Q4 from Take Home 1

4.      Joan borrowed $15000 to buy a car.  She repaid $2000 two months later and $5000 seven months later.  After twelve months, she borrowed an additional $4000, and repaid $3000 after 16 months.  She paid the entire balance, including the interest, after 24 months.  Interest was 7% compounded monthly for the first year and 7.5% compounded monthly for the remaining time.  What was the size of the final payment?

How to approach this problem:

1. Draw a timeline with the dates and amounts, just as we do in class.

2. We solve this by 'rolling the money along'. We cannot bring all the amounts to the end date because the interest rate changes.

3. So to begin with, the PV is 15000. Find the FV after two months. At that point, you repay 2000. So subtract 2000 from the FV you have just found. That is what Joan owes after two months. This amount is now the PV for the next stage. So keep going with the same method. Be very careful to note when the interest rate changes.

Chapter 9 Quiz Q4

This question asks: what are the proceeds of a $5000 eight-year note bearing interest at 8%, compounded quarterly, discounted three and a half years after the date of issue at 6% compounded monthly.

Attack: first find the maturity value after eight years. Then discount.

Step 1: maturity value

The PV is 5000
The PY/CY is 4 because the note is compounded quarterly
The N is 8 * 4 = 32
The IY is 8
CPT FV = 9422.7

Step 2: the note was sold off 3.5 years after issue. That means 4.5 years BEFORE the maturity date (because 8 - 3.5 = 4.5). The 4.5 is the date on which we want the present value. Note carefully that we now use the interest rate for the new terms.

FV 9422.7
The PY/CY is 12 because the new interest rate is compounded monthly
The IY is 6
The N is 4.5 * 12 = 54
CPT PV 7197.94

That is the answer: $7197.94

Finding cost when only the margin is given

Sometimes the information given is limited, as in this question below. Try to make equations using the information in the question. 

A store sells kettles at a markup of 18% of the selling price. The store's margin on a particular model is $6.57.

a) For how much does the store sell the kettles?

We know that 0.18S = 6.57, where ‘margin’ means contribution to fixed costs.

So: S = 6.57/0.18 = 36.5

b) What was the cost of the kettles to the store?

S = C + 0.18S and so 36.5 = C + 6.57. Therefore C = 36.5 – 6.57 = 29.93

c) What is the rate of markup based on cost?

6.57/29.93 = 22% (rounded)

Breakeven when the unit price isn't given

Calculating breakeven when the unit price is given is not difficult. It does seem harder when there is no unit price. In this case, assume that the unit selling price is $1, as in these two example below from the Chapter 6 quiz:

2. The Frogface Bookstore has $105,000 of sales, variable costs of $39,550 and fixed costs of $32,350. What would their sales have to be to break even?

Guided solution: no unit price is given in the question, so assume that it is $1. The contribution of their sales is then 105,000 – 39,550 = 65450. As a share of their total sales, this is 65450/105,000 = 0.62. This means that for every dollar in sales that they get, 62 cents ‘contributes’ towards paying off their fixed costs. To find how many sales of $1 they need, divide the fixed cost by the contribution: 32350/0.62 = 52178 (rounded up).

4. Monk Foods has compiled these estimates for operations:

Sales                                                                          865,000

            Fixed costs                                                    252,100

            Total Variable Cost                                      597,250

a. What's the contribution rate?

(865,000 – 597,250)/865,000 = 267750/865000=0.31

b. Break even point in sales dollars?

Again, assume that the unit selling price is $1. Then how many sales at $1 each do they need to get the fixed costs of $252,1000? It is $252,100/0.31 = $813226.

c. Break even point in units?

Because we specified that the unit selling price would be one dollar, then the number of units we need to sell is the same as the sales dollar amount, or 813,226.

Youtubes

Here are links to some Youtubes I have prepared for you. Now, these are done by me at home with very limited equipment ($20 microphone from London Drugs for example!) so please forgive the quality. We do what we can:

Chapter 7: calculating days, interest etc

Chapter 7: Present Value and Future Value

Chapter 9 Questions answered

Chapter 9, Q9.2B: a loan of $5000 with interest at 7.75% compounded semi-annually is repaid after five years and ten months. What is the amount of interest paid?

Attack: find the FV. Subtract the PV. That is the amount of interest.
PV 5000
IV 7.75
PY/CY 2
N 5.83 * 2 = 11.66
CPT FV 7789.13
Interest is 7789.13 – 5000 = 2789.13

Chapter 9, Q5. scheduled payments of $400 due today and $700 with interest at 4.5% compounded monthly in eight months are to be settled by a payment of $500 six months from now and a final payment in fifteen months. Determine the size of the final payment if money is worth 6% compounded monthly.
Attack: call the 400 E1, 700 E2, 500 E3. Use the final payment date as the focal date. There are 3 FV problems.
E1: PV 400 IY 6 PY/CY 12 N (15/12) * 12 = 15 CPT FV 431.07
E2: First, find the amount that the 700 grows to after 8 months. PV 700 IY 4.5 N (8/12) * 12 = 8 CPT FV 721.28…but not done yet. Find the FV of the 721.28 at the focal date. Note that the focal date is 15 – 8 = 7 months later…here is the second part: PV 721.28 IY 6 CY/PY 12 N (7/12) * 12 = 7 CPT FV 746.91
E3: PV 500 IY 6 N (now the time between six months and 15 months is 9)…(9/12)*12 = 9 CPT FY = 522.96

The final payment is E1 + E2 = E3 + X, so X = 655.02

Chapter 9, 9.5 B 14 page 380: 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.

Attack: first find the values of the 800 and the 1000 at the focal date, which is 4 years from today. Then calculate the two payments.

E1 , the 800: the time is 2 + 4 = 6 years. PV 800 PY/CY 2 N 6 * 2 = 12 IY 12 CPT FV 1609.76
E2, the 1000: this is in five years time, meaning one year AFTER the focal date. So this will be a PV problem: FV 1000 N 1 * 2 = 2 IY 12 CPT PV 890

So the debt on the focal date is 890 + 1609.76 = 2499.76

Call each of the payments X1 and X2. They are equal. So…and here is the key point,
2499.76 = X1 + PV(X2). On the focal date, you pay X1, which has no interest because it is on the focal date. Now, find the present value of $1, which will represent X2.
FV 1 N (the second payment is 4 years after the focal date) 4 * 2 = 8 IY 12 PY/CY 2 CPT PV 0.63
So….2499.76 = X + 0.63X

2499.76 = 1.63X, so X = 1533.6. You make two equal payments of $1533.6

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….