\(\color{brown}{\text{Financial Mathemactis 1}}\)
\(\color{brown}{\text{Group Members}}\)
| Student Name | ID |
|---|---|
| Nguyen Thi Tuong Vy | MAMAIU17034 |
| Doan Ha Anh Thu | MAMAIU17013 |
| Tran Thanh Dat | MAMAIU17036 |
| Bui To Mai | MAMAIU17022 |
\(\color{brown}{\text{Homework}}\)
\(\color{gray}{\text{Homework 1}}\)
Slide 20
Suppose you commit 200,000,000(VND) to invest a project that give you a fixed rate of return 15% per year for the next 10 years. If interest obtained from each year is also reinvested to the project, how many years needed for the accumulated money become triple the original money?
The number of years needed for the accumulated money become triple the original money: \[D\times(1+r)^n = 3 \times D \rightarrow n=\frac{ln(3)}{ln(1+r)}=\frac{ln(3)}{ln(1.15)}\approx 7.86\] Answer: 8 years
Slide 21
Suppose you commit 500,000,000(VND) to invest a project that give you a fixed rate of return -25% per year for the next 10 years. If interest obtained from each year is also reinvested to the project, how many years needed for the accumulated money become half the original money?
The number of years needed for the accumulated money become half the original money: \[D\times(1+r)^n=\frac{1}{2} \times D \rightarrow n=\frac{ln(0.5)}{ln(1+r)}=\frac{ln(0.5)}{ln(0.75)}\approx 2.41\] Answer: 3 years
Slide 22
Fred Derf found his lost passbook for a saving account that he had opened with a $100 deposit 12 years ago. If the bank paid interest at a rate of 5% compounded annually over this period, what should be the balance in the account today?
The balance in the account today should be: \[P \times (1+r)^n= 100 \times (1+5\%)^{12} \approx 179.59\] Answer: $179.59
Slide 30
Suppose a bank offer a loan for 2 years with fixed interest rate 9% per year compounded daily. Assume that a year has 365 days and 12 months, what is the equivalent interest rate (per month) compounded monthly for the loan.
The equivalent interest rate (per month) compounded monthly for the loan: \[(1+\frac{9\%}{365})^{365}=(1+r)^{12} \rightarrow r =^{12}\sqrt{(1+\frac{9\%}{365})^{365}}-1 \approx 0.00753 \] Answer: 0.75%
Slide 32
In two years time, I wish to own 100,000,000 VND. I can invest money at 7% per year with interest compounded quarterly. What amount must I invest today to ensure that I have 100,000,000 VND in two years time?
The amount that you must invest today is: \[P \times (1+\frac{7\%}{4})^8 = 100,000,000 \rightarrow P=\frac{100,000,000}{(1+\frac{7\%}{4})^8} = 87,041,157.31\] Answer: $87,041,157.31
Question 1
Joe invests £2000 at 3.9% per annum with interest compounded twice yearly. What is the equivalent with the interest compounded annually?
The equivalent with the interest compounded annually: \[\left( 1+\frac{3.9\%}{2} \right)^2 = (1+r)^1 \rightarrow r= \left( 1+\frac{3.9\%}{2}\right)^2-1 = 0.03938\] Answer: 3.94%
Question 2
Interest is charged at 3.67% per year, compounded monthly. What is the equivalent annually compounded rate?
The equivalent with the interest compounded annually: \[\left(1+\frac{3.67\%}{12}\right)^{12} = (1+r)^1 \rightarrow r=\left(1+\frac{3.67\%}{12}\right)^{12}-1 = 0.03732\] Answer: 3.73%
Question 3
Sarah can borrow £20,000 and pay interest at 6% per year (compounded annually). What is the equivalent rate when interest is compounded quarterly?
