For an investment ending at time T we denote the net cash flow at time t by c_t and the net rate of cashflow per unit time by ρ(t). The present time is t = 0 and time is measured in years.
An infrastructure fund considers the construction of a new bridge. It estimates that the project will require an initial outlay of £22.475m = £22,475,000 and a further outlay of £10m after one year (m = million). There will be an estimated inflow of toll charges of £1m per annum payable continuously for 47 years, beginning at time t = 3.

Task : Give the yield equation for this problem. Determine whether the yield of this investment is (approximately) 1.1%, 1.3%, 1.5%, 1.7% or 1.9%.

Answer: We obtain, in millions of £

a)NPV(i)=-22.475-10v+∫_0^47(1+i)^(-t)dt at rate i

b)NPV(i)=22.475+10v-∫_0^47(1+i)^(-t)dt at rate i  

c)NPV(i)=-22.475-10v+∫_3^50(1+i)^(-t)dt at rate i

d)NPV(i)=22.475+10v-∫_3^50(1+i)^(-t)dt at rate i     

The net cash flow changes sign   

a).only once   

b). more than once, 

so the yield i₀   

a).exist  

b).does not exist . 

The approximate yield is 

a). 1.1%  

b). 1.3%  

c). 1.7%  

d). 1.9% , 

found by plugging the candidate values into the yield equation 

 a).NPV(i)=0   

b). NPV(i)=T  

c). A(i)=0    

d). A(i)=T