11.3 B10

Imagine you are standing in front of a piece of equipment…something you’ve always wanted. Perhaps a cement mixer? Anyway, the sales guy gives you a choice. Let’s work through the choices:

Case A: PMT 150 PY 12 CY 12 N 1.5 * 12 = 18 IY 13.56 FV 0 CPT PV 2430.76

So you could buy the mixer for 2430.76 cash right now, or finance it.

You look dubious, so the sales guy says ‘hey wait!’. How about you pay nothing for a year and then pay $180 a month for 18 months.

Case B. There are two problems to solve here. First, the PV of the payments which you would have to start making in a year’s time. And then the PV of the money you need to have ready in a year’s time.

First, the PV of your financial obligations in a year’s time:

PMT 180 CPT PV 2916.91 (I’ve left off the other entries, they’re the same).

But you want to compare this 2916.91 with the other offer made for TODAY in Case B.

So now calculate the present value of 2916.91, which is just a straightforward PV problem, no longer an annuity:

FV 2916.91 N 1 * 12 = 12 CPT PV 2548.96.

So it looks like Case A would be best….

Extra Questions Ch 8 -10

These questions are worth trying. I have picked only odd-numbered ones. The solutions are in the book or in the solutions manual on reserve in the library.

8.2 A5, A7
8.3 A1, B3, C1
9.2 B3, B7, B13, B17, C1, C3, C7
9.3 B3, B7, B11, C5, C9
9.5 B1, B3, B5, B15
10.1 B1, B3, B7, B13
10.3 B1, B3, B7, B11, B13

Good luck!

9.5 B14

Link to Youtube
 Scheduled payments of $800 due two years ago and $1000 due in five years are to be replaced by two equal payments. The first replacement payment is due in four years and the second payment is due in eight years. Determine the size of the two replacement payments if interest is 12% compounded semi-annually and the focal date is four years from now.

Draw a timeline:

800 NOW X 1000 X
Two debts, 800 and 1000. The 800 is a FV problem..it will have increased by the focal date. The time is 2 + 4 = 6.

The 800
PY 2
PV 800
IY 12
N 2 * 6 = 12
CPT FV 1609.76

The 1000 is a PV problem, the time is 5 – 4 = 1
N = 1 * 2 = 2
FV 1000
CPT PV 890 So on the focal date your debts are 1609.76 + 890 = 2499.76


2499.76 = X + PV(X). How do we get the PV of an unknown quantity, X? Call X = $1. The time is 8 – 4 = 4
FV = 1
N = 4 * 2 = 8
CPT PV 0.63. Whatever X is, the PV of X will be 0.63 * X
So 2499.76 = X + 0.63 X
2499.76 = 1.63 X, so X = 2499.76/1.63 = 1533.6

Finding the PV of the second X by the formula method...some folks asked me about this, but note you do NOT need to know this.

First, we are using compound interest, so we need to divide the nominal annual interest rate by the number of payments per year. This gives us the periodic interest rate, in this example 0.12/2=0.06. So the PV of $1 will be 1/(1+0.06)^2*4 = 0.63.

Q7.5 B8

Q7.5 B8

1. Bring everything to the focal date, which is 92 days from today.

2. There are two debts which have to be settled, 4000 and 6000

3. This is a future value problem, because all the transactions are in the future.

4. Let’s calculate the debts first. On the focal date you will owe a total of:
4000(1+0.083*292/365) + 6000(1+0.083*155/365)=4265.6 + 6211.48 = 10477.08

5. Today, you credited your account with 5000. That amount will increase until the focal date to 5000(1+0.083*92/365)=5104.6.

6. What you owe on the focal date will be the difference between these: 10477.08-5104.6 = 5372.48

NOTE: it is also possible to calculate the future values of the 4000 and the 6000 as of today, subtract 5000, and then work out the future value of that balance. You’ll get exactly the same answer

Focal date worked example

Link to Youtube
You should have paid someone $500 78 days ago. Someone else should pay you $600 in 95 days. Interest is 10%. What is your net worth today?

