9.5 B14

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 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

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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

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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