The equivalent with the interest compounded quarterly: \[\left(1+\frac{r}{4}\right)^4 = (1+6\%)^1 \rightarrow r=((1+6\%)^\frac{1}{4} -1)\times4 = 0.05869\] Answer: 5.87%
Question 4
An investment company offers investors a rate oof 4.75% per year compounded quarterly. What would be the equivalent rate with interest compounded twice yearly?
The equivalent rate with interest compounded twice yearly: \[\left(1+\frac{r}{2}\right)^2 = \left(1+\frac{4.75\%}{4}\right)^4 \rightarrow r=\left(\left(\left(1+\frac{4.75\%}{4}\right)^4\right)^\frac{1}{2}-1\right)\times2 = 0.04778\] Answer: 4.78%
Question 5
If interest is paid at 5.2% per year compounded annually, what will be the equivalent continuously compounded rate?
The equivalent continuously compounded rate: \[(1+5.2\%)^1 = e^{r} \rightarrow r=ln(1+5.2\%)= 0.05069\] Answer: 5.07%
Question 6
When £5,000 is invested for six months, the interst is £200.
- What is the (annually compounded) rate of interest?
- What whould be the equivalent continuously compounded rate?
The (annually compounded) rate of interest: \[(1+r)^\frac{1}{2} = 1+\frac{200}{5000} \rightarrow r= \left( {1+\frac{200}{5000}} \right)^2-1= 0.0816\] The equivalent continuously compounded rate: \[(1+8.16\%)^1 = e^{r} \rightarrow r= ln(1+8.16\%) = 0.07844\] Answer:
- 8.16%
- 7.84%
Question 7
Which of the following two annual rates would be more attractive to an investor 6.4% compounded daily or 6.395% compounded continuously?
\[1.06608=\left(1+\frac{6.4\%}{365}\right)^{365}>e^{6.395\%}=1.06604\] Answer: 6.4% compounded daily would be more attractive to an investor
Question 8
Amy McPhee wishes to invest £5,000,000 for one month. Which interest rate should she choose?
- AAABank offering 6.13% per year, simply compounded.
- FriendlyBank offering 6.3% per year compounded annually.
- InvestandGrow offering 6.2% per year compounded semi-annually.
- MoneyValue offering 6.11% per year compounded continuously.
| Company | Interest rate per year | Compounded | Calculation |
|---|---|---|---|
| AAABank | 6.13% | simply | \[r_{1}=\frac{6.13\%}{12}=0.005108\] |
| FriendlyBank | 6.3% | annually | \[(1+r_{2})^{12} = 1+6.3\%\] \[\rightarrow r_{2}= \left( {1+6.3\%} \right)^{\frac{1}{12}}-1= 0.005104\] |
| InvestandGrow | 6.2% | semi-annually | \[(1+r_{3})^{12} = \left(1+\frac{6.3\%}{2} \right)^2 \] \[\rightarrow r_{3}=\left(\left(1+\frac{6.3\%}{2} \right)^2 \right)^\frac{1}{12}-1=0.005101\] |
| MoneyValue | 6.11% | continuously | \[(1+r_{4})^{12} = e^{6.11\%} \] \[\rightarrow r_{4}=\left(e^{6.11\%}\right)^{\frac{1}{12}}-1=0.005104\] |
Answer: Amy McPhee should choose AAABank
Question 9
I am offered interest rates of:
- 5.5% per annum compounded quarterly
- 5.49% per year compoounded monthly
- 5.6% per year commpounded semi-annually
- 5.48% per year compounded continuously
Which rate should I choose if I plan to (i) invest money, (ii) borrow money?
| Interest rate per year | Calculation |
|---|---|
| 5.5% per annum compounded quarterly | \[r_{1}=\left(1+\frac{5.5\%}{4}\right)^4-1=0.056144\] |
| 5.49% per year compoounded monthly | \[r_{2}=\left(1+\frac{5.49\%}{12}\right)^{12}-1=0.056302\] |
| 5.6% per year commpounded semi-annually | \[r_{3}=\left(1+\frac{5.6\%}{2}\right)^2-1=0.056784\] |
| 5.48% per year compounded continuously | \[r_{4}=e^{5.48\%}-1=0.056329\] |
Answer:
- To invest money, you should choose 5.6% per year commpounded semi-annually
- To borrow money, you should choose 5.5% per annum compounded quarterly
Question 10
Alan owes £5,000 on his credit card. At the end of the first week, the company charges interest of £20.19. If the company is charging a compounding rate and Alan did not pay off any part of the debt in the meantime, how much did the company charge at the end of the fourth week?