X----------------------------------X-------X------------------------------X

500 Today New Focal 600

FV of 500 = 500(1+0.1*78/365)= 510.68

PV of 600 = 600/(1+0.1*95/365) = 584.78

TODAY (focal date) you owe 510.68, your assets are 584.78, so net worth is – 74.10

But we could pick any date. Let’s try ten days from today. This changes the number of days:

to 88 for the 500

and 85 for the 600

FV of 500 = 500(1+0.1*88/365) = 512.05

PV of 600 = 600/(1+0.1*85/365) = 586.35

Net worth is 512.05 – 586.35 = - 74.30

Quiz Solutions Chapter 7

Quiz on Chapter 7 worked solutions
1. I = Prt so I = 2500*0.08*265/365=141.58

2. t=I/Pr, so t = 5.7/(360*0.12) then multiply by 365 to get days, round up to 49

3. This is a present value problem because you pay less NOW instead of more LATER. First find the number of days which is 182. Then use the PV formula so P=1850/(1+0.132*182/365) = 1735.75

4. Two PV problems. We know how much you would get in the future, want to find out how much it is worth NOW: P = 800/(1+0.065*3/12) + 9500/(1+0.065*5/12) = 787.21 + 9249.49 = 10036.7


5. Draw a timeline. Then find the days between each event and the focal date. Here:

1 Mar and focal: 213
30 April and focal: 153
20 June and focal: 102
10 Aug and focal: 51

Now we want an ‘equation of equivalence’. Call each event ‘E’.

Borrow 1600 is E1
Pay back on 30 April is E2
Pay back on 20 June is E3
Pay back on 10 Aug is E4

What you borrowed has to exactly equal the three payments and the 500 final payment on the focal date.

So E1 = E2 + E3 + E4 + 500

Get the future value of each E

1600(1+0.07*213/365) = (1+0.07*153/365)X + (1+0.07*102/365)X + (1+0.07*51/365)X + 500
which works out to

1665.36 = 1.03X + 1.02X + 1.01X + 500. Take 500 from both sides

1165.36 = 3.06X. Divide through both sides by 3.06 to find X = 380.84

You make three equal payments of 380.84 and a final payment of 500

Example 5.3E

Link to Youtube
This question asks us to find one unknown, in this case the selling price. It is usual to put the unknown on the left side of the equation, which is what we did in class. The example in the book puts the equation the other way round....actually the meaning is the same. But I will stick with putting the unknown on the left.

Recall this important equation:

S = C + M

we know that C = 72, and we also know that M is 40% of C, or we can write M = 0.4 * C

now rewrite the important equation, substituting what we know:

S = 72 + 0.4 * C

now we know that 0.4 * C = 0.4 * 72 = 28.8

so S = 72 + 28.8 = 100.8

The second part of the example gives the markup of 40% based on selling price

Back to the important equation

S = C + M

now M = 0.4 * S

rewrite

S = C + 0.4 * S

subtract 0.4 S from both sides to give

0.6 S = C

we know C = 72 (given in the question)

so

0.6 S = 72

we can find S by dividing both sides by 0.6

S = 72/0.6 = 120

Question 5 Chapter 6 quiz

5. You run a newsstand with your friend Lurch. You buy magazines at the list price of $5 less 25% discount. Your fixed costs, which include a hearing aid for Lurch and special cream for your false teeth (you must smile at the customers!) come to $190 per week. You usually sell the magazines at the list price. Answer these questions independently:

a. If the desired profit is $140, how many magazines must you and Lurch sell each week?

The TR is 5X, TC is 5*0.75X, FC = 190 + 140 (the desired profit is a fixed cost)
5X = 3.75X + 330
1.25X = 330
X = 264

b. If the cost to purchase the magazines is 30% off the list price, and 200 magazines are sold, what is the lowest price they can charge for each magazine and still break even?
Now we are looking for a price, not a quantity of sales. Call the unknown price P.

200P = 5*0.7*200 + 190
200P = 700 + 190
200P = 890
P = 4.45

If you sold at $4.45 you would exactly break even.

Q 7.5 B10

When Ruby borrowed $2300, she agreed to repay the loan in two equal instalments, 90 days and 135 days from the date the money was borrowed. If interest is 9.25%, what is the size of the equal payments if a focal date of today is used?

The 2300 is a present value. You know how much she borrowed today. So we are not asking how much will 2300 increase to, but what amount paid in the future is equal to 2300 today.
Notice that the payments are equal. So if we call one payment ‘x’, then the other one is also x. This is a present value problem…..the only difference from the ones we’ve done in class being that we already know the present value.

