Introduction to Corporate Finance

3: The Time Value of Money

For example, suppose that you borrow $25,000 at 8% annual interest for five years to purchase a new

car. To demonstrate the basic approach, we first amortise this loan assuming that you make payments at

the end of years 1 through 5. We then modify the annual formula to compute the more typical monthly

car loan payments. To find the size of the annual payments, the lender determines the amount of a five

year annuity discounted at 8% that has a present value of $25,000. This process is actually the inverse of

finding the present value of an annuity.

finance in practice

SAVING FOR YOUR RETIREMENT

It is important to begin saving for retirement when you start your first real job. Most people begin later. Let’s

assume that you are in your mid-thirties, and have two children and an annual income of $150,000 before

taxes. You now want to get serious about retirement, and have made the following estimates.

Years till retirement 35 years

Estimated years in retirement (based on actuarial tables) 25 years

Current level of household expenditures $92,000

% of current household expenses needed in retirement 75%

Estimated annual end-of-year income in retirement from:

Government pension $32,000

Employer superannuation contribution 11,000

Personal superannuation savings 20,000

Total $63,000

Expected annual inflation rate during retirement 5%

Expected annual rate of return on investments before retirement 7%

Expected annual rate of return on investments during retirement 9%

Using your estimates, you wish to determine the annual end-of-year savings needed to fund your retirement.

This value can be calculated as follows:

Estimated annual household expenditures in retirement = 0.75 × $92,000 = $69,000

Additional annual retirement income needed = $69,000 – $63,000 = $6,000

Eq. 3.1 Inflation-adjusted annual retirement income needed = $6,000 × (1 + 0.05)^30 = $25,932

Eq. 3.7 Lump sum needed in 30 years to fund additional

annual retirement income = $25,932/0.09 × {1 – [1/(1 + 0.09)^25 ]}

= $288,131 × 0.8840 = $25,470

Eq. 3.15 Annual end-of-year savings required to fund lump sum = $25,470/{[(1 + 0.07)^35 – 1]/0.07}

= $25,470/138.243= $184.24

So, in order to fund your retirement goal over your 25 years of retirement, you need to save $184.24

at the end of each of the next 35 years. Note that your assumed rate of return during the 35 years you are

accumulating funds is 7%, and during retirement, when funds are being distributed, you are assumed to earn

a 9% rate of return. If you earn lower returns, you would need to save more each year.