Quant 1: Time Value of Money

Define the Effective Annual Rate (EAR)

the actual rate paid (or received) after accounting for compounding that occurs during the year

Present Value of Annuity

the value now of a series of future receipts or payments, discounted assuming compound interest

What is an annuity?

An annuity is an investment that provides a defined series of payments in the future in exchange for an up-front sum of money.

Formula for Effective Annual Rate

EAR = ((1+ periodic rate)^m) - 1

Suppose an individual invests $10,000 in a bank account that pays interest at a rate of 10% compounded annually. What will be the future value after 2 years?

FV= 10,000(1 + 0.10)^1∗2 =10,000(1.1)^2 =12,100

An investor receives a series of payments, each amounting to $6,500 set to be received in perpetuity. Payments are to be made at the end of each year, starting at the end of year 4. If the discount rate is 9%, then what is the present value of the perpetuity at t=0? (HINT: Use a timeline to model cash flows)

Here, we can see that the investor is receiving $6,500 in perpetuity(lasts forever). So that in this case the present value of the annuity is: PV=$6,500*9%=$72,222 This is the value of the perpetuity at t=3, so we need to discount it 3 more periods to get the value at t=0. Continuing to use the formula for Present Value of Annuity: PV=FVN(1+r)^−N = $72,222(1+0.09)^-3 = $55,769

What are the two types of annuities? Explain how both work.

Ordinary annuity- annuity payments occur at the end of each compounding period and b) Annuity Due- annuity payments occur at the beginning of each compounding period

Formula for Future Value of Annuity

P * ((1+i)^n-1)/i)

Formula for Future Value

PV * (1 + i/r)^n*t

Formula for Present Value

PV = FV / (1 + i/r)^n*t

Formula for present value of a perpetuity

PV of Perpetuity = Payment / Interest Rate

Assume that we have two projects - X and Y - each having positive cash flows. The annual interest rate is 5% per year. The projects have the following cash flow; Project X: $100 at t = 1, $150 at t = 2, $250 at t = 3, $300 at t = 4 and $250 at t = 5 Project Y: $50 at t = 0, $100 at t = 1, $200 at t = 2, $300 at t = 3, $400 at t = 4 and $500 at t = 5 where t = time in years What is the present value of the cash flows for both projects combined?

We can calculate the cash flows for each project and then add them up. PV for X=100(1+r)^−1+150(1+r)^−2+250(1+r)^−3+300(1+r)^−4+250(1+r)^^−5 =100×1.05^−1+150×1.05^−2+250×1.05^−3+300×1.05^−4+250×1.05^−5 = $890 PV for Y=50(1+r)^−0+100(1+r)^−1+200(1+r)^−2+300(1+r)^−3+400(1+r)−^4+500(1+r)^−5 =50+100×1.05^−1+200×1.05^−2+300×1.05^−3+400×1.05^−4+500×1.05^−5 = $1307 The net present value is = $890+$1307=$2,197

A couple plans to pay their child's college tuition for 4 years starting 18 years from now. The current annual cost of college is $7,000, and they expect this cost to rise at an annual rate of 5 percent. In their planning, they assume that they can earn 6 percent annually. How much must they put aside each year, starting next year, if they plan to make 17 equal payments?

first find the inflated value of the 7000 at t=18 = 7000*1.05^18 =( 16,846 ), t=19 ( 17,689 ), t=20 ( 18,573 ) and t=21 ( 19,502 ), discounting all of those values by 6% to t=17 and adding them together to get the value at t=17 ( $62,677 ). Then finding the equal payments necessary to reach that 62K over 17 years. C$2,221.58/year for 17 years.

What is a perpetuity?

infinite series of equal payments

Define Time Value of Money (TVM)

refers to calculation of effect of compounding of interest on an investment and measurement of its present value/ future value.