Question

Question 4

a. Differentiate between the concepts of discounting and compounding as they relate to cash flow valuations. ( 1 mark)

b. What are the differences among a lump sum, annuity, perpetuity and a growing annuity? Show by way of formulae and example, how the present values and future values of each of the different cash flows listed above can be computed.

c. Illustrate which formulas or a combination of formulas can be used to determine the following: (1) amount loan to be amortized over a period of time, given a loan amount and applicable interest rate, (2) bonds valuation and stock valuation.

d. Show by way examples which approach(es) can be used to calculate the yield to maturity of a bond.

  

e. What are theoretical differences between how bonds are valued and how stocks are valued?

f. What is the difference between the effective annual rate and the annual percentage rate? When do we compute each?

Answer 1

Solution

a.

Discounting is calculating the present value of cash flow expected in the future.

Eg. If I were to receive $110 after an year and required return for me is 10%. The present value of the $110 is = 110/(1+ 0.1) = $100

Compounding is calculating the future value of cash flows. In compounding, you have interest on interest.

Eg. If you invest $100 at 10% interest for two years, after first year you will have 100 + 100*10% = $110. For the next year, you shall receive 10% on 110. So you shall have 110 + 110 *10% = $121

b.

Lump sum:

Lump sum means a single cash flow in future period of time.

Ex. $1000 to be received after five years

Present value of the lumpsum amount is computed using formula = CF/(1+r)^n

Where CF -> Cash Flow, r -> Discount Rate per period, n -> number of periods after which the CF was received.

Ex.

Present value of $1000 to be received after five years. Considering the discount rate of 10%.

= 1000/(1+ 10%)^5 = $620.92

Annuity:

Annuity is same cashflows after same intervals.

Ex. $100 received every year for 10 years is an annuity. $200 received every two years for 10 years is also an annuity.

Present value of the Annuity is computed using formula = CF/r * (1 – 1/(1+r)^n)

Where CF -> Periodic Cash Flow, r -> discount rate per period, n -> number of periods

Ex.

Present value of $100 received every year for 10 years. Considering the discount rate of 10% per year.

= 100/.1*(1- 1/(1+.1)^10) = $614.46

Perpetuity:

Perpetuity means same cash flow for perpetual period of time. It is also called as perpetual annuity.

Ex. $100 received for year for perpetual period.

Present value of the perpetuity is computed using formula = CF/r

Where CF -> Periodic Cash Flow, r -> discount rate per period

Ex.

Present value of $100 received for perpetual period. Considering the discount rate of 10% per year.

= 100/.1 = $1000

Growing Annuity:

Growing annuity are cash flows that grow at a proportionate rate after fixed intervals.

Ex. Rent received from a tenant is $100 at the end of this year and is expected to grow at 10% each year for next 2 years.

Present value of the growing annuity is computed using formula = CF/(r – g) *(1 – (1+g)^n/(1+r)^n)

Where CF -> Periodic Cash Flow, r -> discount rate per period, g à growth rate

Ex.

Present value of Rent received from a tenant is $100 at the end of this year and is expected to grow at 10% each year for next 2 years. Considering the discount rate of 10% per year.

= 100/(0.1- 0.03) * (1 – (1+0.03)^3/(1+0.1)^3) = $255.74

c.

(1) amount loan to be amortized over a period of time, given a loan amount and applicable interest rate

To determine loan instalments, you can use Present Value of annuity formula and solve for periodic payments. Note, in this case, present value of annuity is the loan amount.

(2) bonds valuation and stock valuation.

Bonds have periodic payments which are coupon payment and a lump sum payment at the maturity. Hence you have to use PV of Annuity to get the present value of coupon payment and PV of lump sum amount to arrive at the present value of payment on maturity.

Stock normally give an constant stream of dividends. In that case PV of perpetuity can be used to determine the value of stocks.

If the stock is expected to provide cash flows growing at a certain rate for few years and then the stabilised stream of cash flows for perpetuity, you may use PV of Growing Annuity for the cash flows during the growth period and PV of perpetuity for the perpetual stream. Note, this PV of Perpetuity would need to be discounted further to present day using PV of lump sum formula.

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