The amount that the company charge at the end of the fourth week is: \[R=5,000\left(1+\frac{20.19}{5,000}\right)^4-5,000=81.2505\] Answer: £81.25
Question 11
Oleg owes £100,000. The interest charged is 15% (per year) compounded daily. How many days before Oleg’s debt is more than £1,000,000?
(Assume no repayments are made until the £1,000,000 has been reached)
Days till Obleg’s debt becomes more than £1,000,000 is: \[100,000\times\left(1+\frac{15\%}{365}\right)^t=100,0000 \rightarrow t=\frac{ln(10)}{ln\left(1+\frac{15\%}{365}\right)}=5,604.108\] Answer: 5604 days
Question 12
The interest rate today is 6.5% per year (annually compounded). What is the value today of:
- £5,000 to be received in two years’ time?
- £10,000 to be received in six months’ time?
- £10,000,000 to be received in five years’ time?
| Future value | Duration | Present value |
|---|---|---|
| £5,000 | two years’ time | \[{PV}_{i}=\frac{5,000}{\left(1+6.5\%\right)^2}=4,408.29641\] |
| £10,000 | six months’ time | \[{PV}_{ii}=\frac{10,000}{\left(1+6.5\%\right)^{\frac{1}{2}}}=9,690.03166\] |
| £10,000,000 | five years’ time | \[{PV}_{iii}=\frac{10,000,000}{\left(1+6.5\%\right)^5}=7,298,808.365\] |
Answer:
- £4,408.30
- £9,690.03
- £7,298,808.37
Question 13
PJ Furnishing has to pay $100,000 in two years’ time. The interest rate today (continuously compounded) is 5.5%. How much should the company set aside today?
The company should set aside today: \[PV=\frac{100,000}{e^{5.5\%\times 2}}=89,583.41353\] Answer: $89,583.41
Question 14
Similla is to receive £20,000 in seven years’ time. The interest rate is 6.5%, compounded semi-annually. What is the value today of this legacy?
The value today of this legacy is: \[PV=\frac{20,000}{\left(1+\frac{6.5\%}{2}\right)^{7\times2}}=12,781.12702\] Answer: £12,781.13
Question 15
Andrew will be paid £8,500 in two years’ time. What is the value of this amount today if the interest rate (compounded quarterly) is 6.8% per year?
The value of this amount today is: \[PV=\frac{8,500}{\left(1+\frac{6.8\%}{4}\right)^{2\times4}}=7,427.64780\] Answer: £7,427.65
Question 16
FirstInvestors wants to invest a sum of money today to ensure it will have £100,000 in two years’ time. Several interest rate are available:
Interest rate (per year) Compounded 6.88 Annually 6.75 Semi- annually 6.68 Monthly 6.65 Continuously Which interest rate should it choose?
| Interest rate per year | Compounded | Present Value |
|---|---|---|
| 6.88 | Annually | \[PV=\frac{100,000}{\left(1+\frac{6.88\%}{1}\right)^{2\times1}}=87,540.11438\] |
| 6.75 | Semi- annually | \[PV=\frac{100,000}{\left(1+\frac{6.75\%}{2}\right)^{2\times2}}=87,566.48352\] |
| 6.68 | Monthly | \[PV=\frac{100,000}{\left(1+\frac{6.68\%}{12}\right)^{2\times12}}=87,526.41776\] |
| 6.65 | Continuously | \[PV=\frac{100000}{e^{6.65\%\times2}}=87546.50921\] |
Answer: FirstInvestors should choose 6.68% compounded monthly
Question 17
An investment company will receive $15,000 in one year, $17,000 in two years, $21,000 in three years, $5,000 in four years and $3,000 in five years. The interest rates with these maturities are as shown in the table:
Maturity (years) Interest rates (per year) compounded annually 1 6.6 2 6.8 3 6.95 4 7.1 5 7.2 What is the value today of these future payments?