2300 = x/(1 + 0.0925*90/365) + x(1 + 0.0925*135/365)
2300 = 0.98x + 0.97x
2300 = 1.95x
x = 1179.49

So Ruby can pay off her debt in two equal payments of $1179.49

Q7.5 B9

In this question you know the present value of two payments which you are going to have to make in the future. The total is 1100. You are going to have to make two equal payments, one in 4 months and another in 6 months. Interest is 8.5%. Two things: you know the present value of two payments which are in the future, so we use the present value formula P = S/(1 + rt). AND since the two payments are the same, we can call each payment 'x'.

So...

1100 = x/(1+ 0.085*4/12) + x/(1 + 0.085*6/12)
1100 = x/1.0283 + x/1.0425
1100 = 0.972 x + 0.959 x
1100 = 1.932 x
x = 1100/1.932 = 569.36

Note that my answer is slightly different from the book's answer just because of rounding.

Think through this problem carefully and make sure you 'get it'. Working out whether an interest problem is 'future value' or 'present value' isn't easy. So work at it. Example 7.5 H covers the same ground as the question above.

When there's no selling price

Sometimes the selling price isn’t provided. In these cases, assume that the selling price is $1 per unit. So in the case of the question below from the quiz:
(Chapter 6 quiz Q4) Monk Foods has compiled these estimates for operations:
Sales 865,000
Fixed costs 252,100
Total variable cost 597,250
The margin is 865000 – 597250 = 267750. Now we calculate the margin per dollar of sales. Divide the margin by the total sales: 267750/865000 = 0.31. That means for every dollar in sales, we get to keep 31 cents to use to pay off the fixed costs. So how many units do we need to sell at $1 each to pay off the fixed costs of 252,100?
252,100/0.31 = 813,225.8. Now, in units we would want to round UP to 813,226. For amount of sales required to break even, leave the amount at 813,225.8.

See Example 6.18 on page 234 of the textbook.

Partial payments

Sometimes we can’t settle a complete invoice, but we can pay at least some of it and can get a discount for what we do pay. In the Quiz on Chapter 5, there is a question on this. Basically, how much do you need to pay to reduce an invoice of 9600 to 7000. So, we want the creditor (the company you owe money to) to record a payment of 2600. But your partial payment occurs within the discount period. So you don’t have to pay the full 2600. You pay less. The 2600 is the ‘amount given credit for’. The discount is 5%. So we take off 5% from what we want to have appear in the creditor company accounts. That’s 0.95*2600 = 2470. So you pay the creditor 2470 but they record your payment as being 2600. The textbook (8th edition) has a good example on page 193.

Q5.4 B8

Here is the solution to Q5.4 B8.

First, calculate the full cost, meaning what the folks would have paid if they didn't have the discount. That is 225 + 2 x 12 + .75 x 2 x 20 = 279

Now, the package was sold at the discounted price of 199. So the markdown was 279 - 199 = 80.

The rate of markdown was 80/279 = 28.67%

Points to note: the rate of markdown is calculated based on the list price.

The Nalini word problem

Here is the question: Nalini invested a total of $24,000 in two mutual funds. Her investment in the Equity Fund is $4000 less than three times her investment in the Bond Fund. How much did Nalini invest in the Equity Fund?

Solution: first, assign a letter to each of the unknown quantities. We have two unknown quantities, let's call the Equity Fund 'E' and the Bond Fund 'B'. Now, because we have two unknown quantities, we will need two equation to find them.

First off, her investment is E + B = 24000. That is from the first sentence.

Now,  the tougher second sentence. Recall that the point of an equation is that both sides must be equal. So, we can write

E = 3B - 4000

Go  back to our first equation. Because an equation is...well....an equation, we can substitute
the rearranged second equation for E into the first equation to get

3B - 4000 + B = 24000 ( make sure you get this bit)

so 4B = 28000
so B = 7000

We can get E like this

E + 7000 = 24000, so E = 17000.

Check your work. Go back to the sentence "Her investment in the Equity Fund is $4000 less than three times her investment in the Bond Fund". Three times the Bond Fund would be 3 * 7 = 21000. Take away 4000 to get 21000 - 4000 = 17000. We're done!