Instructions Valuation The very basic of Financial Mathematics Financial Mathematic Lecture 1 Luděk Benada, Dagmar Linnertova Department of Finance - 402, benada.esf@gmail.com October 13, 2017 The Study materials prepared by Mikhail Dmitrievich Balyka Luděk Benada BPF_AFMT Simple and compound interest Luděk Benada BPF_AFMT Interest is the cost of borrowing money Depending on how we calculate it, can be defined as simple interest or compound interest Example: Suppose you deposite 1000$ into a bank at 10% per annum (the interest is calculated annually). How much do you have 3 years later using: a)Simple interest b)Compound interest Simple and compound interest Luděk Benada BPF_AFMT a)Simple interest is calculated on the principal (amount you deposit). Since at the end of the 1st year we have: 1000 (our principal) + 1000*10% (interest)=1100 1000+1000*10%+1000*10%=1200 – 2nd year 1000+1000*10%+1000*10%+1000*10%=1300 – 3rd year Or, using formula: PV, FV – present and future value r – interest rate t – time in years Derivation of the formula: as it was mentioned, to find the amount of money in an account, one should add an interest to a principal. Hence, we have: PV+Interest=PV+PVr=PV(1+r) -1^st year PV+PVr+PVr=PV(1+2r) – 2^nd year PV+PVr+PVr+PVr=PV(1+3r) – 3^rd year So, at the and of t-th year we have PV(1+tr) Simple and compound interest Luděk Benada BPF_AFMT b)Compound interest is calculated on the principal plus the accumulated interest of previous periods: 1000+1000*10%=1100 – 1st year 1100+1100*10%=1210 – 2nd year 1210+1210*10%=1331 – 3rd year Or, using formula: n – number of compounding periods Derivation of the formula: FV=PV+PVr=PV(1+r) – 1^st year FV=PV(1+r)+rPV(1+r)=PV(1+r)[1+r]=PV(1+r)^2 – 2^nd year FV=PV(1+r)^2+rPV(1+r)^2=PV(1+r)^2[1+r]=PV(1+r)^3 – 3^rd year So, at the end of t-th year we have FV=PV(1+r)^n Simple and compound interest Luděk Benada BPF_AFMT Money invested at compound interest grows faster than money left to grow at simple interest If we have non-integer number of interest periods, then we can maximise our investment using a compound interest for an integer number of interest periods and simple interest for the rest. Simple and compound interest Luděk Benada BPF_AFMT Combining simple and compound interest example Let’s assume that we deposit 15000 at 7% p.a. into a bank which calculates interest 3 times a year (4 months). Also we know that we will have 20000 in a given time. So, what is the time, assuming we maximising our investment? FV=PV*(1+r)n 20000=15000*(1+0,07/3)n 4/3=1,0233n t=ln(4/3)/ln1,0233=12,4725. (1) Hence, we have 12 full interest periods, then we can write down the equation that combines two types of interest: 20000=15000(1+0,07/3)12*(1+0,07/3*t) t=0,4696 (2) Note, that result 2 (0,4696) is less than the decimal part of result 1 (0,4725). It shows us that usage of simple interest is more effective than compounding in case of non-integer IP. Answer: 12 full interest periods (i.e. 12/3=4years)+0,4696 IP (i.e 0,4696*120days=56days) If we have non-integer number of interest periods, then we can maximise our investment using a compound interest for an integer number of interest periods and simple interest for the rest. Simple and compound interest Luděk Benada BPF_AFMT Effective rate It’s anually converted rate that gives the same interest earnings as the rate r(m) converted m times per year, where m≠1 Assume that the nominal compounded rate r(m) yields the same future value for 1$ invested for one year as the annually compound rate re 1+re=[1+r(m)/m]m Simple and compound interest Luděk Benada BPF_AFMT Effective rate Find the annual effective rates for 8% compounded: a)quarterly; re=(1+0,08/4)4-1=8,24% b)monthly; re=(1+0,08/12)12-1=8,3% c)daily; re=(1+0,08/365)365-1=8,33% Simple and compound interest Luděk Benada BPF_AFMT Continuous compounding What if