CHAPTER 1: The time value of money
We are going to link the present and the future by using the notion of interest rate that could be called discount rate, required rate of return or cost of capital. Finance is all about cash flows but more precisely about the exact date of the realization of the cash flow.
I) PRESENT VALUE
What is the value today of $110 to be received in one year?
- suppose the interest rate , r =10%
- if you had the money today you could :
o put in the bank at 19%
o wait one year
o receive PV * (1,10) = $110
- PV = $110/(1,10) = $100
We can’t compare cash flows of different maturities also we bring all the cash flows down to a common period. That is why most of the time we use the Present Value to compare cash flows and to relate future cash flows with the present.
What is the value today of $1100 to be received in two years and $100 in one year?
PV = 100/(1,10)+1100/(1,10)²
A) General Formula:
PV = Present ValueCFi = cash flow for period i
PV = CF1 / (1+r1) + CF2 / (1+r2) + CF3 / (1+r3) + CFn / (1+rn)
PV = ∑ CFt / (1+rt) t
B) Constant Annuity:
CF is paid at the end of every year for n years. What is the present value?
PV= CF*[(1-(1+r)-n)/ r]
C) Perpetual Constant Annuity:
CF is paid at the end of every year forever. What is the present value?
PV= CF / r
D) Other Helpful Formulas
Constant growth of the cash flow for n periods:
PV = CF1 / r-g * [1-((1+g))n/(1+r))]
Constant growth of the cash flow forever:
PV = CF / r-g
II) TERMINAL VALUE (Future value):
A) General Formula
TV = CF1*(1+r1,n)n-1 + CF2*(1+r2,n)n-2 +…+ CFn-1*(1+rn-1,n)+ CFn
[pic] for an ordinary annuity (constant end of period)
Summary of Class 1:
- Determine the date that each cash flow will be received - Estimate the cash flow for each date
- Discount ( compound) each cash flow at the appropriate rate for the appropriate number of periods - Take the sum
1) The interest rate is 5%. You will receive $500 at the end of 3 years. What is the present value?
PV = 500/(1,05)3 = 431,92
2) If you can lend $100 at an interest rate of 7%, what will it be worth in 6 years?
FV = 100*(1,07)6 = 150,07
3) Your bank says that if you put up $1000 now it will pay you 1685 at the end of 5 year. What return do you earn on the money?
FV= 1000 (1+r)5 = 1685
(1+r)5 = 1685/1000
(1+r) = (1685/1000)1/5
1+r = 1,11
r = 0,11 = 11%
4) A Savings bond pays $50 per year for two years and $1050 in the third year. The interest rate is 7%. How much should you pay for the bond today?
PV = 50/(1,07)1 + 50/(1,07)2 + 1050/(1,07)3 = 947,51
How do you calculate the effective annual interest rate for a period of time?
3month = 12% p.a but 3 month is 3/12 also 3/12*12% = 3% for 3 month 100*(1,03)4 = 112, 551 and it gives 12,55 % p.a. which is the effective annual interest rate
A security is a financial asset. Each type of security, even each cash flow of a security can have a different discount factor. The reason is that the discount rate depends of the riskiness of the cash flow. The higher the risk is, the higher the return will be. The lowest risk is called the risk free rate. Government securities are considered as risk free.
CHAPTER 2: Stock and bond valuation
Def: A bond is a contract that gives the right to a fixed set of cash payoffs. The market value of a bond changes with the interest rate. A bond is called in French Obligation.
Example 1: £75 per year in interest payments for 5 years and the principle payments at the end of the 5th year
1000= 75/(1+r) + 75/(1+r)² +……+75/(1+r)^5
There are different types of bonds:
▪ Secured (by...