Linear growth

• dw/dt = k
• dw = k*dt
• w(t) = w0 + k*t

• dw/dt = kw
• dw/w = kdt
• ln(w) = dt
• w(t) = wo * e^kt
• FV = PV * e^kt

• w(t) = w0*(1+k)^t
• FV = PV*(1+k)^t

SOME USEFUL FORMS OF THE STEPWISE GROWTH EQUATION

Future value of an annuity

• FV = PMT*(1+k)^(n-1) + PMT*(1+k)^(n-2)
• + ...+ PMT*(1+k)^2+ PMT*(1+k)^1+ PMT
• multiply this equation by (1+k) and get
• FV*(1+k) = PMT*(1+k)^n + PMT*(1+k)^(n-1)
• + ...+ PMT*(1+k)^3+ PMT*(1+k)^2+ PMT*(1+k)
• now subtract the first equation from the second
• FV*(1+k) - FV = PMT*(1+k)^n - PMT
• k*FV = PMT*[(1+k)^n - 1]
• FV = PMT*[(1+k)^n - 1]/k

• PV = PMT*[(1+k)^n - 1]/[k*(1+k)^n]
• As n approaches infinity [(1+k)^n - 1] approaches (1+k)^n
• PV appraoches PMT/k

• A bond contract is an annuity until maturity and a lump sum paid at maturity
• the annuity pays $PMT per period. PMT = coupon*face value
• normally payments are made every six months
• the lump sum portion is the stated redemption value on the contract at maturity
• the future value of the bond at maturity is the sum of the future value of the annuity and the redemption value
• the value of the bond contract at k is the present value of this sum

• normally the compounding period is one year
• if it is not one year, then there are two ways to state the equivalent annual rate: the nominal rate and the annualized rate
• when period is less than one year
  • there are d days in the period and 365 days per year: the nominal rate is quoted as k%
  • the number of periods per year is p=365/d
  • this means that the periodic rate is kd=k/p % by definition of "nominal rate"
  • the n
  • the future value of $1 one year from now is FV = $1*(1+kd)^p
  • the present value of the future value at the annualized rate is = FV/(1+ka) where ka is the annualized rate
  • solve for the annualized rate
• when the period is more than one year = n years
  • the future value n years from now is known
  • solve for k in the PV equation
  • example: a fund grows by 50% in 4 years
  • FV of $1 is $1.50 and so $1.50 = $1*(1+k)^4: solve for k

Stock Valuation with constant dividend

• If dividend constant then perpetuity
• Po = Sum( (D/(1+k)^t) + Pn/(1+k)^n
• but Pn itself can also be expressed as above. Therefore.
• Po = D/k, a perpetuity

• But what if dividends are expected to grow?
• assume constant growth rate of g% per year
• Dt = Do*(1+g)^t
• Po = Do* [(1+g)/(1+k)]^1 + [(1+g)/(1+k)]^2
• +...+ [(1+g)/(1+k)]^(n-1) + [(1+g)/(1+k)]^n
• multiply by (1+k)/(1+g)
• Po*(1+k)/(1+g) = Do* [1 + [(1+g)/(1+k)]^1
• +...+ [(1+g)/(1+k)]^(n-2) + [(1+g)/(1+k)]^(n-1)
• Subtract the first equation
• Po*(1+k)/(1+g) - Po = Do* [1 - [(1+g)/(1+k)]^n]
• g will always be less than k (WHY?)
• Therefore, as n approaches infinity [(1+g)/(1+k)]^n will approach zero
• Po*(1+k)/(1+g) - Po = Do
• Po*[(1+k)/(1+g) - 1] = Do
• Po*[1+k-1-g]/(1+g) = Do
• Po*(k-g)/(1+g) = Do
• Po = Do*(1+g)/(k-g)
• this is the dcf model of stock valuation
• remember the assumption of constant dividend growth

Textbooks provide "PV, PVA, PVIF" and other tables to facilitate these computations for people who do not have calculators. If you do have a calculator it is simpler to use the equations presented here rather than tables.

All terms used in this handout will be explained in the lecture.