The value today of these future payments is: \[PV=\frac{15,000}{\left(1+\frac{6.6\%}{1}\right)^1}+\frac{17,000}{\left(1+\frac{6.8\%}{2}\right)^2}+\frac{21,000}{\left(1+\frac{6.95\%}{3}\right)^3}+\frac{5,000}{\left(1+\frac{7.1\%}{4}\right)^4}+\frac{3,000}{\left(1+\frac{7.2\%}{5}\right)^5} \\ =14,071+14,904+17,166+3,800+2,119=52,060\] Answer: $52,060
Question 18
Mary invests £9,956 today and in three months’ time she will have £10,000. What semi-annually compounded interst rate has she used?
She has used semi-annually compounded interst rate at: \[\left(1+\frac{r}{2}\right)^{2\times\frac{1}{4}}=\frac{10,000}{9,956} \rightarrow r=\left(\left(\frac{10,000}{9,956}\right)^2-1\right)\times2=0.01772 \] Answer: 1.77%
Question 19
A bank will receive $10,000 in two years’ time and $20,000 in four years’ time. The interest rate with a two-year maturity is 5.8% per year with quarterly compounding. The bank would like to borrow $24,000 today and use the money it is to receive in two years and in four years to pay off the loan. What interest rate (compounded quarterly) with a maturity of four years will it need?
It will need interest rate (compounded quarterly) with a maturity of four years at: \[24,000=\frac{10,000}{\left(1+\frac{5.8\%}{4}\right)^{2\times4}}+\frac{20,000}{\left(1+\frac{r}{4}\right)^{4\times4}} \rightarrow r=0.07109\] Answer: 7.11%
Question 20
A share in XAY company costs £7.38 on the London Stock Exchange. This share is selling for $13.14 on the New York Stock Exchange. The exchange rate is £1 = $1.775. What should I do? What assumptions have you made to calculate your answer?
\[ 7.38 \times 1.775 = 13.0995 < 13.14\] Answer: You should buy in London Stock Exchange
\(\color{gray}{\text{Homework 2}}\)
Slide 50
A star-up forecasts that, with an initial investment of $1000, it will generate net cash flow of $1320 1 year from now, $1452 2 years from now. These net cash flows are not required to reinvest in the star-up. The mean annual growth rate of other similar investment is 10%. What are the net present value and the internal rate of return? Should an investor takese part in this star-up?
Net present value: \[NPV=-1000+\frac{1320}{\left(1+10\%\right)}+\frac{1452}{\left(1+10\%\right)^2}=1400\] Internal rate of return: \[1000=\frac{1320}{\left(1+r\right)}+\frac{1452}{\left(1+r\right)^2} \rightarrow r=1.039 \] Answer: NPV=$1400, IRR=1.039, investors should take these part in this start-up
Slide 57
Consider two investment proposals A and B competing for funding. The expected cash flow streams produced by A and B are as follows:
Year 0 1 A -5000 7500 B -50000 62500 The mean annual growth rate of other similar investments is 15%. According to the net present value criterion, which proposal is better to invest?
Net present value A: \[NPV_A=-5000+\frac{7500}{\left(1+15\%\right)}=1522\] Net present value B: \[NPV_B=-50000+\frac{62500}{\left(1+15\%\right)}=4348\] Answer: \(NPV_A\)=1522, \(NPV_B\)=4348