

4.  P  =  $975, I  = $36.73, t  =220/365  years

[latex]r=\frac{I}{Pt}=\frac{$36.73}{$975\left(\frac{220}{365}\right)} = 0.062500932 = 6.25%[/latex]

Knowledge Check 3.2

1. P = $4,000, r  = 8%  = 0.08,  t = 210/365  years

[latex]FV=$4,000 \left[1 +0.08\left(\frac{210}{365}\right)\right][/latex]

2. P = $1,250,  r = 6.75% = 0.0675; [latex]t = (251 - 69)+365 =\frac{182}{365}[/latex] years (from Table 3- 1)

[latex]FV=$1,250\left[1 +0.0675\left(\frac{182}{365}\right)\right]=$1,292.07[/latex]

3. P = $2,500, r = 3.75% = 0.0375, t = 2 years

[latex]FV =$2,500 (1 +0.0375\times 2)= $2,687.50[/latex]

Knowledge Check 3.3

1. P = $2,000, FV = $2,210, t = 1.5 years

Either of the two following approaches is acceptable:

Approach A:

[latex]I = FV - P =$2,210 -$2,000 =$210[/latex]

So

[latex]r=\frac{I}{Pt}=\frac{$210}{$2,000\times1.5}=0.07=7%[/latex]

Approach B:

[latex]r = \frac{\frac{FV}{P}-1}{t} = \frac{\frac{$2,210}{$2,000}-1}{1.5}= 0.07[/latex]

2. FV = $1,871.25, r = 9% = 0.09, t = 33 months =33/12  years.

[latex]P=\frac{FV}{1+rt}=\frac{$1,871.25}{1+0.09\times \left(\frac{33}{12}\right)}=$1,500.00[/latex]

Knowledge Check 3.4

a. Find FV at 7%:

[latex]FV = P(1+rt) = $10,000 \left(1+0.07×\frac{6}{12}\right) = $10,350[/latex]

Conclusion: The value of $10,000 now, in six months’ time is $10,350. Since this is $25 greater than $10,325, you would prefer $10,000 now from a purely financial point of view.

b. Find FV at 6%:

[latex]FV = P(1+rt) = $10,000 \left(1+0.06×\frac{6}{12}\right) = $10,300[/latex]

Conclusion: The value of $10,000 now, in six months time is $10,300. Since this is less than $10,325, you would prefer $10,325 in six months time.

Knowledge Check 3.5

[latex]\begin{align*} &FV_1=$20,000\left(1+0.08\times\frac{15}{12}\right)=$22,000.00\\ &FV_2=$5,000\left(1+0.08\times\frac{10}{12}\right)=$5,333.33\\ &FV_3=$10,000\left(1+0.08\times \frac{6}{12}\right)=$10,400.00\\ \end{align*}[/latex]

Total Debt Outstanding = $37,733.33(6 months from now)

Knowledge Check 3.6

1.

[latex]PV=\frac{$5,000}{1+0.09\times \frac{19}{12}}=$4,376.37[/latex]

[latex]\begin{align*} &PV_1=\frac{$2,000}{1+0.07\times \frac{7}{12}}=$1,921.54\\ &PV_2=\frac{$4,000}{1+0.07\times \frac{13}{12}}=$3,718.05 \end{align*}[/latex]

Total Equivalent Debt Now = $5,639.59

Knowledge Check 3.8

Value of “old” payments at the focal date :

[latex]$3,500\left(1+0.08\times \frac{8}{12}\right)+$5,500\left(1+0.08\times \frac{1}{12}\right)[/latex]

[latex]=$3,686.67+$5,536.67=$9,223.33[/latex]

Value of “replacement” payments at the focal date:

[latex]$3,000\left(1+0.08\times \frac{6}{12}\right)+x=$3,120+x[/latex]

Therefore:

[latex]$3,120+x=$9,223.33[/latex]

[latex]x =$9,223.33 -$3,120.00=$6,103.33[/latex]

Knowledge Check 3.9

Let the amount of two equal payments be x. Value of “old” payments at the focal date:

[latex]$10,000\left(1+0.09\times \frac{4}{12}\right)+\frac{$9,000}{\left(1+0.09\times \frac{2}{12}\right)}[/latex]

Value of “replacement” payments at the focal date:

[latex]x+\frac{x}{\left(1+0.09\times \frac{6}{12}\right)}[/latex]

And we can set these to be equal:

[latex]x+\frac{x}{\left(1+0.09\times \frac{6}{12}\right)} = $10,000\left(1+0.09\times \frac{4}{12}\right)+\frac{$9,000}{\left(1+0.09\times \frac{2}{12}\right)}[/latex]

And solve, remembering to store all intermediate values in the calculator:

[latex]\begin{align*} x+\frac{x}{1.045} &=$10,000(1.03)+\frac{$9,000}{1.015}\\ x\left(1+\frac{1}{1.045}\right)&=$10,300+$8866.995074\\ x&=\frac{$19,166.995074}{1.956937799}=$9,794.38 \end{align*}[/latex]

Therefore: The size of the two equal payments is $9,794.38.