we let the value of m in this formula become infinitely large? This means that interest compounds more often than every second; in fact we say it’s compounded continuously. Suppose you invest 1$ at 100% p.a. Let’s calculate the future value of this investment when interest compounds: a) yearly b) quarterly c)monthly d)daily Simple and compound interest Luděk Benada BPF_AFMT e number is equal to 2,718281828 there is a tip how to memorise it: it is just 2,7 and then two times we have the birth year of Leo Tolstoy, which is 1828 probably, it is the best way to memorise the writer’s year of birth, but nevertheless, it works Simple and compound interest Luděk Benada BPF_AFMT How we derive FV=PV*e^ft: 1) FV=PV(1+r)^n - future value formula 2)FV=PV(1+e^r -1)^t - replace r by the effective rate (e^r – 1) and n by the number of years t 3)FV=PV*e^rt - simplify Simple and compound interest Luděk Benada BPF_AFMT Continuous compounding Example: 1) Find annual effective rate for 8% compounded continuously re=e0,8-1=8,3287% 2) Find the nominal rate r compounded continuously that will produce an effective rate of 8% re=ef-1 , that is f=ln(1+re) f=ln(1,08)=7,696% 3) Find the future value of 4000 invested for 42 months (3,5 years) at 8% compounded continuously FV=PV*eft=4000*e0,08*3,5=5292,52 Simple and compound interest Luděk Benada BPF_AFMT Discount interest rate Let’s consider the concept of discount rate on the following example: If you borrow 500$ for a year at a 10% discount rate, the banker would give you 450$ and expect you to pay back 500$ at the end of the year (i.e. Interest collecting is up front). If it were a simple interest, you would get the entire 500$ but pay back 550$. Hence, with simple interest 500$ - PV, but with discount interest 500$ - FV Simple and compound interest Luděk Benada BPF_AFMT Discount interest rate As we know, FV=PV(1+rt) or PV=FV/(1+rt), so we can express d in terms of r : Simple and compound interest Luděk Benada BPF_AFMT Compound interest and taxation An interest earned on an investment is a taxable income. And we need to deduct this tax amount from the interest. The way we calculate it depends on a tax period (TP) and interest period (IP). There are three possible situations: a)IP=TP b)TP>IP c)TP – just once, at the end of your investment Let’s calculate it using the following example: We deposit 1000$ into a bank at 5% p.a. For 10 years. Tax rate is 10%. Calculate the future value after tax FVtax. Simple and compound interest Luděk Benada BPF_AFMT Compound interest and taxation a)IP=TP=1 year FV=PV(1+r)n to calculate the tax we need to understand that every year our interest is reduced by the amount of tax or in other words our interest rate reduced by the tax rate i.e it is r*(1-tax) Hence, after the 1st year we have FVtax=PV[1+r*(1-t)] or after n periods: FVtax=1000[1+0,05*(1-0,1)]10=1552,97 Simple and compound interest Luděk Benada BPF_AFMT Compound interest and taxation b) IP=3 months TP=1 year TP>IP FV=PV(1+r)n to calculate the tax we need to understand that before we deduct the tax amount, we obtain some interest for several IP in one year: FV=PV(1+r/m)m by subtracting “1” from FV we obtain interest which we can use to calculate the interest after tax: PV[(1+r/m)m -1]*(1-tax). Now we can move back “1” to get FV and raise it to the power of years: Simple and compound interest Luděk Benada BPF_AFMT Compound interest and taxation FVtax=1000[((1+0,05/4)4-1)*0,9+1]10=1565,66 Simple and compound interest Luděk Benada BPF_AFMT Compound interest and taxation c) TP=once per 10 years IP=1 year FV=PV(1+r)n to calculate the tax we need to find the interest obtained during these 10 years I=PV(1+r)n-1 then we can define the interest after tax multiplying it by (1-tax) and finally add back “1” and obtaining the future value after tax: FVtax=1000[(1+0,05)10-1]*0,9+1=1566,01